function spaces, approximation theory, and their...

Journal of Function Spaces

Function Spaces, Approximation Theory, and Their Applications

Guest Editors: Carlo Bardaro, Ioan Rasa, Rudolf L. Stens, and Gianluca Vinti

Function Spaces, Approximation Theory,and Their Applications

Journal of Function Spaces

Function Spaces, Approximation Theory,and Their Applications

Guest Editors: Carlo Bardaro, Ioan Rasa, Rudolf L. Stens,and Gianluca Vinti

Copyright © 2016 Hindawi Publishing Corporation. All rights reserved.

This is a special issue published in “Journal of Function Spaces.” All articles are open access articles distributed under the Creative Com-mons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work isproperly cited.

Editorial Board

John R. Akeroyd, USARichard I. Avery, USAIsmat Beg, PakistanBjorn Birnir, USAMessaoud Bounkhel, KSAHuy Qui Bui, New ZealandVictor I. Burenkov, ItalyJ. Chung, Republic of KoreaRaúl E. Curto, USALuisa Di Piazza, ItalyLars Diening, GermanyDragan Djordjevic, SerbiaM. Engliš, Czech RepublicJose A. Ezquerro, SpainDashan Fan, USAXiang Fang, USAHans G. Feichtinger, AustriaAlberto Fiorenza, ItalyAjda Fošner, SloveniaEva A. G. Gutiérrez, SpainAurelian Gheondea, TurkeyAntonio S. Granero, SpainYongsheng S. Han, USASeppo Hassi, FinlandStanislav Hencl, Czech Republic

Henryk Hudzik, PolandGennaro Infante, ItalyKrzysztof Jarosz, USAAnna Kamińska, USAValentin Keyantuo, Puerto RicoV. M. Kokilashvili, GeorgiaAlois Kufner, Czech RepublicDavid R. Larson, USAYuri Latushkin, USAYoung J. Lee, Republic of KoreaHugo Leiva, VenezuelaGuozhen Lu, USADag Lukkassen, NorwayGiuseppe Marino, ItalyMiguel Martín, SpainMark A. McKibben, USAMihail Megan, RomaniaA. Montes-Rodriguez, SpainGisele Mophou, FranceDumitru Motreanu, FranceSivaram K. Narayan, USAKasso A. Okoudjou, USAGestur Ólafsson, USACarlos Pérez, SpainLars E. Persson, Sweden

Adrian Petrusel, RomaniaKonrad Podczeck, AustriaGelu Popescu, USAMaria Alessandra Ragusa, ItalyNatasha Samko, PortugalYoshihiro Sawano, JapanSimone Secchi, ItalySergei Silvestrov, SwedenMitsuru Sugimoto, JapanTomonari Suzuki, JapanRodolfo H. Torres, USAWilfredo Urbina, USAN. L. Vasilevski, MexicoP. Veeramani, IndiaIgor E. Verbitsky, USACalogero Vetro, ItalyPasquale Vetro, ItalyAnthony Weston, USAShanhe Wu, ChinaGen Qi Xu, ChinaQuanhua Xu, FranceKari Ylinen, FinlandRuhan Zhao, USAKehe Zhu, USAWilliam P. Ziemer, USA

Contents

Function Spaces, ApproximationTheory, andTheir ApplicationsCarlo Bardaro, Ioan Rasa, Rudolf L. Stens, and Gianluca VintiVolume 2016, Article ID 7423041, 1 page

A Review on Approximation Results for Integral Operators in the Space of Functions ofBounded VariationLaura Angeloni and Gianluca VintiVolume 2016, Article ID 3843921, 11 pages

Mappings of Type Special Space of SequencesAwad A. BakeryVolume 2016, Article ID 4372471, 6 pages

On the Stability of Alternative Additive Equations in Multi-𝛽-Normed SpacesXiuzhong Yang, Jing Ma, and Guofen LiuVolume 2016, Article ID 2534597, 7 pages

SPDIEs and BSDEs Driven by Lévy Processes and Countable BrownianMotionsPengju DuanVolume 2016, Article ID 5916132, 9 pages

Optimal Bounds for Gaussian Arithmetic-Geometric Mean with Applications to Complete EllipticIntegralHuaWang, Wei-Mao Qian, and Yu-Ming ChuVolume 2016, Article ID 3698463, 6 pages

AbstractTheorems on Exchange of Limits and Preservation of (Semi)continuity of Functions andMeasures in the Filter Convergence SettingAntonio Boccuto and Xenofon DimitriouVolume 2016, Article ID 4237423, 10 pages

Dual 𝐿𝑝-Mixed Geominimal Surface Area and Related InequalitiesTongyi Ma and Yibin FengVolume 2016, Article ID 1481793, 10 pages

Classes of 𝐼-Convergent Double Sequences over 𝑛-Normed SpacesNazneen KhanVolume 2016, Article ID 7594031, 7 pages

A Generalization of Uniformly Extremely Convex Banach SpacesSuyalatu Wulede, Wurichaihu Bai, and Wurina BaoVolume 2016, Article ID 9161252, 6 pages

Products of Composition and Differentiation Operators from Bloch into 𝑄𝐾 SpacesShunlai Wang and Taizhong ZhangVolume 2016, Article ID 5084794, 5 pages

EditorialFunction Spaces, Approximation Theory, and Their Applications

Carlo Bardaro,1 Ioan Rasa,2 Rudolf L. Stens,3 and Gianluca Vinti1

1Department of Mathematics and Computer Sciences, University of Perugia, Via Vanvitelli 1, 06123 Perugia, Italy2Technical University of Cluj-Napoca, Cluj-Napoca, Romania3Lehrstuhl A für Mathematik, RWTH Aachen, Templergraben 55, 52056 Aachen, Germany

Correspondence should be addressed to Carlo Bardaro; [email protected]

Received 8 August 2016; Accepted 9 August 2016

Copyright © 2016 Carlo Bardaro et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The purpose of this special issue was to present new devel-opments in the theory of function spaces, along with thedeep interconnections with approximation theory and theapplications in various fields of pure and applied mathemat-ics. The reaction of the mathematical community was verysatisfactory. We collected thirty-five submissions, coveringa wide range of mathematical topics, ten of which werefound to be suitable for publications in this issue. Themajor part of the accepted papers treats function spacesand their applications. In this respect, in the article by XYang et al. a new class of function spaces, named “multi-𝛽-normed spaces”, is introduced, in connection with stabilityproperties of certain type of functional equations, while, inthe paper by A. A. Bakery, sequential spaces of Orlicz type arestudied and connected with the theory of summability. In thereview paper by L. Angeloni and G. Vinti, the approximationtheory in the space of functions with bounded variationis developed, in view of applications to signal processing.Different notions of variation are considered and severalapproximation theorems for families of integral or discretetype operators are given. In the more theoretical article by S.Wulede et al., a new class of Banach spaces which generalizesthe class of uniformly extremely convex Banach spaces isintroduced, and some characterizations of these spaces aregiven. Another paper by N. Khan treats the convergence ofnew type of double sequences, here introduced, in 𝑛-normedspaces. An interesting abstract approach to the theory offilter convergence is given in the article by A. Boccuto andX. Dimitriou, in which the links with function spaces andapproximation theory are also dealt with. Other aspects ofthe theory of function spaces and their interconnections with

calculus of variations, numerical analysis, complex variables,and stochastic processes are discussed, respectively, in thearticles by T. Ma and Y. Feng, H. Wang et al., S. Wang andT. Zhan, and finally P. Duan.These four papers point out howcertain methods of general approximation theory in functionspaces can be employed in order to solve problems comingfrom a large variety of mathematical fields. We think thatthese contributions may represent starting points for newresearches in the field of function spaces and approximationtheory.

Acknowledgments

The guest editors wish to express their deep gratitude to allthe contributors for the interest showed for our special issueand their interesting articles.

Carlo BardaroIoan Rasa

Rudolf L. StensGianluca Vinti

Hindawi Publishing CorporationJournal of Function SpacesVolume 2016, Article ID 7423041, 1 pagehttp://dx.doi.org/10.1155/2016/7423041

http://dx.doi.org/10.1155/2016/7423041

Review ArticleA Review on Approximation Results for Integral Operators inthe Space of Functions of Bounded Variation

Laura Angeloni and Gianluca Vinti

Dipartimento di Matematica e Informatica, Università degli Studi di Perugia, Via Vanvitelli 1, 06123 Perugia, Italy

Correspondence should be addressed to Gianluca Vinti; [email protected]

Received 16 May 2016; Accepted 20 July 2016

Academic Editor: Henryk Hudzik

Copyright © 2016 L. Angeloni and G. Vinti. This is an open access article distributed under the Creative Commons AttributionLicense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properlycited.

We present a review on recent approximation results in the space of functions of bounded variation for some classes of integraloperators in the multidimensional setting. In particular, we present estimates and convergence in variation results for bothconvolution andMellin integral operators with respect to the Tonelli variation. Results with respect to a multidimensional conceptof 𝜑-variation in the sense of Tonelli are also presented.

1. Introduction

The aim of the present paper is to give a review on recentresults about convergence of integral operators of convolu-tion type with respect to some concepts of multidimensionalvariation. We will consider the case of classical convolutionintegral operators of the form

(𝑇𝑤𝑓) (s) = ∫

R𝑁𝐾𝑤(t) 𝑓 (s − t) 𝑑t,

s ∈ R𝑁

, 𝑤 > 0,

(I)

where 𝑓 ∈ 𝐿1(R𝑁) and {𝐾𝑤}𝑤>0

is a family of approximateidentities, as well as the case of Mellin integral operators ofthe form

(𝑀𝑤𝑓) (s) = ∫

R𝑁+

𝐾𝑤(t) 𝑓 (st) ⟨t⟩

−1

𝑑t,

s ∈ R𝑁

+, 𝑤 > 0,

(II)

where st fl (𝑠1𝑡1, . . . , 𝑠

𝑁𝑡𝑁), s, t ∈ R𝑁

+, and ⟨t⟩ fl

∏𝑁

𝑖=1𝑡𝑖. The above operators (II) are of convolution type

with respect to the homothetic operator and the measure𝜇(𝐴) = ∫

𝐴

(𝑑x/⟨x⟩), where 𝐴 is a Borel subset of R𝑁+(which

is an invariant measure with respect to the multiplicativeoperation).

An important tool in order to frame the results of thepaper is the setting of the functional spaces we deal with.TheBV-spaces, apart from the well-known importance from themathematical point of view, also play an important role inproblems of Image Reconstruction where some of the variousapproaches make use of integral operators of convolutiontype (see, e.g., sampling-type operators).

The working space will be the space of functions ofbounded multidimensional variation in the sense of Tonelli(defined in Section 2) and, as further extension, in order todeal with a larger class of functions, we will consider thespace BV𝜑, where 𝜑 is a 𝜑-function (see Section 2). We pointout that, due to the necessary assumptions on the 𝜑-function𝜑 (Assumption ii), the case of BV cannot be obtained asparticular case of BV𝜑.This is the reason why the two settingshave to be treated independently.

For the above classes of operators we will provide esti-mates, convergence results, and also a characterization ofthe absolutely (𝜑-absolutely) continuous functions in termsof the respective convergence in variation. Moreover, therate of approximation has been considered and examples ofkernel functions to which the results can be applied are alsofurnished. Finally, also the nonlinear case for both the con-volution and the Mellin-type operators has been considered.

Apart from the well-known importance of the classicalconvolution integral operators, the Mellin operators are very

Hindawi Publishing CorporationJournal of Function SpacesVolume 2016, Article ID 3843921, 11 pageshttp://dx.doi.org/10.1155/2016/3843921

http://dx.doi.org/10.1155/2016/3843921

2 Journal of Function Spaces

interesting and widely studied in approximation theory (forthe basic theory see [1, 2] while, for results about similarhomothetic-type operators, see, e.g., [3–16]), also because oftheir important applications in several fields. Among them,for example, we recall thatMellin analysis is deeply connectedwith some problems of Signal Processing, in particular withthe so-called Exponential Sampling, which have applicationsin various problems of engineering and optical physics (see,e.g., [17–20]).

Concerning now the multidimensional concept of 𝜑-variation introduced in [21], we point out that, due to thelack of an integral representation of the 𝜑-variation for 𝜑-absolutely continuous functions as happens for the classicalvariation, the major results about convergence require adifferent approach and suitable techniques. In particular, thisholds for the convergence of the 𝜑-modulus of continuity.Again, a similar problem occurs in case of Mellin integraloperators on BV(R𝑁

+), where the Tonelli integrals are defined

via the log-measure. Moreover, in the latter case, in orderto prove that the operators are absolutely continuous, incase of regular kernels (this is a crucial point to obtainthe characterization of absolute continuity), one has to passthrough an equivalent notion of absolute continuity (for thelog-absolute continuity, see Section 4) compatible with thesetting of R𝑁

+equipped with the log-measure.

We finally remark that, of course, all the results of thepaper contain, in particular, the one-dimensional case (see[22–27]).

2. Preliminaries and SomeConcepts of Variation

Wewill now recall the multidimensional concept of variationin the sense of Tonelli. Such definition was introduced byTonelli [28] for functions of two variables and then extendedto dimension𝑁 > 2 by Radó [29] and Vinti [30].

Let us introduce some notations. If we are interested inthe 𝑗th coordinate of a vector x = (𝑥

1, . . . , 𝑥

𝑁) ∈ R𝑁, we will

write

x

𝑗= (𝑥1, . . . , 𝑥

𝑗−1, 𝑥𝑗+1

, . . . , 𝑥𝑁) ∈ R𝑁−1,

x = (x

𝑗, 𝑥𝑗) ,

(1)

so that, for a function 𝑓 : R𝑁 → R, there holds

𝑓 (x) = 𝑓 (x

𝑗, 𝑥𝑗) . (2)

Given an 𝑁-dimensional interval 𝐼 = ∏𝑁𝑖=1

[𝑎𝑖, 𝑏𝑖], by 𝐼

𝑗=

[a𝑗, b𝑗] we will denote the (𝑁 − 1)-dimensional interval

obtained by deleting the 𝑗th coordinate from 𝐼; namely,

𝐼 = [a

𝑗, b

𝑗] × [𝑎

𝑗, 𝑏𝑗] , 𝑗 = 1, . . . , 𝑁. (3)

Definition 1. A function𝑓 : R𝑁 → R is said to be of boundedvariation if the sections of 𝑓 are a.e. of bounded variation onR and their variation is summable; that is, 𝑉R[𝑓(x

𝑗, ⋅)] (the

usual Jordan one-dimensional variation of the 𝑗th section of

𝑓) is finite a.e. x𝑗∈ R𝑁−1 and ∫

R𝑁−1𝑉R[𝑓(x

𝑗, ⋅)]𝑑x

𝑗< +∞,

for every 𝑗 = 1, . . . , 𝑁.

In order to compute the variation of 𝑓 on an interval 𝐼,the first step is to define the (𝑁 − 1)-dimensional integrals(the so-called Tonelli integrals)

Φ𝑗(𝑓, 𝐼) fl ∫

b𝑗

a𝑗

𝑉[𝑎𝑗 ,𝑏𝑗]

[𝑓 (x

𝑗,⋅)] 𝑑x

𝑗, 𝑗 = 1, . . . , 𝑁. (4)

Let now Φ(𝑓, 𝐼) be the Euclidean norm of the vector(Φ1(𝑓, 𝐼), . . . , Φ

𝑁(𝑓, 𝐼)); that is,

Φ(𝑓, 𝐼) fl{

{

{

𝑁

∑

𝑗=1

Φ2

𝑗(𝑓, 𝐼)

}

}

}

1/2

, (5)

where we put Φ(𝑓, 𝐼) = ∞ if Φ𝑗(𝑓, 𝐼) = ∞ for some 𝑗 =

1, . . . , 𝑁.The variation of 𝑓 on an interval 𝐼 ⊂ R𝑁 is defined as

𝑉𝐼[𝑓] fl sup

𝑚

∑

𝑘=1

Φ(𝑓, 𝐽𝑘) , (6)

where the supremum is taken over all the families of 𝑁-dimensional intervals {𝐽

1, . . . , 𝐽

𝑚} which form partitions of 𝐼.

Finally, the variation of 𝑓 over the wholeR𝑁 is defined as

𝑉R𝑁 [𝑓] fl sup𝐼⊂R𝑁

𝑉𝐼[𝑓] , (7)

where the supremum is taken over all the intervals 𝐼 ⊂ R𝑁.By

BV (R𝑁) fl {𝑓 ∈ 𝐿1 (R𝑁) : 𝑉R𝑁 [𝑓] < +∞} (8)

we will denote the space of functions of bounded variation onR𝑁.

We recall that it can be proved that if 𝑓 ∈ BV(R𝑁) then∇𝑓 exists a.e. in R𝑁 and ∇𝑓 ∈ 𝐿1(R𝑁) (see, e.g., [29, 30]).

We point out that, in the multidimensional setting, it isnatural to consider functions of bounded variation withinthe Lebesgue space 𝐿1(R𝑁): indeed this is analogous to thedistributional concept of variation given byCesari [31] and, inequivalent forms, by Krickeberg [32], De Giorgi [33], Giusti[34], and Serrin [35].Wenotice that the definition of variationin the sense of Tonelli is equivalent to the distributional onein the class of functions which satisfy some approximatecontinuity properties (see, e.g., [30]).

We will now recall the concept of absolute continuity insense of Tonelli.

Definition 2. A function 𝑓 : R𝑁 → R is locally absolutelycontinuous (𝑓 ∈ ACloc(R

𝑁

)) if, for every interval 𝐼 =∏𝑁

𝑖=1[𝑎𝑖, 𝑏𝑖] and for every 𝑗 = 1, 2, . . . , 𝑁, the 𝑗th section

𝑓(x𝑗, ⋅) : [𝑎

𝑗, 𝑏𝑗] → R is (uniformly) absolutely continuous

for almost every x𝑗∈ [a𝑗, b𝑗].

Journal of Function Spaces 3

Similarly to the one-dimensional case, it is possible toprove that if 𝑓 ∈ BV(R𝑁) ∩ ACloc(R

𝑁

), then

𝑉R𝑁 [𝑓] = ∫R𝑁

∇𝑓 (x) 𝑑x (9)

(see, e.g., [29, 30]).We will denote by AC(R𝑁) the space of all the functions

𝑓 ∈ BV(R𝑁) ∩ ACloc(R𝑁

).In the following we will present also results about con-

vergence for a family of Mellin integral operators. In orderto study such kind of operators the most natural way is toconsiderR𝑁

+fl ]0, +∞)𝑁 (the domain of the functionswhere

Mellin operators act), as a group with themultiplicative oper-ation (instead of the additive operation onR𝑁) and equippedwith the logarithmic Haar-measure 𝜇(𝐴) = ∫

𝐴

(𝑑x/⟨x⟩) (𝐴 isa Borel subset of R𝑁

+and ⟨x⟩ fl ∏𝑁

𝑖=1𝑥𝑖, x = (𝑥

1, . . . , 𝑥

𝑁) ∈

R𝑁+), instead of the usual Lebesgue measure. For this reason,

in order to obtain results in BV-spaces for such kind ofoperators, it seems natural to adapt the definition of theTonelli variation to this frame: we therefore introduced in[36] a new concept of multidimensional variation in which,in the Tonelli integrals, the Lebesgue measure is replaced bythe logarithmic measure 𝜇.

Definition 3. One will say that 𝑓 ∈ �̃�1(R𝑁+) is of bounded

variation on R𝑁+

if the sections 𝑓(x𝑗, ⋅) are of bounded

variation onR+a.e. x𝑗∈ R𝑁−1+

and𝑉R+[𝑓(x

𝑗, ⋅)] ∈ �̃�

1

(R𝑁−1+

),where �̃�1(R𝑁

+) denotes the space of the functions𝑓 : R𝑁

+→ R

such that ∫R𝑁+

|𝑓(t)|⟨t⟩−1 𝑑t < +∞.

In order to define the multidimensional variation onR𝑁+,

for a fixed interval 𝐼 fl ∏𝑁𝑖=1

[𝑎𝑖, 𝑏𝑖] ⊂ R𝑁

+we consider the

(𝑁 − 1)-dimensional integrals

Φ𝑗(𝑓, 𝐼) fl ∫

b𝑗

a𝑗

𝑉[𝑎𝑗 ,𝑏𝑗]

[𝑓 (x

𝑗, ⋅)]

𝑑x𝑗

⟨x𝑗⟩, (10)

where ⟨x𝑗⟩ denotes the product ∏𝑁

𝑖=1,𝑖 ̸=𝑗𝑥𝑖. The remaining

steps for the definition of the variation follow as before.For the sake of simplicity, we will use the same notations

for the variation in both the cases of R𝑁 and R𝑁+: as it

is natural, when one works on R𝑁+, it is intended that the

measure used is the logarithmic one.The classical definition of Jordan variation [37] was

extended in the literature in several directions: one of thefirst generalizations was the quadratic variation introducedby Wiener [38], extended to the 𝑝-variation, 𝑝 ≥ 1 [39, 40],and later to the concept of 𝜑-variation. The 𝜑-variation wasfirst introduced by Young [41] and then extensively studied byMusielak and Orlicz and their school (see, e.g., [22, 42–49]).

From now on we will assume that 𝜑 : R+0

→ R+0is as

follows.

Assumption i. 𝜑 is a convex 𝜑-function, where a 𝜑-functionis a continuous, nondecreasing function on R+

0, such that

𝜑(0) = 0, 𝜑(𝑢) > 0 for 𝑢 > 0, and lim𝑢→+∞

𝜑(𝑢) = +∞.

Assumption ii. 𝑢−1𝜑(𝑢) → 0 as 𝑢 → 0+.

We recall that (see [22]) the 𝜑-variation of 𝑓 : R → R on[𝑎, 𝑏] ⊂ R is defined as

𝑉𝜑

[𝑎,𝑏][𝑓] fl sup

𝐷

𝑛

∑

𝑖=1

𝜑 (𝑓 (𝑠𝑖) − 𝑓 (𝑠𝑖−1)

) , (11)

where the supremum is taken over all the partitions𝐷 = {𝑠0=

𝑎, 𝑠1, . . . , 𝑠

𝑛= 𝑏} of the interval [𝑎, 𝑏], and

𝑉𝜑

R [𝑓] fl sup[𝑎,𝑏]⊂R

𝑉𝜑

[𝑎,𝑏][𝑓] . (12)

Definition 4. A function 𝑓 : R → R is said to be of bounded𝜑-variation (𝑓 ∈ BV𝜑(R)) if 𝑉𝜑R[𝜆𝑓] < +∞, for some 𝜆 > 0.

The Musielak-Orlicz 𝜑-variation was generalized to themultidimensional frame in [50] following the approach ofVitali. However, in order to study approximation problems,the approach of the Tonelli variation seems to be the mostnatural in this context. For such reason in [21] we introduceda concept of multidimensional 𝜑-variation inspired by theTonelli and C. Vinti approach.

Again, the crucial point is to define, for 𝑗 = 1, . . . , 𝑁, theTonelli integrals: in this case we put

Φ𝜑

𝑗(𝑓, 𝐼) fl ∫

b𝑗

a𝑗

𝑉𝜑

[𝑎𝑗 ,𝑏𝑗][𝑓 (x

𝑗, ⋅)] 𝑑x

𝑗, (13)

where 𝑉𝜑[𝑎𝑗 ,𝑏𝑗]

[𝑓(x𝑗, ⋅)] is the (one-dimensional) Musielak-

Orlicz 𝜑-variation of the jth section of 𝑓.Putting now

Φ𝜑

(𝑓, 𝐼) fl{

{

{

𝑁

∑

𝑗=1

[Φ𝜑

𝑗(𝑓, 𝐼)]

2}

}

}

1/2

, (14)

the multidimensional 𝜑-variation of 𝑓 : R𝑁 → R on aninterval 𝐼 ⊂ R𝑁 is defined as

𝑉𝜑

𝐼[𝑓] fl sup

𝑚

∑

𝑘=1

Φ𝜑

(𝑓, 𝐽𝑘) , (15)

(the supremum is taken over all the partitions {𝐽1, . . . , 𝐽

𝑚} of

𝐼) and, finally,

𝑉𝜑

R𝑁[𝑓] fl sup

𝐼⊂R𝑁𝑉𝜑

𝐼[𝑓] . (16)

By

BV𝜑 (R𝑁)

= {𝑓 ∈ 𝐿1

(R𝑁

) : ∃𝜆 > 0 s.t. 𝑉𝜑 [𝜆𝑓] < +∞}(17)

we will denote the space of functions of bounded 𝜑-variationover R𝑁.

Similarly to the classical variation, it is natural to intro-duce a concept of multidimensional 𝜑-absolute continuity.


Definition 5. One says that 𝑓 : R𝑁 → R is locally 𝜑-absolutely continuous (AC𝜑loc(R

𝑁

)) if it is 𝜑-absolutely con-tinuous in the sense of Tonelli; that is, for every 𝐼 =∏𝑁

𝑖=1[𝑎𝑖, 𝑏𝑖] ⊂ R𝑁 and 𝑗 = 1, 2, . . . , 𝑁, the section 𝑓(x

𝑗, ⋅) :

[𝑎𝑗, 𝑏𝑗] → R is (uniformly) 𝜑-absolutely continuous for

almost every x𝑗∈ [a𝑗, b𝑗].

Here (see [22]) a function 𝑔 : [𝑎, 𝑏] ⊂ R → R is 𝜑-absolutely continuous if there exists 𝜆 > 0 such that thefollowing property holds:

for every 𝜀 > 0, there exists 𝛿 > 0 for which

𝑛

∑

]=1𝜑 (𝜆

𝑔 (𝛽]) − 𝑔 (𝛼

])) < 𝜀, (18)

for all the finite collections of nonoverlapping inter-vals [𝛼], 𝛽]] ⊂ [𝑎, 𝑏], ] = 1, . . . , 𝑛, such that

𝑛

∑

]=1𝜑 (𝛽

]− 𝛼

]) < 𝛿. (19)

By AC𝜑(R𝑁) we will denote the space of all the functions𝑓 ∈ BV𝜑(R𝑁) ∩ AC𝜑loc(R

𝑁

).As before, in order to obtain results for Mellin integral

operators in BV𝜑-spaces in the multidimensional frame, weadapted the previous definition of 𝜑-variation in the sense ofTonelli to the case of functions defined onR𝑁

+equipped with

the logarithmic measure 𝜇. In such concept of multidimen-sional 𝜑-variation, introduced in [51], the Tonelli integrals(13) are replaced by

Φ𝜑

𝑗(𝑓, 𝐼) fl ∫

b𝑗

a𝑗

𝑉𝜑

[𝑎𝑗 ,𝑏𝑗][𝑓 (x

𝑗, ⋅)]

𝑑x𝑗

⟨x𝑗⟩. (20)

Again, we will use the same notations, as in the case of thevariation, for functions defined on both R𝑁 and R𝑁

+.

3. Approximation Results for ConvolutionIntegral Operators

In this section we will present results about approximationin variation by means of the convolution integral operators,namely, (I), for 𝑓 ∈ 𝐿1(R𝑁). Here {𝐾

𝑤}𝑤>0

is a family ofapproximate identities (see, e.g., [52]); that is,

(𝐾𝑤.1) 𝐾𝑤

: R𝑁 → R is a measurable essentiallybounded function such that 𝐾

𝑤∈ 𝐿1

(R𝑁), ‖𝐾𝑤‖1≤

𝐴 for an absolute constant 𝐴 > 0 and ∫R𝑁

𝐾𝑤(t)𝑑t =

1, for every 𝑤 > 0;

(𝐾𝑤.2) for any fixed 𝛿 > 0, ∫

|t|>𝛿|𝐾𝑤(t)|𝑑t → 0, as

𝑤 → +∞.

In the following we will say that {𝐾𝑤}𝑤>0

⊂ K𝑤if (𝐾𝑤.1)

and (𝐾𝑤.2) are satisfied.

Of course the operators (I) are well-defined for every 𝑓 ∈𝐿1

(R𝑁) and therefore in particular for every function 𝑓 ∈BV(R𝑁), since

(𝑇𝑤𝑓) (x) ≤

𝑓1

𝐾𝑤∞ , ∀x ∈ R

𝑁

. (21)

We first recall that the family of operators (I) mapBV(R𝑁) into itself. Indeed, the following estimate holds.

Proposition6 (see [25]). Let𝑓 ∈ BV(R𝑁). If {𝐾𝑤}𝑤>0

satisfies(𝐾𝑤.1), then

𝑉R𝑁 [𝑇𝑤𝑓] ≤ 𝐴𝑉R𝑁 [𝑓] , (22)

𝑤 > 0, where 𝐴 is the constant of Assumption (𝐾𝑤.1).

Remark 7. In the case of nonnegative kernels {𝐾𝑤}𝑤>0

, Propo-sition 6 gives the “variation diminishing property” for theoperators 𝑇

𝑤𝑓: indeed in this case 𝐴 = ‖𝐾

𝑤‖1= 1, 𝑤 > 0.

In order to obtain the main result about convergence invariation, the following estimate of the error of approxima-tion is essential.

Proposition 8 (see [25]). If 𝑓 ∈ BV(R𝑁) then, for every 𝑤 >0,

𝑉R𝑁 [𝑇𝑤𝑓 − 𝑓]

≤ ∫R𝑁

𝑉R𝑁 [𝑓 (⋅ − t) − 𝑓 (⋅)]𝐾𝑤 (t)

𝑑t.(23)

Another important tool is a characterization of theconvergence for themodulus of smoothness of 𝑓, defined as

𝜔 (𝑓, 𝛿) fl sup|t|≤𝛿

𝑉R𝑁 [𝜏t𝑓 − 𝑓] , (24)

where (𝜏t𝑓)(s) fl 𝑓(s − t), for every s, t ∈ R𝑁, is the

translation operator (see, e.g., [6, 53]).

Theorem 9 (see [25]). Let 𝑓 ∈ BV(R𝑁). Then 𝑓 ∈ 𝐴𝐶(R𝑁) ifand only if

lim𝛿→0

𝜔 (𝑓, 𝛿) = 0. (25)

The proof of the sufficient part of this result is a con-sequence of integral representation (9) of the variation forabsolutely continuous functions and of the continuity in 𝐿1of the translation operator. For the necessary part, in [25]it is proved that if the kernel functions 𝐾

𝑤are absolutely

continuous (as it happens in the most common cases), thenalso the integral operators 𝑇

𝑤𝑓 belong to AC(R𝑁). Then,

since AC(R𝑁) is a closed subspace of BV(R𝑁) with respectto the convergence in variation [25] and by estimate (23), incase of regular kernels, the convergence of the modulus ofsmoothness implies that 𝑓 ∈ AC(R𝑁).

By means of Proposition 8 and Theorem 9 it is possibleto obtain the main result about convergence for absolutelycontinuous functions.


Theorem 10 (see [25]). If 𝑓 ∈ 𝐴𝐶(R𝑁) and {𝐾𝑤}𝑤>0

⊂ K𝑤,

then

lim𝑤→+∞

𝑉R𝑁 [𝑇𝑤𝑓 − 𝑓] = 0. (26)

Remark 11. We point out that the assumption of absolutecontinuity of the function is crucial to obtain the mainconvergence theorem and such result does not hold, ingeneral, if, for example, 𝑓 ∈ BV(R𝑁) \AC(R𝑁). For example,in the case𝑁 = 1, let us consider 𝑓 : R → R defined by

𝑓 (𝑥) ={

{

{

1, |𝑥| ≤ 1,

0, |𝑥| > 1.

(27)

First of all we point out that 𝑉[𝜏𝑡𝑓 − 𝑓] 0 as 𝑡 → 0. Let us

now consider the Poisson-Cauchy kernel defined as

𝐾𝑤(𝑡) = √

2

𝜋

𝑤

1 + 𝑤2𝑡2, 𝑤 > 0, 𝑡 ∈ R. (28)

Then

(𝑇𝑤𝑓) (𝑠)

= √2

𝜋[arctan (𝑤 (𝑠 + 1)) − arctan (𝑤 (𝑠 − 1))] ,

𝑠 ∈ R,

(29)

and therefore

𝑉R [𝑇𝑤𝑓 − 𝑓] ≥ 𝑉(−∞,−1] [𝑇𝑤𝑓 − 𝑓]

=

(𝑇𝑤𝑓) (−1) − lim

𝑠→−∞

(𝑇𝑤𝑓) (𝑠)

= −√2

𝜋arctan (−2𝑤) .

(30)

This implies that

lim inf𝑤→+∞

𝑉R [𝑇𝑤𝑓 − 𝑓] ≥ √𝜋

2, (31)

and hence 𝑉R[𝑇𝑤𝑓 − 𝑓] 0 as 𝑤 → +∞.

In case of regular kernels, by the closure of AC(R𝑁) inBV(R𝑁), the converse of Theorem 10 is also true. Hencewe obtain the following characterization of the space of theabsolutely continuous functions, similarly to what happensin the one-dimensional case for the Jordan variation.

Theorem 12 (see [25]). Assume that 𝑓 ∈ BV(R𝑁) and{𝐾𝑤}𝑤>0

⊂ K𝑤

∩ 𝐴𝐶(R𝑁). Then 𝑓 ∈ 𝐴𝐶(R𝑁) if and onlyif

lim𝑤→+∞

𝑉R𝑁 [𝑇𝑤𝑓 − 𝑓] = 0. (32)

The previous results were generalized to the case of themultidimensional 𝜑-variation in [21]. In particular, besides akind of variation diminishing property, in [21] the followingestimate for the error of approximation is obtained.

Proposition 13 (see [21]). Let 𝑓 ∈ 𝐵𝑉𝜑(R𝑁) and let {𝐾𝑤}𝑤>0

be such that (𝐾𝑤.1) is satisfied.Then, for every 𝜆, 𝛿 > 0 and for

every 𝑤 > 0,

𝑉𝜑

R𝑁[𝜆 (𝑇𝑤𝑓 − 𝑓)]

≤ 𝐴−1

{𝜔𝜑

(𝜆𝐴𝑓, 𝛿) ∫|t|≤𝛿

𝐾𝑤 (t) 𝑑t

+ 𝑉𝜑

R𝑁[2𝜆𝐴𝑓]∫

|t|>𝛿

𝐾𝑤 (t) 𝑑t} ,

(33)

where 𝜔𝜑(𝑓, 𝛿) fl sup|t|≤𝛿𝑉

𝜑

R𝑁[𝜏t𝑓 − 𝑓] is the 𝜑-modulus of

smoothness of 𝑓.

Using the previous estimate, the main convergence resultfollows by the singularity assumption on the kernel functions(𝐾𝑤.2) and by the convergence for the 𝜑-modulus of smooth-

ness.

Theorem 14 (see [54]). Let 𝑓 ∈ 𝐵𝑉𝜑(R𝑁). Then there exists𝜆 > 0 such that

lim𝛿→0+

𝜔𝜑

(𝜆𝑓, 𝛿) = 0, (34)

if and only if 𝑓 ∈ 𝐴𝐶𝜑loc(R𝑁

).

The convergence for the modulus of smoothness, in caseof the Tonelli variation, is a direct consequence of the integralrepresentation of the variation for absolutely continuousfunctions (9). On the contrary, for the 𝜑-variation there areno results of this kind and, in order to get the convergence in𝜑-variation of the 𝜑-modulus of smoothness, it is necessaryto use a different technique. In particular (see [54]), thecrucial point is to construct a kind of “step” functions thatapproximate the function 𝑓 and for which a convergenceresult can be proved.

UsingTheorem 14, it is possible to obtain the main resultof convergence in 𝜑-variation for the convolution integraloperators (I).

Theorem 15 (see [21]). If 𝑓 ∈ 𝐴𝐶𝜑(R𝑁) and {𝐾𝑤}𝑤>0

⊂ K𝑤,

then there exists 𝜆 > 0 such thatlim𝑤→+∞

𝑉𝜑

R𝑁[𝜆 (𝑇𝑤𝑓 − 𝑓)] = 0. (35)

To give a sketch of the proof, the starting point is theestimate of Proposition 13 for the error of approximation𝑉𝜑

R𝑁[𝜆(𝑇𝑤𝑓−𝑓)]. ByTheorem 14, we have that𝜔𝜑(𝜆𝑓, 𝛿) tends

to 0, for sufficiently small 𝛿 > 0, while, by Assumption (𝐾𝑤.2)

on the kernel functions, in correspondence with such small𝛿, ∫|t|>𝛿

|𝐾𝑤(t)|𝑑t converges to 0 for 𝑤 large enough; hence

the result follows, taking into account (𝐾𝑤.1) and the fact that

𝑓 ∈ BV𝜑(R𝑁).As before, in case of regular kernels the converse of

Theorem 15 is also true.

Theorem 16 (see [21]). Let 𝑓 ∈ 𝐵𝑉𝜑(R𝑁) and let {𝐾𝑤}𝑤>0

⊂

K𝑤∩𝐴𝐶𝜑

(R𝑁). Then 𝑓 ∈ 𝐴𝐶𝜑(R𝑁) if and only if there exists𝜆 > 0 such that

lim𝑤→+∞

𝑉𝜑

R𝑁[𝜆 (𝑇𝑤𝑓 − 𝑓)] = 0. (36)


Remark 17. We point out that it is easy to find examples ofkernel functions to which the previous results can be applied.Among them, for example, the Gauss-Weierstrass kernel isdefined as

𝐺𝑤(t) =

𝑤𝑁

𝜋𝑁/2𝑒−𝑤2|t|2

, t ∈ R𝑁

, 𝑤 > 0, (37)

or the Picard kernel

𝑃𝑤(t) =

𝑤𝑁

Γ (𝑁/2)

2𝜋𝑁/2Γ (𝑁)𝑒−𝑤|t|

, t ∈ R𝑁

, 𝑤 > 0, (38)

where Γ is the Gamma Euler function (see Figure 1).

4. Approximation Results for MellinIntegral Operators

We now turn our attention to Mellin integral operators,defined as (II), where st fl (𝑠

1𝑡1, . . . , 𝑠

𝑁𝑡𝑁), s, t ∈ R𝑁

+. On

the kernel functions {𝐾𝑤}𝑤>0

we assume that

(�̃�𝑤.1) 𝐾𝑤

: R𝑁+

→ R is a measurable essen-tially bounded function such that 𝐾

𝑤∈ �̃�1

(R𝑁+),

‖𝐾𝑤‖�̃�

1 ≤ 𝐴 for an absolute constant 𝐴 > 0 and∫R𝑁+

𝐾𝑤(t)⟨t⟩−1𝑑t = 1, for every 𝑤 > 0;

(�̃�𝑤.2) for every fixed 0 < 𝛿 < 1,

∫|1−t|>𝛿 |𝐾𝑤(t)|⟨t⟩

−1

𝑑t → 0, as 𝑤 → +∞,

that is, the assumptions of approximate identities, adaptedto the present setting of R𝑁

+. If {𝐾

𝑤}𝑤>0

satisfy (�̃�𝑤.1) and

(�̃�𝑤.2), we will write {𝐾

𝑤}𝑤>0

⊂ K̃𝑤.

For Mellin integral operators it is possible to develop an“approximation theory” similar to the case of the convolutionintegral operators; however, one of themain differences is thehomothetic structure of R𝑁

+which leads to the choice of the

logarithmicmeasure 𝜇 and also to the necessity to adapt somedefinitions. For example, the modulus of smoothness of 𝑓 ∈BV(R𝑁

+) has to be now defined as

𝜔 (𝑓, 𝛿) fl sup|1−t|≤𝛿

𝑉R𝑁+

[𝜎t𝑓 − 𝑓] , (39)

0 < 𝛿 < 1, where (𝜎t𝑓)(s) fl 𝑓(st), for every s, t ∈ R𝑁

+,

is the homothetic operator and 1 = (1, . . . , 1) is the unitvector of R𝑁

+. Such notion of modulus of smoothness is the

natural generalization, in the present frame of BV(R𝑁+), of the

classical modulus of continuity (see, e.g., [6, 21, 25]).Of course, due to the presence of the logarithmicmeasure,

we cannot use the integral representation of the Tonellivariation; nevertheless, it is possible to directly prove a resultof convergence in variation for themodulus of smoothness incase of AC-functions, using a kind of “separated” variations(𝑉𝑗[𝑓], 𝑗 = 1, . . . , 𝑁) which take into account just a singledirection instead of all the𝑁 directions.

Theorem 18 (see [36]). If 𝑓 ∈ 𝐴𝐶(R𝑁+), then

lim𝛿→0+

𝜔 (𝑓, 𝛿) = 0. (40)

By means of the previous result and an estimate for theerror of approximation (𝑀

𝑤𝑓 − 𝑓) analogous to Proposi-

tion 13, it is possible to prove the following convergence result.

Theorem 19. Let 𝑓 ∈ 𝐴𝐶(R𝑁+) and {𝐾

𝑤}𝑤>0

⊂ K̃𝑤. Then

lim𝑤→+∞

𝑉R𝑁+

[𝑀𝑤𝑓 − 𝑓] = 0. (41)

A natural question now is whether, at least in case of AC-kernels, the converse of the previous result holds, as in thecase of convolution operators. Actually, due to the form of theoperators𝑀

𝑤𝑓, such question is nowmuchmore delicate and

direct approach cannot be used. In order to solve the problem,it is necessary to use another concept of absolute continuity(the log-absolute continuity), equivalent to the classical one,which takes into account the logarithmic measure 𝜇. We firstpresent the definition in the one-dimensional case.

Definition 20 (see [55]). One says that 𝑓 : R+

→ R is log-absolutely continuous on [𝑎, 𝑏] ⊂ R

+(𝑓 ∈ AClog([𝑎, 𝑏])) if for

every 𝜀 > 0 there exists 𝛿 > 0 such that, for every collectionof nonoverlapping intervals [𝛼], 𝛽]]𝑛]=1 in [𝑎, 𝑏] such that

𝑛

∑

]=1

log (𝛽]) − log (𝛼]) < 𝛿, (42)

then𝑛

∑

]=1

𝑓 (𝛽]) − 𝑓 (𝛼

]) < 𝜀. (43)

By AClog(R+)we will denote the space of functions which areof bounded variation on R

+and log-absolutely continuous

on [𝑎, 𝑏], for every [𝑎, 𝑏] ⊂ R+.

Now, in the general multidimensional frame, 𝑓 : R𝑁+

→

R is log-absolutely continuous on 𝐼 = ∏𝑁𝑖=1

[𝑎𝑖, 𝑏𝑖] ⊂ R𝑁

+if, for

every 𝑗 = 1, 2, . . . , 𝑁, the 𝑗th sections of𝑓,𝑓(x𝑗, ⋅) : [𝑎

𝑗, 𝑏𝑗] →

R, are (uniformly) log-absolutely continuous for almost everyx𝑗∈ R𝑁−1+

.By means of the definition of the log-absolute continuity,

in [55] it is proved that Mellin integral operators, as theclassical convolution operators, preserve absolute continuity:this, together with the fact that the set of the absolutelycontinuous functions is a closed subspace of the set of the BV-functions, allows us to obtain the following characterization.

Theorem21 (see [55]). Let𝑓 ∈ BV(R𝑁+) and {𝐾

𝑤}𝑤>0

⊂ K̃𝑤∩

𝐴𝐶(R𝑁+).Then𝑓 ∈ 𝐴𝐶(R𝑁

+) if and only if lim

𝑤→+∞𝑉R𝑁+

[𝑀𝑤𝑓

−𝑓] = 0.

In [56] approximation properties for Mellin integraloperators were studied in the frame of BV𝜑-spaces, using themultidimensional version of the 𝜑-variation on R𝑁

+intro-

duced in [51]. In particular the following theorem is obtained.

Theorem 22 (see [56]). Let 𝑓 ∈ 𝐴𝐶𝜑(R𝑁+) and {𝐾

𝑤}𝑤>0

⊂

K̃𝑤. Then there exists a constant 𝜇 > 0 such that

lim𝑤→+∞

𝑉𝜑

R𝑁+

[𝜇 (𝑀𝑤𝑓 − 𝑓)] = 0. (44)


−0.4−0.2

00.2

0.4 −0.4

−0.2

0

0.2

0.4

x

y

30

10

20

0

(a)

−0.4−0.2

00.2

0.4 −0.4−0.2

00.2

0.4

x

y

12.5

7.5

2.5

10

15

5

(b)

Figure 1: Examples of Gauss-Weierstrass kernel 𝐺𝑤(𝑥, 𝑦) and Picard kernel 𝑃

𝑤(𝑥, 𝑦) with 𝑤 = 10.

The proof of Theorem 22 is based on the estimate of theerror of approximation [56],

𝑉𝜑

[𝜆 (𝑀𝑤𝑓 − 𝑓)]

≤ 𝜔𝜑

(𝜆𝐴𝑓, 𝛿)

+ 𝐴−1

𝑉𝜑

[2𝜆𝐴𝑓]∫|1−t|>𝛿

𝐾𝑤 (t) ⟨t⟩−1

𝑑t,

(45)

and on a convergence result for the𝜑-modulus of smoothness

𝜔𝜑

(𝑓, 𝛿) fl sup|1−t|≤𝛿

𝑉𝜑

R𝑁+

[𝜎t𝑓 − 𝑓] , (46)

𝛿 > 0.This last result was obtained in [51] bymeans of a directapproach: the function 𝑓 is approximated in 𝜑-variation bytwo auxiliary functions, constructed on a grid on which theirsections are piecewise constant.

In order to prove the converse of Theorem 22, it isagain necessary to use a concept of logarithmic 𝜑-absolutecontinuity, which takes into account the homothetic structureofR𝑁+.We report below the definition in the one-dimensional

case, while for the multidimensional case it is sufficient toproceed as for the log-absolute continuity.

Definition 23 (see [57]). One says that 𝑓 : [𝑎, 𝑏] → R is log-𝜑-absolutely continuous on [𝑎, 𝑏] ⊂ R

+if there exists 𝜆 > 0

such that, for every 𝜀 > 0, there exists 𝛿 > 0 for which

𝑛

∑

]=1𝜑 (𝜆

𝑓 (𝛽]) − 𝑓 (𝛼

])) < 𝜀, (47)

for all finite collections of nonoverlapping intervals [𝛼], 𝛽]] ⊂[𝑎, 𝑏], ] = 1, . . . , 𝑛, such that

𝑛

∑

]=1𝜑 (log (𝛽]) − log (𝛼])) < 𝛿. (48)

The log-𝜑-absolute continuity is equivalent to the 𝜑-absolute continuity and allows obtaining the characterizationof AC𝜑(R𝑁

+) in terms of convergence in 𝜑-variation of Mellin

integral operators.

Theorem 24 (see [57]). Let 𝑓 ∈ 𝐵𝑉𝜑(R𝑁+) and {𝐾

𝑤}𝑤>0

⊂

K̃𝑤∩𝐴𝐶𝜑

(R𝑁+). Then 𝑓 ∈ 𝐴𝐶𝜑(R𝑁

+) if and only if there exists

𝜆 > 0 such that lim𝑤→+∞

𝑉𝜑

R𝑁+

[𝜆(𝑀𝑤𝑓 − 𝑓)] = 0.

Remark 25. We point out that taking 𝑁 = 1 as particularcase of Theorem 24 we obtain the characterization of 𝜑-absolute continuity in the one-dimensional case, namely, forthe classical Musielak-Orlicz 𝜑-variation.

Remark 26. It is not difficult to find examples of kernelfunctions which fulfill Assumptions (�̃�

𝑤.1) and (�̃�

𝑤.2).

Among them, for example, themoment-type kernels (or aver-age kernels) are defined as

𝐴𝑤(t) fl 𝑤𝑁 ⟨t⟩𝑤 𝜒

]0,1[𝑁 (t) , t ∈ R

𝑁

+, 𝑤 > 0. (49)

It is easy to see that they fulfill Assumption (�̃�𝑤.1). Moreover,

for every 𝛿 ∈]0, 1[, |1 − t| > 𝛿 implies that there exists anindex 𝑗 = 1, . . . , 𝑁 such that |1 − 𝑡

𝑗| > 𝛿/√𝑁; hence {t ∈

]0, 1[𝑁

: |1 − t| > 𝛿} ⊂ ⋃𝑁𝑗=1

{t ∈ R𝑁+

: 0 < 𝑡𝑗< 1 − 𝛿/√𝑁, 0 <

𝑡𝑖< 1, ∀𝑖 ̸= 𝑗}.Therefore

∫|1−t|>𝛿

𝐴𝑤 (t) ⟨t⟩−1

𝑑t

≤

𝑁

∑

𝑗=1

{

{

{

(∏

𝑖 ̸=𝑗

∫

1

0

𝑤𝑡𝑤−1

𝑖𝑑𝑡𝑖)∫

1−𝛿/√𝑁

0

𝑤𝑡𝑤−1

𝑗𝑑𝑡𝑗

}

}

}

= 𝑁(1 −𝛿

√𝑁)

𝑤

→ 0,

(50)

as 𝑤 → +∞; that is, also (�̃�𝑤.2) is satisfied.


Other families of kernel functions to which the previousresults can be applied are the Mellin-Gauss-Weierstrass ker-nels, defined as

�̃�𝑤(t) fl

𝑤𝑁

𝜋𝑁/2𝑒−𝑤2| log t|2

, t ∈ R𝑁

+, 𝑤 > 0, (51)

or the Mellin-Picard kernels, which are defined as

�̃�𝑤(t) fl

𝑤𝑁

2𝜋𝑁/2

Γ (𝑁/2)

Γ (𝑁)𝑒−𝑤| log t|

, t ∈ R𝑁

+, 𝑤 > 0. (52)

We point out that these definitions are the natural refor-mulations, in the present multiplicative setting of R𝑁

+, of

the classical Gauss-Weierstrass kernels and Picard kernels,respectively (see Remark 17).

5. Further Results

We will now give some hints about further approximationresults that were obtained in BV-spaces.

First of all, an interesting problem is to study the nonlin-ear versions of operators (I) and (II). We point out that thenonlinear case is much more delicate than the linear one andrequires some ad hoc assumptions; on the other side, it notonly is interesting from amathematical point of view, being ofcoursemore general than the linear one, but also is importantfrom the point of view of the applications. Indeed, there areseveral applicative problems that cannot be faced bymeans oflinear processes; an example is furnished by some problemsof Signal Processing.

The nonlinear version of the convolution integral opera-tors (II) is

(𝑇𝑤𝑓) (s) = ∫

R𝑁𝐾𝑤(t, 𝑓 (s − t)) 𝑑t,

𝑤 > 0, s ∈ R𝑁

,

(III)

where {𝐾𝑤}𝑤>0

is a family of measurable functions𝐾𝑤

: R𝑁×

R → R of the form

𝐾𝑤(t, 𝑢) = 𝐿

𝑤(t)𝐻𝑤(𝑢) (53)

for every t ∈ R𝑁, 𝑢 ∈ R. Here 𝐿𝑤

: R𝑁 → R and 𝐻𝑤

: R →

R with 𝐻𝑤(0) = 0 is a Lipschitz kernel for every 𝑤 > 0; that

is, there exists 𝐾 > 0 such that𝐻𝑤 (𝑢) − 𝐻𝑤 (V)

≤ 𝐾 |𝑢 − V| , ∀𝑢, V ∈ R. (54)

Moreover we assume that

(𝐾𝑤.1) 𝐿𝑤

: R𝑁 → R is a measurable function suchthat 𝐿

𝑤∈ 𝐿1

(R𝑁), ‖𝐿𝑤‖1≤ 𝐴, for some𝐴 > 0 and for

every 𝑤 > 0, and ∫R𝑁

𝐿𝑤(t)𝑑t = 1, for every 𝑤 > 0;

(𝐾𝑤.2) for any fixed 𝛿 > 0, ∫

|t|>𝛿|𝐿𝑤(t)|𝑑t → 0, as

𝑤 → +∞;(𝐾𝑤.3) denoted by𝐺

𝑤(𝑢) fl 𝐻

𝑤(𝑢) − 𝑢, 𝑢 ∈ R,𝑤 > 0,

𝑉𝐽[𝐺𝑤]

𝑚 (𝐽)→ 0, as 𝑤 → +∞, (55)

H3(u)H5(u)H7(u)

H9(u)u

0.5 1 1.5 20

u

0

0.5

1

1.5

2

Figure 2: Example of kernels 𝐻𝑤(𝑢), 𝑤 = 3, 5, 7, 9.

uniformly with respect to every (not trivial) boundedinterval 𝐽 ⊂ R; that is, for every 𝜀 > 0 there exists𝑤 > 0 such that 𝑉

𝐽[𝐺𝑤]/𝑚(𝐽) < 𝜀, for every 𝑤 ≥ 𝑤

and for every bounded interval 𝐽 ⊂ R.

Remark 27. We point out that Assumption (𝐾𝑤.3) is due to

the nonlinear frame and of course it is obviously satisfiedin the linear case (𝐻

𝑤(𝑢) = 𝑢, 𝑢 ∈ R). Moreover it is

not difficult to provide examples of kernels which fulfill allthe previous assumptions. For example, we can consider thekernel functions 𝐾

𝑤(𝑡, 𝑢) = 𝐿

𝑤(𝑡)𝐻𝑤(𝑢), 𝑡 ∈ R+

0, 𝑢 ∈ R, 𝑤 >

0, where {𝐿𝑤}𝑤>0

are approximate identities,

𝐻𝑤(𝑢) =

{{

{{

{

𝑢 + log(1 + 𝑢𝑤

) , 0 ≤ 𝑢 < 1,

𝑢 + log(1 + 1𝑤𝑢

) , 𝑢 ≥ 1,

(56)

and the definition of𝐻𝑤(𝑢) is extended in odd way for 𝑢 < 0

(see Figure 2).

The problem of the convergence in variation for thenonlinear integral operators (III) was faced in [58, 59]; inparticular, the main convergence result reads as follows.

Theorem 28 (see [58]). If 𝑓 ∈ 𝐴𝐶(R𝑁) and {𝐾𝑤}𝑤>0

satisfy(𝐾𝑤.𝑖), 𝑖 = 1, 2, 3, then

lim𝑤→+∞

𝑉R𝑁 [𝑇𝑤𝑓 − 𝑓] = 0. (57)

Similar approximation results were obtained in [60] forthe nonlinear convolution integral operators (III) in theframe of BV𝜑(R𝑁) and in [61] for the nonlinear version ofthe Mellin integral operators (II) in BV(R𝑁

+).

We finally point out that, besides the problem of con-vergence, the rate of approximation has been also studied inall the previously mentioned settings. In order to do it, asit is natural, one has to introduce suitable Lipschitz classes


which take into account the variation functional. We pointout that, in order to approach the mentioned problem, theassumptions on kernels have to be slightly modified.

For example, let us consider the case of the convolutionintegral operators (I) in the setting of BV(R𝑁). In this framethe Lipschitz class is defined as

𝑉Lip𝑁 (𝛼) fl {𝑓 ∈ BV (R𝑁) : 𝑉R𝑁 [𝜏t𝑓 − 𝑓]

= 𝑂 (|t|𝛼

) , as |t| → 0} ,(58)

𝛼 > 0, and Assumption (𝐾𝑤.2) has to be replaced by the

following:

(𝐾

𝑤.2) for any fixed 𝛿 > 0, ∫

|t|>𝛿|𝐿𝑤(t)|𝑑t = 𝑂(𝑤−𝛼),

as 𝑤 → +∞.

Moreover we will say that {𝐾𝑤}𝑤>0

is an 𝛼-singular kernel,for 0 < 𝛼 ≤ 1, if

∫|t|>𝛿

𝐾𝑤 (t) 𝑑t = 𝑂 (𝑤

−𝛼

) , as 𝑤 → +∞, (59)

for every 𝛿 > 0. Then it is possible to obtain the followingresult about the order of approximation for the convolutionintegral operators (I).

Theorem 29 (see [25]). Let 𝑓 ∈ 𝑉𝐿𝑖𝑝𝑁(𝛼) and let {𝐾𝑤}𝑤>0

⊂

K𝑤be an 𝛼-singular kernel satisfying (𝐾

𝑤.1) and (𝐾

𝑤.2).

Moreover assume that there exists 0 < �̃� < 1 such that

∫|t|≤̃𝛿

𝐾𝑤 (t) |t|𝛼

𝑑t = 𝑂 (𝑤−𝛼

) , as 𝑤 → +∞. (60)

Then

𝑉R𝑁 [𝑇𝑤𝑓 − 𝑓] = 𝑂 (𝑤−𝛼

) , (61)

as 𝑤 → +∞.

Similar results in the nonlinear casewere obtained in [58],while, for results about the rate of approximation for convolu-tion integral operators with respect to the multidimensional𝜑-variation, see [21] and [60] (nonlinear case).

The case of Mellin integral operators was studied in [36]and in [61] (nonlinear case) with respect to the Tonelli varia-tion, while the case of the multidimensional 𝜑-variation wasstudied in [56]. We finally refer to [62, 63] for approximationresults in the slightly different setting of BV𝜑((R+

0)𝑁

).

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this article.

Acknowledgments

The authors are Members of the Gruppo Nazionale perl’Analisi Matematica, la Probabilitá e le loro Applicazioni(GNAMPA) of the Istituto Nazionale di Alta Matematica

(INdAM). This work was supported by Department ofMathematics and Computer Sciences, University of Perugia.Moreover, the first author of the paper has been partiallysupported within the GNAMPA-INdAM Project 2016 “Prob-lemi di Regolarità nel Calcolo delle Variazioni e di Approssi-mazione.”

References

[1] R. G. Mamedov, The Mellin Transform and ApproximationTheory, Elm, Baku, 1991 (Russian).

[2] P. L. Butzer and S. Jansche, “A direct approach to the Mellintransform,” Journal of Fourier Analysis and Applications, vol. 3,no. 4, pp. 325–376, 1997.

[3] W. Z. Chen and S. S. Guo, “On the rate of convergence of thegamma operator for functions of bounded variation,” Approxi-mationTheory and its Applications, vol. 1, no. 5, pp. 85–96, 1985.

[4] C. Bardaro and G. Vinti, “Modular estimates of integral oper-ators with homogeneous kernels in Orlicz type classes,” Resultsin Mathematics, vol. 19, no. 1-2, pp. 46–53, 1991.

[5] I. Mantellini and G. Vinti, “Modular estimates for nonlinearintegral operators and applications in fractional calculus,”Numerical Functional Analysis and Optimization, vol. 17, no. 1-2,pp. 143–165, 1996.

[6] C. Bardaro, J. Musielak, and G. Vinti, Nonlinear Integral Opera-tors and Applications, vol. 9 of De Gruyter Series in NonlinearAnalysis and Applications, Walter de Gruyter, New York, NY,USA, 2003.

[7] G. Vinti, “Approximation in Orlicz spaces for linear integraloperators and Applications,” Rendiconti del Circolo Matematicodi Palermo, supplement 76, no. 2, pp. 103–127, 2005.

[8] H. Karsli, “Rate of convergence of new Gamma type operatorsfor functionswith derivatives of bounded variation,”Mathemat-ical and Computer Modelling, vol. 45, no. 5-6, pp. 617–624, 2007.

[9] C. Bardaro, G. Vinti, and H. Karsli, “On pointwise convergenceof linear integral operators with homogeneous kernels,” IntegralTransforms and Special Functions, vol. 19, no. 5-6, pp. 429–439,2008.

[10] H. Karsli, V. Gupta, and A. Izgi, “Rate of pointwise convergenceof a new kind of gamma operators for functions of boundedvariation,” Applied Mathematics Letters, vol. 22, no. 4, pp. 505–510, 2009.

[11] L. Angeloni and G. Vinti, “Approximation with respect toGoffman-Serrin variation by means of non-convolution typeintegral operators,” Numerical Functional Analysis and Opti-mization, vol. 31, no. 4–6, pp. 519–548, 2010.

[12] C. Bardaro, G. Vinti, and H. Karsli, “Nonlinear integral oper-ators with homogeneous kernels: pointwise approximationtheorems,” Applicable Analysis, vol. 90, no. 3-4, pp. 463–474,2011.

[13] A. Boccuto, D. Candeloro, and A. R. Sambucini, “Vitali-typetheorems for filter convergence related to vector lattice-valuedmodulars and applications to stochastic processes,” Journal ofMathematical Analysis and Applications, vol. 419, no. 2, pp. 818–838, 2014.

[14] V. Gupta and R. P. Agarwal, Convergence Estimates in Approxi-mation Theory, Springer, 2014.

[15] G. Vinti and L. Zampogni, “Approximation results for a generalclass of Kantorovich type operators,” Advanced Nonlinear Stud-ies, vol. 14, no. 4, pp. 991–1011, 2014.


[16] A. Boccuto, D. Candeloro, and A. R. Sambucini, “𝐿𝑃 spaces invector lattices and applications,” Mathematica Slovaca, https://arxiv.org/abs/1604.07570.

[17] M. Bertero and E. R. Pike, “Exponential-sampling method forLaplace and other dilationally invariant transforms: I. Singu-lar-system analysis. II. Examples in photon correlation spec-troscopy and Fraunhofer diffraction,” Inverse Problems, vol. 7,pp. 1–41, 1991.

[18] P. L. Butzer and S. Jansche, “The exponential sampling theoremof signal analysis,” Atti del Seminario Matematico e Fisicodell’Università di Modena e Reggio Emilia, vol. 46, pp. 99–122,A Special Issue of the International Conference in Honour ofProf. Calogero Vinti, 1998.

[19] C. Bardaro, P. L. Butzer, and I. Mantellini, “The exponentialsampling theorem of signal analysis and the reproducing kernelformula in the Mellin transform setting,” Sampling Theory inSignal and Image Processing, vol. 13, no. 1, pp. 35–66, 2014.

[20] C. Bardaro, P. L. Butzer, and I. Mantellini, “TheMellin-Parsevalformula and its interconnections with the exponential samplingtheorem of optical physics,” Integral Transforms and SpecialFunctions, vol. 27, no. 1, pp. 17–29, 2016.

[21] L. Angeloni and G. Vinti, “Convergence and rate of approx-imation for linear integral operators in 𝐵𝑉𝜑-spaces in mul-tidimensional setting,” Journal of Mathematical Analysis andApplications, vol. 349, no. 2, pp. 317–334, 2009.

[22] J. Musielak and W. Orlicz, “On generalized variations (I),”Studia Mathematica, vol. 18, pp. 11–41, 1959.

[23] I. Mantellini and G. Vinti, “Φ-variation and nonlinear integraloperators,” in Atti del Seminario Matematico e Fisico dell’Università di Modena e Reggio Emilia, vol. 46 of InternationalConference in Honour of Prof. Calogero Vinti, pp. 847–862, 1998.

[24] S. Sciamannini and G. Vinti, “Convergence and rate of approxi-mation in 𝐵𝑉𝜑(R) for a class of integral operators,” Approxima-tion Theory and its Applications, vol. 17, pp. 17–35, 2001.

[25] C. Bardaro, P. L. Butzer, R. L. Stens, and G. Vinti, “Convergencein variation and rates of approximation for Bernstein-typepolynomials and singular convolution integrals,” Analysis, vol.23, no. 4, pp. 299–340, 2003.

[26] L. Angeloni and G. Vinti, “Approximation by means of nonlin-ear integral operators in the space of functions with bounded𝜑-variation,” Differential and Integral Equations, vol. 20, no. 3,pp. 339–360, 2007.

[27] L. Angeloni and G. Vinti, “Errata Corrige to: approximation bymeans of nonlinear integral operators in the space of functionswith bounded 𝜑-variation,” Differential and Integral Equations,vol. 23, no. 7-8, pp. 795–799, 2010.

[28] L. Tonelli, “Su alcuni concetti dell’analisi moderna,”Annali dellaScuola Normale Superiore di Pisa, vol. 11, no. 2, pp. 107–118, 1942.

[29] T. Radó, Length and Area, American Mathematical Society Col-loquium Publications, vol. 30, American Mathematical Society,New York, NY, USA, 1948.

[30] C. Vinti, “Perimetro-variazione,” Annali della Scuola NormaleSuperiore di Pisa, vol. 18, no. 3, pp. 201–231, 1964.

[31] L. Cesari, “Sulle funzioni a variazione limitata,” Annali dellaScuola Normale Superiore di Pisa, vol. 5, pp. 299–313, 1936.

[32] K. Krickeberg, “Distributionen, Funktionen beschränkter Vari-ation und Lebesguescher Inhalt nichtparametrischer Flächen,”Annali di Matematica Pura ed Applicata. Serie Quarta, vol. 44,no. 1, pp. 105–133, 1957.

[33] E. DeGiorgi, “Su una teoria generale dellamisura (𝑟−1) dimen-sionale in uno spazio adr dimensioni,” Annali di MatematicaPura ed Applicata, vol. 36, no. 1, pp. 191–213, 1954.

[34] E. Giusti,Minimal Surfaces and Functions of Bounded Variation,vol. 80 of Monographs in Mathematics, Birkhäuser, Basel,Switzerland, 1984.

[35] J. Serrin, “On the differentiability of functions of severalvariables,” Archive for Rational Mechanics and Analysis, vol. 7,pp. 359–372, 1961.

[36] L. Angeloni and G. Vinti, “Variation and approximation inmultidimensional setting for Mellin integral operators,” in NewPerspectives on Approximation and Sampling Theory: Festschriftin Honor of Paul Butzer’s 85th Birthday, pp. 299–317, Birkhauser,2014.

[37] C. Jordan, “Sur la serie de Fourier,” Comptes Rendus del’Académie des Sciences Paris, vol. 92, pp. 228–230, 1881.

[38] N.Wiener, “The quadratic variation of a function and its fouriercoefficients,” Journal ofMathematics andPhysics, vol. 3, no. 2, pp.72–94, 1924.

[39] L. C. Young, “An inequality of the Hölder type, connected withStieltjes integration,” Acta Mathematica, vol. 67, no. 1, pp. 251–282, 1936.

[40] E. R. Love and L. C. Young, “Sur une classe de fonctionnelleslinéaires,” FundamentaMathematicae, vol. 28, no. 1, pp. 243–257,1937.

[41] L. C. Young, “Sur une généralisation de la notion de variationde puissance 𝑝𝑖𝑒𝑚𝑒 bornée au sens de M. Wiener, et sur la con-vergence des séries de Fourier,” Comptes Rendus de l’Académiedes Sciences Paris, vol. 204, pp. 470–472, 1937.

[42] J. Musielak, Orlicz Spaces and Modular Spaces, vol. 1034 ofLecture Notes in Mathematics, Springer, Berlin, Germany, 1983.

[43] J. Szelmeczka, “On convergence of singular integrals in thegeneralized variationmetric,” Functiones et Approximatio, Com-mentarii Mathematici, vol. 15, pp. 53–58, 1986.

[44] W.Matuszewska andW.Orlicz, “Onproperty B1 for functions ofbounded 𝜑-variation,” Bulletin of the Polish Academy of SciencesMathematics, vol. 35, no. 1-2, pp. 57–69, 1987.

[45] A.-R. K. Ramazanov, “On approximation of functions in termsof Φ-variation,” Analysis Mathematica, vol. 20, no. 4, pp. 263–281, 1994.

[46] J. A. Adell and J. de la Cal, “Bernstein-type operators diminishthe 𝜑-variation,” Constructive Approximation, vol. 12, no. 4, pp.489–507, 1996.

[47] V. V. Chistyakov and O. E. Galkin, “Mappings of bounded 𝜙-variation with arbitrary function 𝜙,” Journal of Dynamical andControl Systems, vol. 4, no. 2, pp. 217–247, 1998.

[48] J. Appell, J. Banaś, and N. Merentes, Bounded Variation andAround, vol. 17 of De Gruyter Series in Nonlinear Analysis andApplications, De Gruyter, Berlin, Germany, 2014.

[49] J. Musielak, “Sequences of bounded 𝜑-variation and weightedunconditional convergence of series,” Demonstratio Mathemat-ica, vol. 38, no. 1, pp. 153–162, 2005.

[50] G. Vinti, “The generalized 𝜑-variation in the sense of vitali: esti-mates for integral operators and applications in fractionalcalculus,” Commentationes mathematicae: Prace matematyczne,vol. 34, pp. 199–213, 1994.

[51] L. Angeloni and G. Vinti, “A sufficient condition for the conver-gence of a certain modulus of smoothness in multidimensionalsetting,”Communications onAppliedNonlinearAnalysis, vol. 20,no. 1, pp. 1–20, 2013.

[52] P. L. Butzer and R. J. Nessel, Fourier Analysis and ApproximationI, Academic Press, New York, NY, USA, 1971.

[53] J. Musielak, “Nonlinear approximation in some modular func-tion spaces. I,” Mathematica Japonica, vol. 38, no. 1, pp. 83–90,1993.


[54] L. Angeloni, “A characterization of a modulus of smoothness inmultidimensional setting,” Bollettino della Unione MatematicaItaliana. Serie IX, vol. 4, no. 1, pp. 79–108, 2011.

[55] L. Angeloni and G. Vinti, “A characterization of absolute conti-nuity bymeans ofMellin integral operators,”Zeitschrift für Ana-lysis und Ihre Anwendungen, vol. 34, no. 3, pp. 343–356, 2015.

[56] L. Angeloni and G. Vinti, “Convergence and rate of approxi-mation in 𝐵𝑉𝜑(R𝑁

+) for a class of Mellin integral operators,”

Atti della Accademia Nazionale dei Lincei—Rendiconti Lincei:Matematica e Applicazioni, vol. 25, pp. 217–232, 2014.

[57] L. Angeloni and G. Vinti, “A concept of absolute continuity andits characterization in terms of convergence in variation,”Math-ematische Nachrichten, 2016.

[58] L. Angeloni and G. Vinti, “Convergence in variation and rate ofapproximation for nonlinear integral operators of convolutiontype,” Results in Mathematics, vol. 49, no. 1-2, pp. 1–23, 2006.

[59] L. Angeloni and G. Vinti, “Erratum to: convergence in variationand rate of approximation for nonlinear integral operators ofconvolution type,” Results in Mathematics, vol. 57, no. 3-4, pp.387–391, 2010.

[60] L. Angeloni, “Approximation results with respect to multidi-mensional 𝜑-variation for nonlinear integral operators,” Zeits-chrift für Analysis und Ihre Anwendungen, vol. 32, no. 1, pp. 103–128, 2013.

[61] L. Angeloni and G. Vinti, “Convergence in variation and a cha-racterization of the absolute continuity,” Integral Transforms andSpecial Functions, vol. 26, no. 10, pp. 829–844, 2015.

[62] L. Angeloni, “Convergence in variation for a homothetic mod-ulus of smoothness in multidimensional setting,” Communica-tions on Applied Nonlinear Analysis, vol. 19, no. 1, pp. 1–22, 2012.

[63] L. Angeloni andG.Vinti, “Approximation in variation by homo-thetic operators in multidimensional setting,” Differential andIntegral Equations, vol. 26, no. 5-6, pp. 655–674, 2013.

Research ArticleMappings of Type Special Space of Sequences

Awad A. Bakery1,2

1Department of Mathematics, Faculty of Science and Arts, University of Jeddah (UJ), P.O. Box 355, Khulais 21921, Saudi Arabia2Department of Mathematics, Faculty of Science, Ain Shams University, P.O. Box 1156, Abbassia, Cairo 11566, Egypt

Correspondence should be addressed to Awad A. Bakery; awad [email protected]

Received 12 April 2016; Revised 26 June 2016; Accepted 28 July 2016

Academic Editor: Gianluca Vinti

Copyright © 2016 Awad A. Bakery. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We give sufficient conditions on a special space of sequences defined by Mohamed and Bakery (2013) such that the finite rankoperators are dense in the complete space of operators whose approximation numbers belong to this sequence space. Hence, undera few conditions, every compact operator would be approximated by finite rank operators.We apply it on the sequence space definedby Tripathy and Mahanta (2003). Our results match those known for 𝑝-absolutely summable sequences of reals.

1. Introduction and Basic Definitions

By 𝜔 and 𝐿(𝑉,𝑊), we will denote the spaces of all realsequences and all bounded linear operators between twoBanach spaces𝑉 into𝑊, respectively. In [1], Pietsch, by usingthe approximation numbers and p-absolutely summablesequences of real numbers, formed the operator ideals. In [2],Mohamed and Bakery have considered the space ℓ

𝑀, when

𝑀(𝑡) = 𝑡𝑝(0 < 𝑝 < ∞), which matches especially ℓ𝑝.

A subclass 𝑈 of 𝐿 = {𝐿(𝑉,𝑊)} is an operator ideal if itscomponents verify the following conditions:

(i) The space 𝐹(𝑉,𝑊) of all finite rank operators is asubset of 𝑈(𝑉,𝑊).

(ii) The space 𝑈(𝑉,𝑊) is linear.

(iii) For two Banach spaces𝑉0and𝑊

0, if𝑇 ∈ 𝐿(𝑉

0, 𝑉), 𝑆 ∈

𝑈(𝑉,𝑊), and 𝑅 ∈ 𝐿(𝑊,𝑊0), then 𝑅𝑆𝑇 ∈ 𝑈(𝑉

0,𝑊0).

See [3, 4].

An Orlicz function is a function 𝑀 : [0,∞) → [0,∞)which is convex, positive, nondecreasing, and continuous,where𝑀(0) = 0 and lim

𝑥→∞𝑀(𝑥) = ∞. An Orlicz function

𝑀 is said to satisfy Δ2-condition for all values of 𝑥 ≥ 0 if

there exists a constant 𝑘 > 0, such that 𝑀(2𝑥) ≤ 𝑘𝑀(𝑥).

Lindenstrauss and Tzafriri [5] used the idea of an Orliczfunction to define Orlicz sequence spaces as follows:

ℓ𝑀= {𝑥 ∈ 𝜔 : ∃𝜆 > 0 with 𝜌 (𝜆𝑥) =

∞

∑

𝑘=1

𝑀(|𝜆𝑥 (𝑘)|)

< ∞} .

(1)

(ℓ𝑀, ‖𝑥‖) is a Banach space, where ‖𝑥‖ = inf {𝜆 > 0 :

𝜌(𝑥/𝜆) ≤ 1}. The space ℓ𝑝 is an Orlicz sequence space with𝑀(𝑥) = 𝑥

𝑝 for 1 ≤ 𝑝 < ∞.

Remark 1. For any Orlicz function 𝑀, we have 𝑀(𝜆𝑥) ≤𝜆𝑀(𝑥), for all 𝜆 with 0 < 𝜆 < 1.

Let 𝑃𝑠be the class of all subsets ofN = {0, 1, 2, . . .} that do

not contain more than 𝑠 number of elements and let {𝜙𝑛} be

a nondecreasing sequence of positive reals such that 𝑛𝜙𝑛+1

≤

(𝑛+1)𝜙𝑛, for all 𝑛 ∈ N. Tripathy andMahanta [6] defined and

studied the following sequence space:

𝑚(𝜙,𝑀) = {𝑥 = (𝑥𝑘) ∈ 𝜔 : ∃𝜁

> 0 with sup𝑠≥1,𝜎∈𝑃

𝑠

1

𝜙𝑠

∑

𝑘∈𝜎

𝑀(

𝑥𝑘

𝜁) < ∞} ,

(2)

Hindawi Publishing CorporationJournal of Function SpacesVolume 2016, Article ID 4372471, 6 pageshttp://dx.doi.org/10.1155/2016/4372471

http://dx.doi.org/10.1155/2016/4372471


with the norm

𝜌 (𝑥) = inf {𝜁 > 0 : sup𝑠≥1,𝜎∈𝑃

𝑠

1

𝜙𝑠

∑

𝑘∈𝜎

𝑀(

𝑥𝑘

𝜁) ≤ 1} . (3)

Lemma 2. (i) ℓ𝑀⊆ 𝑚(𝜙,𝑀).

(ii) ℓ𝑀= 𝑚(𝜙,𝑀) if and only if sup

𝑠𝜙𝑠< ∞.

As of late, different classes of sequences have been pre-sented using Orlicz functions by Braha [7], Raj and SharmaSunil [8], Raj et al. [9], and many others ([10–13]).

Definition 3 (see [14, 15]). A special space of sequences (sss)is a linear space 𝐸 with the following:

(1) 𝑒𝑛∈ 𝐸 for all 𝑛 ∈ N, where 𝑒

𝑛= {0, 0, . . . , 1, 0, 0, . . .}

with 1 appearing at 𝑛th place for all 𝑛 ∈ N.

(2) 𝐸 is solid.

(3) (𝑥0, 𝑥0, 𝑥1, 𝑥1, . . .) ∈ 𝐸, if (𝑥

𝑛)𝑛∈N ∈ 𝐸.

A premodular (sss) 𝐸𝜌is a (sss) and there is a function 𝜌 :

𝐸 → [0,∞[ with the following:

(i) 𝜌(𝑥) ≥ 0, for each 𝑥 ∈ 𝐸 and 𝜌(𝑥) = 0 ⇔ 𝑥 = 𝜃,where 𝜃 is the zero element of 𝐸.

(ii) 𝜌 satisfies Δ2-condition.

(iii) For each 𝑥, 𝑦 ∈ 𝐸, 𝜌(𝑥+𝑦) ≤ 𝑘(𝜌(𝑥) +𝜌(𝑦)) holds forsome 𝑘 ≥ 1.

(iv) The space 𝐸 is 𝜌-solid; that is, 𝜌((𝑥𝑛)) ≤ 𝜌((𝑦

𝑛)),

whenever |𝑥𝑛| ≤ |𝑦𝑛|, for all 𝑛 ∈ N.

(v) For some numbers 𝑘0≥ 1, the inequality 𝜌((𝑥

𝑛)𝑛∈N) ≤

𝜌((𝑥0, 𝑥0, 𝑥1, 𝑥1, . . .)) ≤ 𝑘

0𝜌((𝑥𝑛)𝑛∈N) holds.

(vi) 𝐹 = 𝐸𝜌; that is, the set of all finite sequences 𝐹 is 𝜌-

dense in 𝐸.

(vii) For each 𝜆 > 0, there is a constant 𝜉 > 0 such that𝜌(𝜆, 0, 0, 0, . . .) ≥ 𝜉𝜆𝜌(1, 0, 0, 0, . . .).

Condition (ii) says that 𝜌 is continuous at 𝜃. The function𝜌 defines a metrizable topology in 𝐸 and the linear space 𝐸enriched with this topology is denoted by 𝐸

𝜌.

Definition 4 (see [16]). Consider the following:

𝑈app𝐸

fl {𝑈app𝐸

(𝑉,𝑊)} , (4)

where

𝑈app𝐸

(𝑉,𝑊) fl {𝑇 ∈ 𝐿 (𝑉,𝑊) : (𝛼𝑛 (𝑇))𝑛∈N ∈ 𝐸} . (5)

Theorem 5 (see [2]). If 𝐸 is a (sss), then 𝑈𝑎𝑝𝑝𝐸

is an operatorideal.

We explain some results related to the operator spaces.

2. Main Results

In this part, we give sufficient conditions on 𝐸 such thatthe finite rank operators are dense in the complete space ofoperators 𝑈app

𝐸(𝑉,𝑊).

Lemma 6. If 𝐸𝜌is a premodular (sss) and (𝑥

𝑛) ∈ 𝐸

𝜌is a

decreasing sequence of positive reals, then

𝜌(

2𝑛⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞0, 0, 0, . . . , 0, 𝑥

𝑛, 𝑥𝑛+1

, 𝑥𝑛+2

, . . .)

≤ 𝑘0𝜌(

𝑛⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞0, 0, 0, . . . , 0, 𝑥

𝑛, 𝑥𝑛+1

, 𝑥𝑛+2

, . . .) .

(6)

Proof. By using Definition 3, conditions (iv) and (v), andsince the elements of 𝐸 are decreasing, we get

𝜌(

2𝑛⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞0, 0, 0, . . . , 0, 𝑥

𝑛, 𝑥𝑛+1

, 𝑥𝑛+2

, . . .)

≤ 𝜌(

2𝑛⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞0, 0, 0, . . . , 0, 𝑥

𝑛, 𝑥𝑛, 𝑥𝑛+1

, 𝑥𝑛+1

, . . .)

≤ 𝑘0𝜌(

𝑛⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞0, 0, 0, . . . , 0, 𝑥

𝑛, 𝑥𝑛+1

, 𝑥𝑛+2

, . . .) .

(7)

Theorem 7. Let 𝐸𝜌be a premodular (sss); then 𝐹(𝑉,𝑊)𝑔 =

𝑈𝑎𝑝𝑝

𝐸𝜌

(𝑉,𝑊), where 𝑔(𝑇) = 𝜌(𝛼𝑛(𝑇)𝑛∈N).

Proof. To prove that 𝐹(𝑉,𝑊) ⊆ 𝑈app𝐸

(𝑉,𝑊), since 𝑒𝑚∈ 𝐸 for

each 𝑚 ∈ N, from the linearity of 𝐸 and 𝑇 ∈ 𝐹(𝑉,𝑊), thenfinitely many elements of (𝛼

𝑛(𝑇))𝑛∈N are different from zero.

Hence,𝑇 ∈ 𝑈app𝐸

(𝑉,𝑊). For the other inclusion𝑈app𝐸

(𝑉,𝑊) ⊆

𝐹(𝑉,𝑊), let 𝑇 ∈ 𝑈app𝐸

(𝑉,𝑊) and, from the definition ofapproximation numbers, there is 𝑁 ∈ N, 𝑁 > 0, 𝐴

𝑁with

rank(𝐴𝑁) ≤ 𝑁 and also

𝑇 − 𝐴𝑁 ≤ 2𝛼𝑁 (𝑇) . (8)

Since 𝛼𝑁(𝑇) → 0 as 𝑁 → ∞, then ‖𝑇 − 𝐴

𝑁‖ → 0 as

𝑁 → ∞; we have to prove that 𝜌((𝛼𝑛(𝑇 − 𝐴

𝑁))𝑛∈N) → 0 as


𝑁 → ∞, by taking 𝑁 = 8𝜂, where 𝜂 is a natural number.From Definition 3, condition (iii), we have

𝑑 (𝑇, 𝐴𝑁) = 𝜌 ((𝛼

𝑛(𝑇 − 𝐴

𝑁))𝑛∈N

)

= 𝜌[(𝛼0(𝑇 − 𝐴

𝑁) , 𝛼1(𝑇 − 𝐴

𝑁) , . . . ,

𝛼8𝜂−1

(𝑇 − 𝐴𝑁) , 0, 0, 0, . . .) + (

8𝜂

⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞0, 0, 0, . . . , 0,

𝛼8𝜂(𝑇 − 𝐴

𝑁) , 𝛼8𝜂+1

(𝑇 − 𝐴𝑁) , . . . , 𝛼

12𝜂−1(𝑇 − 𝐴

𝑁) ,

0, 0, 0, . . .) + (

12𝜂

⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞0, 0, 0, . . . , 0, 𝛼

12𝜂(𝑇 − 𝐴

𝑁) ,

𝛼12𝜂+1

(𝑇 − 𝐴𝑁) , . . .)] ≤ 𝑘

2[𝜌 (𝛼

0(𝑇 − 𝐴

𝑁) ,

𝛼1(𝑇 − 𝐴

𝑁) , . . . , 𝛼

8𝜂−1(𝑇 − 𝐴

𝑁) , 0, 0, 0, . . .)

+ 𝜌(

8𝜂

⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞0, 0, 0, . . . , 0, 𝛼

8𝜂(𝑇 − 𝐴

𝑁) , 𝛼8𝜂+1

(𝑇 − 𝐴𝑁) ,

. . . , 𝛼12𝜂−1

(𝑇 − 𝐴𝑁) , 0, 0, 0, . . .) + 𝜌(

12𝜂

⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞0, 0, 0, . . . , 0,

𝛼12𝜂

(𝑇 − 𝐴𝑁) , 𝛼12𝜂+1

(𝑇 − 𝐴𝑁) , . . .)] = 𝑘

2[𝐼1 (𝑁)

+ 𝐼2(𝜂) + 𝐼

3(𝜂)] .

(9)

Since 𝛼𝑛(𝐴𝑁) = 0 for 𝑛 ≥ 𝑁, then

𝛼𝑛(𝑇 − 𝐴

𝑁) ≤ 𝛼𝑛−𝑁 (𝑇) . (10)

By using Lemma 6, inequality (10), and Definition 3, condi-tion (v), we get

𝐼3(𝜂) = 𝜌(

12𝜂

⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞0, 0, 0, . . . , 0, 𝛼

12𝜂(𝑇 − 𝐴

𝑁) ,

𝛼12𝜂+1

(𝑇 − 𝐴𝑁) , . . .) ≤ 𝜌(

12𝜂

⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞0, 0, 0, . . . , 0, 𝛼

4𝜂 (𝑇) ,

𝛼4𝜂+1 (𝑇) , . . .) ≤ 𝑘0𝜌(

6𝜂

⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞0, 0, 0, . . . , 0, 𝛼

4𝜂 (𝑇) ,

𝛼4𝜂+1 (𝑇) , . . .) ≤ 𝑘

2

0𝜌(

3𝜂

⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞0, 0, 0, . . . , 0, 𝛼

4𝜂 (𝑇) ,

𝛼4𝜂+1 (𝑇) , . . .) ≤ 𝑘

2

0𝜌(

3𝜂

⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞0, 0, 0, . . . , 0, 𝛼

3𝜂 (𝑇) ,

𝛼3𝜂+1 (𝑇) , . . .) = 𝑘

2

0𝜌 ((𝛼𝑛 (𝑇))

∞

𝑛=3𝜂) → 0

as 𝜂 → ∞.

(11)

Now, using Lemma 6 andDefinition 3, condition (v), we have

𝐼2(𝜂) = 𝜌(

8𝜂

⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞0, 0, 0, . . . , 0, 𝛼

8𝜂(𝑇 − 𝐴

𝑁) ,

𝛼8𝜂+1

(𝑇 − 𝐴𝑁) , . . . , 𝛼

12𝜂−1(𝑇 − 𝐴

𝑁) , 0, 0, 0, . . .)

≤ 𝑘0𝜌(

4𝜂

⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞0, 0, 0, . . . , 0, 𝛼

8𝜂(𝑇 − 𝐴

𝑁) ,

𝛼8𝜂+1

(𝑇 − 𝐴𝑁) , . . . , 𝛼

12𝜂−1(𝑇 − 𝐴

𝑁) , 0, 0, 0, . . .)

≤ 𝑘0𝜌 (𝛼0(𝑇 − 𝐴

𝑁) , 𝛼1(𝑇 − 𝐴

𝑁) , . . . ,

𝛼8𝜂−1

(𝑇 − 𝐴𝑁) , 0, 0, 0, . . .) = 𝑘

0𝐼1 (𝑁) .

(12)

Finally, we have to show that 𝐼1(𝑁) → 0 as𝑁 →∞. Since𝑇 ∈

𝑈app𝐸

(𝑉,𝑊) and𝜌 is continuous at 𝜃, we have𝜌(𝛼𝑘(𝑇))∞

𝑘=𝑛→ 0

as 𝑛 → ∞. Then, for each 𝜀 > 0, there exists𝑁0(𝜀) such that

for all 𝑛 ≥ 𝑁0(𝜀) we have

𝜌 ((𝛼𝑘 (𝑇))

∞

𝑘=𝑛) < 𝜀. (13)

By taking 𝜀1= 𝜀/3𝑙𝑘 for each 𝑛 ≥ 𝑁

0(𝜀1) and using inequality

(13) and Definition 3, conditions (ii) and (iii), then we have

𝐼1 (𝑁) = 𝜌 (𝛼0 (𝑇 − 𝐴𝑁) , 𝛼1 (𝑇 − 𝐴𝑁) , . . . , 𝛼𝑁−1 (𝑇

− 𝐴𝑁) , 0, 0, 0, . . .) ≤ 𝑘 [𝜌 (𝛼

0(𝑇 − 𝐴

𝑁) ,

𝛼1(𝑇 − 𝐴

𝑁) , . . . , 𝛼

𝑁0−1(𝑇 − 𝐴

𝑁) , 0, 0, 0, . . .)

+ 𝜌(

𝑁0

⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞0, 0, 0, . . . , 0, 𝛼

𝑁0

(𝑇 − 𝐴𝑁) , 𝛼𝑁0+1(𝑇 − 𝐴

𝑁) ,

. . . , 𝛼𝑁−1

(𝑇 − 𝐴𝑁) , 0, 0, 0, . . .)] ≤ 𝑘[𝜌 (

𝑇 − 𝐴𝑁 ,

𝑇 − 𝐴𝑁 , . . . ,

𝑇 − 𝐴𝑁 , 0, 0, 0, . . .)

+ 𝜌(

𝑁0

⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞0, 0, 0, . . . , 0, 2𝛼

𝑁 (𝑇) , 2𝛼𝑁 (𝑇) , . . . , 2𝛼𝑁 (𝑇) ,

0, 0, 0, . . .)] ≤ 𝑘[𝑇 − 𝐴𝑁

𝑙𝜌(

𝑁0

⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞1, 1, 1, . . . , 1, 0, 0, 0,

. . .) + 2𝑙𝜌(

𝑁0

⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞0, 0, 0, . . . , 0, 𝛼

𝑁0

(𝑇) , 𝛼𝑁0+1 (𝑇) , . . . ,

𝛼𝑁 (𝑇) , 0, 0, 0, . . .)] ≤ 𝑘 [

𝑇 − 𝐴𝑁 𝑙𝑘1 (𝜀) + 2𝑙𝜀1] ,

(14)

where 𝑘1(𝜀) = 𝜌(

𝑁0

⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞1, 1, 1, . . . , 1, 0, 0, 0, . . .), and since ‖𝑇 −

𝐴𝑁‖ → 0 as 𝑁 → ∞, then for each 𝜀 > 0 there exists 𝑁


such that ‖𝑇 − 𝐴𝑁‖𝑘1(𝜀) ≤ 𝜀

1, for that we have 𝐼

1(𝑁) ≤

𝑘[𝑙𝜀1+ 2𝑙𝜀1] = 3𝑘𝑙𝜀

1= 𝜀.This completes the proof.

We give here the sufficient conditions on the sequencespaces𝑚(𝜙,𝑀) such that the class of all bounded linear oper-ators between any arbitrary Banach spaces with (𝛼

𝑛(𝑇))𝑛∈N in

these sequence spaces form an ideal operator; the ideal of thefinite rank operators in the class of Banach spaces is dense in𝑈

app𝑚(𝜙,𝑀)

(𝑉,𝑊).

Theorem 8. Let 𝑀 be an Orlicz function satisfying Δ2-

condition. Then,

(a) 𝑈𝑎𝑝𝑝𝑚(𝜙,𝑀)

is an operator ideal,

(b) 𝐹(𝑉,𝑊) = 𝑈𝑎𝑝𝑝𝑚(𝜙,𝑀)

(𝑉,𝑊).

Proof. We first prove that the space𝑚(𝜙,𝑀) is a (sss).(1) Let 𝜆

1, 𝜆2∈ R and 𝑥, 𝑦 ∈ 𝑚(𝜙,𝑀); then there exist

𝜁1> 0 and 𝜁

2> 0 such that

sup𝑠≥1,𝜎∈𝑃

𝑠

1

𝜙𝑠

∑

𝑘∈𝜎

𝑀(

𝑥𝑘

𝜁1

) < ∞,


𝑠

1

𝜙𝑠

∑

𝑘∈𝜎

𝑀(

𝑥𝑘

𝜁2

) < ∞.

(15)

Let 𝜁3= max (2|𝜆

1|𝜁1, 2|𝜆2|𝜁2). Since 𝑀 is nondecreasing

convex function with Δ2-condition, we have

∑

𝑘∈𝜎

𝑀(

𝜆1𝑥𝑘 + 𝜆2𝑦𝑘

𝜁3

)

≤1

2[∑

𝑘∈𝜎

𝑀(

𝑥𝑘

𝜁1

) + ∑

𝑘∈𝜎

𝑀(

𝑦𝑘

𝜁2

)] .

(16)

So, we get


𝑠

1

𝜙𝑠

∑

𝑘∈𝜎

𝑀(

𝜆1𝑥𝑘 + 𝜆2𝑦𝑘

𝜁3

)

≤1

2[ sup𝑠≥1,𝜎∈𝑃

𝑠

1

𝜙𝑠

∑

𝑘∈𝜎

𝑀(

𝑥𝑘

𝜁1

)

+ sup𝑠≥1,𝜎∈𝑃

𝑠

1

𝜙𝑠

∑

𝑘∈𝜎

𝑀(

𝑦𝑘

𝜁2

)] .

(17)

Thus, 𝜆1𝑥+𝜆2𝑦 ∈ 𝑚(𝜙,𝑀). Hence,𝑚(𝜙,𝑀) is a linear space

over the field of real numbers. Also, since 𝑒𝑛∈ ℓ𝑀and ℓ𝑀⊆

𝑚(𝜙,𝑀), we have 𝑒𝑛∈ 𝑚(𝜙,𝑀) for all 𝑛 ∈ N.

(2) Let 𝑥 ∈ 𝜔 and 𝑦 = (𝑦𝑘)∞

𝑘=0∈ 𝑚(𝜙,𝑀) with |𝑥

𝑘| ≤ |𝑦𝑘|

for each 𝑘 ∈ N; since𝑀 is nondecreasing, then we get


𝑠

1

𝜙𝑠

∑

𝑘∈𝜎

𝑀(

𝑥𝑘

𝜁) ≤ sup𝑠≥1,𝜎∈𝑃

𝑠

1

𝜙𝑠

∑

𝑘∈𝜎

𝑀(

𝑦𝑘

𝜁)

< ∞;

(18)

then 𝑥 = (𝑥𝑘)∞

𝑘=0∈ 𝑚(𝜙,𝑀).

(3) Let 𝑥 = (𝑥𝑘)∞

𝑘=0∈ 𝑚(𝜙,𝑀); then we have


𝑠

1

𝜙𝑠

∑

𝑘∈𝜎

𝑀(

𝑥[𝑘/2]

𝜁)

≤ 2 sup𝑠≥1,𝜎∈𝑃

𝑠

1

𝜙𝑠

∑

𝑘∈𝜎

𝑀(

𝑥𝑘

𝜁) < ∞;

(19)

then 𝑥 = (𝑥[𝑘/2]

)∞

𝑘=0∈ 𝑚(𝜙,𝑀).

Finally, we have proved that the space𝑚(𝜙,𝑀) with 𝜌(𝑥)is a premodular (sss).

(i) Clearly, 𝜌(𝑥) ≥ 0 for all 𝑥 ∈ 𝑚(𝜙,𝑀) and 𝜌(𝑥) = 0 ⇔𝑥 = 𝜃.

(ii) Let 𝜆 ∈ R and 𝑥 ∈ 𝑚(𝜙,𝑀); then for 𝜆 ̸= 0 we have

𝜌 (𝜆𝑥)

= inf {𝜁 > 0 : sup𝑠≥1,𝜎∈𝑃

𝑠

1

𝜙𝑠

∑

𝑘∈𝜎

𝑀(

𝜆𝑥𝑘

𝜁) ≤ 1}

= inf {|𝜆| 𝜇 > 0 : sup𝑠≥1,𝜎∈𝑃

𝑠

1

𝜙𝑠

∑

𝑘∈𝜎

𝑀(

𝑥𝑘

𝜇) ≤ 1} ,

(20)

where 𝜇 = 𝜁/|𝜆|. Thus, 𝜌(𝜆𝑥) = |𝜆| inf {𝜇 > 0 :sup𝑠≥1,𝜎∈𝑃

𝑠

(1/𝜙𝑠) ∑𝑘∈𝜎

𝑀(|𝑥𝑘|/𝜇) ≤ 1} = |𝜆|𝜌(𝑥).

Also, for 𝜆 = 0, we have 𝜌(𝜆𝑥) = 𝜆𝜌(𝑥) = 0.(iii) Let 𝑥, 𝑦 ∈ 𝑚(𝜙,𝑀); then there exist 𝜁

1> 0 and 𝜁

2> 0

such that


𝑠

1

𝜙𝑠

∑

𝑘∈𝜎

𝑀(

𝑥𝑘

𝜁1

) ≤ 1,


𝑠

1

𝜙𝑠

∑

𝑘∈𝜎

𝑀(

𝑥𝑘

𝜁2

) ≤ 1.

(21)

Let 𝜁 = 𝜁1+ 𝜁2, and since 𝑀 is nondecreasing and convex,

then we have


𝑠

1

𝜙𝑠

∑

𝑘∈𝜎

𝑀(

𝑥𝑘 + 𝑦𝑘


𝑠

1

𝜙𝑠

⋅ ∑

𝑘∈𝜎

𝑀(

𝑥𝑘 +

𝑦𝑘

𝜁1+ 𝜁2

) ≤ sup𝑠≥1,𝜎∈𝑃

𝑠

1

𝜙𝑠

⋅ ∑

𝑘∈𝜎

[(𝜁1

𝜁1+ 𝜁2

)𝑀(

𝑥𝑘

𝜁1

)

+ (𝜁2

𝜁1+ 𝜁2

)𝑀(

𝑦𝑘

𝜁2

)] ≤ (𝜁1

𝜁1+ 𝜁2

) sup𝑠≥1,𝜎∈𝑃

𝑠

1

𝜙𝑠

⋅ ∑

𝑘∈𝜎

𝑀(

𝑥𝑘

𝜁1

) + (𝜁2

𝜁1+ 𝜁2

) ∑

𝑘∈𝜎

𝑀(

𝑦𝑘

𝜁2

) ≤ 1.

(22)


Since 𝜁’s are nonnegative, we have

𝜌 (𝑥 + 𝑦)


𝑠

1

𝜙𝑠

∑

𝑘∈𝜎

𝑀(

𝑥𝑘 + 𝑦𝑘

𝜁) ≤ 1}

≤ inf {𝜁1> 0 : sup𝑠≥1,𝜎∈𝑃

𝑠

1

𝜙𝑠

∑

𝑘∈𝜎

𝑀(

𝑥𝑘

𝜁1

) ≤ 1}

+ inf {𝜁2> 0 : sup𝑠≥1,𝜎∈𝑃

𝑠

1

𝜙𝑠

∑

𝑘∈𝜎

𝑀(

𝑦𝑘

𝜁2

) ≤ 1}

= 𝜌 (𝑥) + 𝜌 (𝑦) .

(23)

(iv) Let |𝑥𝑘| ≤ |𝑦

𝑘| for each 𝑘 ∈ N, and since 𝑀 is

nondecreasing, then we get


𝑠

1

𝜙𝑠

∑

𝑘∈𝜎

𝑀(

𝑥𝑘


𝑠

1

𝜙𝑠

∑

𝑘∈𝜎

𝑀(

𝑦𝑘

𝜁) ; (24)

thus,

inf {𝜁 > 0 : sup𝑠≥1,𝜎∈𝑃

𝑠

1

𝜙𝑠

∑

𝑘∈𝜎

𝑀(

𝑥𝑘

𝜁)}

≤ inf {𝜁 > 0 : sup𝑠≥1,𝜎∈𝑃

𝑠

1

𝜙𝑠

∑

𝑘∈𝜎

𝑀(

𝑦𝑘

𝜁)} .

(25)

So, 𝜌(𝑥) ≤ 𝜌(𝑦).(v) Since


𝑠

1

𝜙𝑠

∑

𝑘∈𝜎

𝑀(

𝑥[𝑘/2]

𝜁)

≤ 2 sup𝑠≥1,𝜎∈𝑃

𝑠

1

𝜙𝑠

∑

𝑘∈𝜎

𝑀(

𝑥𝑘

𝜁) ,

(26)

we have

inf {𝜁 > 0 : sup𝑠≥1,𝜎∈𝑃

𝑠

1

𝜙𝑠

∑

𝑘∈𝜎

𝑀(

𝑥[𝑘/2]

𝜁)}

≤ 2 inf {𝜁 > 0 : sup𝑠≥1,𝜎∈𝑃

𝑠

1

𝜙𝑠

∑

𝑘∈𝜎

𝑀(

𝑥𝑘

𝜁)} .

(27)

So, 𝜌((𝑥𝑘)) ≤ 𝜌((𝑥

[𝑘/2])) ≤ 2𝜌((𝑥

𝑘)).

(vi) For each 𝑥 = (𝑥𝑘)∞

𝑘=0∈ 𝑚(𝜙,𝑀), then

𝜌 ((𝑥𝑘)∞

𝑘=0)


𝑠

1

𝜙𝑠

∑

𝑘∈𝜎

𝑀(

𝑥[𝑘/2]

𝜁) < ∞} ;

(28)

we can find 𝑡 ∈ N such that 𝜌((𝑥𝑘)∞

𝑘=𝑡) < ∞. This means the

set of all finite sequences is 𝜌-dense in𝑚(𝜙,𝑀).

(vii) For any 𝜆 > 0, there exists a constant 𝜁 ∈ ]0, 1] suchthat

𝜌 (𝜆, 0, 0, 0, . . .) ≥ 𝜁𝜆𝜌 (1, 0, 0, 0, . . .) . (29)

By usingTheorems 5 and 7, the proof follows.

As special cases of the above theorem, we obtain thefollowing corollaries.

Corollary 9. If sup𝑠𝜙𝑠< ∞, one gets that

(a) 𝑈𝑎𝑝𝑝ℓ𝑀

is an operator ideal,

(b) 𝐹(𝑉,𝑊) = 𝑈𝑎𝑝𝑝ℓ𝑀

(𝑉,𝑊).

Corollary 10. If sup𝑠𝜙𝑠< ∞ and𝑀(𝑡) = 𝑡𝑝 with 0 < 𝑝 < ∞,

one gets that

(a) 𝑈𝑎𝑝𝑝ℓ𝑝 is an operator ideal,

(b) 𝐹(𝑉,𝑊) = 𝑈𝑎𝑝𝑝ℓ𝑝 (𝑉,𝑊). See [1].

Theorem 11. If 𝐸𝜌is a premodular (sss), then 𝑈𝑎𝑝𝑝

𝐸𝜌

(𝑉,𝑊) iscomplete.

Proof. Let (𝑇𝑚) be a Cauchy sequence in 𝑈app

𝐸𝜌

(𝑉,𝑊); then,by using Definition 3, condition (vii), and since𝑈app

𝐸𝜌

(𝑉,𝑊) ⊆

𝐿(𝑉,𝑊), we get

𝜌 ((𝛼𝑛(𝑇𝑖− 𝑇𝑗))𝑛∈N

) ≥ 𝜌 (𝛼0(𝑇𝑖− 𝑇𝑗) , 0, 0, 0, . . .)

= 𝜌 (𝑇𝑖− 𝑇𝑗

, 0, 0, 0, . . .)

≥ 𝜉𝑇𝑖− 𝑇𝑗

𝜌 (1, 0, 0, 0, . . .) ;

(30)

then (𝑇𝑚) is also a Cauchy sequence in 𝐿(𝑉,𝑊). Since the

space 𝐿(𝑉,𝑊) is a Banach space, then there exists 𝑇 ∈𝐿(𝑉,𝑊) such that ‖𝑇

𝑚− 𝑇‖ → 0, as 𝑚 → ∞, and since

(𝛼𝑛(𝑇𝑚))𝑛∈N ∈ 𝐸, for each 𝑚 ∈ N, then from Definition 3,

conditions (iii) and (v), and since𝜌 is continuous at 𝜃, we have

𝜌 ((𝛼𝑛 (𝑇))𝑛∈N) = 𝜌 (𝛼𝑛 (𝑇 − 𝑇𝑚 + 𝑇𝑚))𝑛∈N

≤ 𝑘�

function spaces, approximation theory, and their...

Documents