polynomial wavelet type expansions on the...

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......Polynomial Wavelet Type Expansions on the Spline

A. Askari-Hemmat1, M. A. Dehghan2, M. A. Skopina3

1Department of Mathematics, Graduate University of Advanced Technology, Kerman, Iran1Department of Mathematics, Shahid Bahonar University, Kerman, Iran

2Department of Mathematics, Vali-E-Asr University, Rafsanjan, Iran3Department of Mathematics, St. Petersburg University, Russia

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Jan., 06, 2014Askari-Hemmat ([email protected]) Polynomial Wavelet Type Expansions on the Spline Jan., 06, 2014 1 / 22

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Introduction

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We present a polynomial wavelet-type system on Sd such that anycontinuous function can be expanded with respect to these wavelets.The order of the growth of the degree of the polynomials is optimal.The coefficients in the expansion are the inner product of the function andthe corresponding element of a dual wavelet system.The dual wavelet system is also a polynomial system with the same growthof degree of polynomials. The system is redundant.

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A construction of a polynomial basis is also presented. In constrast to ourwavelet-type system, this basis is not suitable for implementation, becauseof two drawbacks: first there are no explicit formoula for the coefficientfunctionals and, secend, the growth of the degree of polynomials is toorapid.

Askari-Hemmat ([email protected]) Polynomial Wavelet Type Expansions on the Spline Jan., 06, 2014 2 / 22

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Introduction

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We present a polynomial wavelet-type system on Sd such that anycontinuous function can be expanded with respect to these wavelets.The order of the growth of the degree of the polynomials is optimal.The coefficients in the expansion are the inner product of the function andthe corresponding element of a dual wavelet system.The dual wavelet system is also a polynomial system with the same growthof degree of polynomials. The system is redundant.

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......

A construction of a polynomial basis is also presented. In constrast to ourwavelet-type system, this basis is not suitable for implementation, becauseof two drawbacks: first there are no explicit formoula for the coefficientfunctionals and, secend, the growth of the degree of polynomials is toorapid.

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History

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1961, Foias and Singer found the first polynomial basis for C[a, b].

1987, Privalov, consrucrtd optimal polynomial basis for the space ofcontinuous functions in both the trigonometric and the algebraiccases.

1994, Lorentz and Saakyan , proved that packets of periodic Meyerwavelets are such bases. That is orthogonal.

1988, Freeden and Schreiner, proposed the wavelet-type polynomialsystems on the two-dimensional sphere.

1988, Farkov extended their construction to the multidimensionalcase.

2001, Skopina, found a similar wavelet system in L2[a, b] and provedthat the corresponding wavelet packets are optimal plynomialorthogonal bases for the space C[a, b].

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History

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We show that, for any function f ∈ C(Sd), certain spherical waveletsprovide a uniformly convergent polynomial expansion.

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The following notations will be used:

The inner product

x.y = x1y1 + · · ·+ xdyd.

πdn : is the space of polynomials in d variables of degree at most n.

Bd ={x ∈ Rd|∥x∥ ≤ 1

}, Sd = ∂Bd+1.

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The inner product

x.y = x1y1 + · · ·+ xdyd.

Bd ={x ∈ Rd|∥x∥ ≤ 1

}, Sd = ∂Bd+1.

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The inner product

x.y = x1y1 + · · ·+ xdyd.

Bd ={x ∈ Rd|∥x∥ ≤ 1

}, Sd = ∂Bd+1.

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The inner product

x.y = x1y1 + · · ·+ xdyd.

Bd ={x ∈ Rd|∥x∥ ≤ 1

}, Sd = ∂Bd+1.

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The inner product

x.y = x1y1 + · · ·+ xdyd.

Bd ={x ∈ Rd|∥x∥ ≤ 1

}, Sd = ∂Bd+1.

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G(λ)n : Denotes the nth Gegenbauer polynomial of order λ.

Jacobi’s polynomials, {P (α,β)n (x)}, are defined as orthogonalpolynomials with respect to the weight function (1− x)α(1 + x)β on(−1, 1). (α > −1, β > −1)If we put α = β in the above definition and define A

(λ)0 = 1,

A(0)n =2n+1(n− 1)

1.3. . . . .(2n− 1), n ≥ 1

A(λ)n =2n(2λ)(2λ+ 1) · · · (2λ+ n− 1)

(2λ+ 1) · · · (2(λ+ n)− 1), n ≥ 1, λ ̸= 0

then Gegenbauer polynomials are defined as

G(λ)n = A(λ)n P

(α,α)n (x), λ = (α+

1

2), n ≥ 0.

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G(λ)n : Denotes the nth Gegenbauer polynomial of order λ.

Jacobi’s polynomials, {P (α,β)n (x)}, are defined as orthogonalpolynomials with respect to the weight function (1− x)α(1 + x)β on(−1, 1). (α > −1, β > −1)If we put α = β in the above definition and define A

(λ)0 = 1,

A(0)n =2n+1(n− 1)

1.3. . . . .(2n− 1), n ≥ 1

A(λ)n =2n(2λ)(2λ+ 1) · · · (2λ+ n− 1)

(2λ+ 1) · · · (2(λ+ n)− 1), n ≥ 1, λ ̸= 0

then Gegenbauer polynomials are defined as

G(λ)n = A(λ)n P

(α,α)n (x), λ = (α+

1

2), n ≥ 0.

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The area of d-dimensional sphere (Sd):

ωd =

∫sdds(x) =

2π(d+12

)

Γ(d+12 ).

For functions F,G ∈ L2(Sd), the inner product is defined as follows.

< F,G >=

∫Sd

F (x)Ḡ(x)ds(x).

The restriction to Sd of a homogeneous harmonic polynomial ofdegree n is called a spherical harmonic of degree n.

For a fixed integer n, Hdn denotes the class of spherical harmonics ofdegree n.

The spaces Hdn are mutually orthogonal.

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ωd =

∫sdds(x) =

2π(d+12

)

Γ(d+12 ).

< F,G >=

∫Sd

F (x)Ḡ(x)ds(x).

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ωd =

∫sdds(x) =

2π(d+12

)

Γ(d+12 ).

< F,G >=

∫Sd

F (x)Ḡ(x)ds(x).

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ωd =

∫sdds(x) =

2π(d+12

)

Γ(d+12 ).

< F,G >=

∫Sd

F (x)Ḡ(x)ds(x).

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ωd =

∫sdds(x) =

2π(d+12

)

Γ(d+12 ).

< F,G >=

∫Sd

F (x)Ḡ(x)ds(x).

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⊕nl=1H

dl comprises the restrictions to of all algebraic polynomials in

d+ 1 variables of degree not exceeding n.

Dimension of Hdn is given by:

δdn := dimHdn =

{2n+d+1n+d−1

(n+d−1

n

)if n ≥ 1

1 if n = 0

⊕∞n=1H

dn is dense in L2(S

d).

Muller’s formula (1966): Let{Ynk, k = 1, · · · δdn be an orthonormal

basis for Hdn, then

δdn∑k=0

Ynk(x)Ynk(y) =(2n+ d− 1)ωd(d− 1)

Gd−12

n (x.y) (1)

for all x, y ∈ Sd and for all n = 0, 1, . . . .

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⊕nl=1H

δdn := dimHdn =

{2n+d+1n+d−1

(n+d−1

n

)if n ≥ 1

1 if n = 0

⊕∞n=1H

dn is dense in L2(S

d).

basis for Hdn, then

δdn∑k=0

Gd−12

n (x.y) (1)

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⊕nl=1H

δdn := dimHdn =

{2n+d+1n+d−1

(n+d−1

n

)if n ≥ 1

1 if n = 0

⊕∞n=1H

dn is dense in L2(S

d).

basis for Hdn, then

δdn∑k=0

Gd−12

n (x.y) (1)

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⊕nl=1H

δdn := dimHdn =

{2n+d+1n+d−1

(n+d−1

n

)if n ≥ 1

1 if n = 0

⊕∞n=1H

dn is dense in L2(S

d).

basis for Hdn, then

δdn∑k=0

Gd−12

n (x.y) (1)

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.Theorem 1: F. Mahaskar and others (2001)..

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There is constants Nd and Ad depending only on d so that for any finiteset {ηl}l∈Ω of distinct points ηl ∈ Sd and for any positive integer N ≥ Ndsatisfying

N maxx∈Sd

minl∈Ω

|x− ηl| ≤ Ad

there exist nonnegative weights al, l ∈ Ω such that∫Sd

P (x)ds(x) =∑l∈Ω

alP (ηl)

for all P ∈ πd+1N .

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Because of this theorem, to each positive integer j we can assign a set

{η(j)l }l∈Ωj of distinct points η(j)l ∈ S

d and a set {a(j)l }l∈Ωj of nonnegativeweights the following properties:card(Ωj)∼ 2dj , ∫

Sd

P (x)ds(x) =∑l∈Ωj

a(j)l P (η

(j)l )

for any P ∈ πd+12j

. Additionally, we introduce the set Ω0 := {0} and puta(0)0 = ωd.

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Due to this theorem to each positive integer j we can assign a set

{η(j)l } of distinct points η(j)l ∈ S

d and a set {a(j)l }l∈Ωj of nonnegativeweights with the following properties:

card(Ωj)∼ 2dj . ∫Sd

a(j)l P (η

(j)l ) (2)

.

Additionally we introduce the set Ω0 := {0} and put a(0)0 = ωd.

Let hj(n) =(2

j−n+d−1d )

(2j+d−12j−1 )

for n = 0, ..., 2j − 1, hj(n) = 0 for

n = 2j , 2j + 1, ....

An inequality due to kogbeliantz states that

n∑n=0

(N − n+ d

d

)(2n+d−1)G(

d−12

)n (t) ≥ 0, for all t ∈ [−1, 1]. (3)

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card(Ωj)∼ 2dj .

∫Sd

a(j)l P (η

(j)l ) (2)

.

Let hj(n) =(2

j−n+d−1d )

(2j+d−12j−1 )

for n = 0, ..., 2j − 1, hj(n) = 0 for

n = 2j , 2j + 1, ....

n∑n=0

(N − n+ d

d

)(2n+d−1)G(

d−12

)n (t) ≥ 0, for all t ∈ [−1, 1]. (3)

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a(j)l P (η

(j)l ) (2)

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Let hj(n) =(2

j−n+d−1d )

(2j+d−12j−1 )

for n = 0, ..., 2j − 1, hj(n) = 0 for

n = 2j , 2j + 1, ....

n∑n=0

(N − n+ d

d

)(2n+d−1)G(

d−12

)n (t) ≥ 0, for all t ∈ [−1, 1]. (3)

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a(j)l P (η

(j)l ) (2)

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Let hj(n) =(2

j−n+d−1d )

(2j+d−12j−1 )

for n = 0, ..., 2j − 1, hj(n) = 0 for

n = 2j , 2j + 1, ....

n∑n=0

(N − n+ d

d

)(2n+d−1)G(

d−12

)n (t) ≥ 0, for all t ∈ [−1, 1]. (3)

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a(j)l P (η

(j)l ) (2)

.

Let hj(n) =(2

j−n+d−1d )

(2j+d−12j−1 )

for n = 0, ..., 2j − 1, hj(n) = 0 for

n = 2j , 2j + 1, ....

n∑n=0

(N − n+ d

d

)(2n+d−1)G(

d−12

)n (t) ≥ 0, for all t ∈ [−1, 1]. (3)

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Now we are going to define our wavelet-type functions.Set for j = 1, ..., n = 0, 1, ...:

gj(n) = hj(n) + hj−1(n)

g̃j(n) = hj(n)− hj−1(n)

g0(0) = h0(0) + 1, g̃0(0) = h0(0)− 1, g0(n) = g̃0(n) = 0, for n = 1, 2, ....

For each nonnegative integer j and for each l ∈ Ωj+1, define the waveletfunction Ψjl, the dual wavelet function Ψ̃jl and the scaling functionΦ(j+1)l by

Ψjl(x) =1

ωd(d− 1)∑n∈z+

gj(n)(2n+ d− 1)×G( d−1

2)

n (η(j+1)l . x)

Ψ̃jl(x) =1

ωd(d− 1)∑n∈z+

g̃j(n)(2n+ d− 1)×G( d−1

2)

n (η(j+1)l . x)

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Now we are going to define our wavelet-type functions.Set for j = 1, ..., n = 0, 1, ...:

gj(n) = hj(n) + hj−1(n)

g̃j(n) = hj(n)− hj−1(n)

g0(0) = h0(0) + 1, g̃0(0) = h0(0)− 1, g0(n) = g̃0(n) = 0, for n = 1, 2, ....For each nonnegative integer j and for each l ∈ Ωj+1, define the waveletfunction Ψjl, the dual wavelet function Ψ̃jl and the scaling functionΦ(j+1)l by

Ψjl(x) =1

ωd(d− 1)∑n∈z+

gj(n)(2n+ d− 1)×G( d−1

2)

n (η(j+1)l . x)

Ψ̃jl(x) =1

ωd(d− 1)∑n∈z+

g̃j(n)(2n+ d− 1)×G( d−1

2)

n (η(j+1)l . x)

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Φ(j+1)l(x) =1

ωd(d− 1)∑n∈z+

h(n)(2n+ d− 1)×G(d−12

)n (η

(j+1)l . x)

Complete this collection by the function Φ0,0 =1ωd

and set Φ0 =√ωdΦ0,0.

For F ∈ C(Sd), we will study the convergence of the series

< F,Φ0 > Φ0 +∞∑i=0

∑l∈Ωi+1

a(i+1)l < F, Ψ̃il > Ψil.

Set

Λj,ω(F ) =< F,Φ0 > Φ0+

j−1∑i=0

∑l∈Ωi+1

a(i+1)l < F, Ψ̃il > Ψil

+∑l∈ω

a(j+1)l < F, Ψ̃jl > Ψjl

where ω is a subset of Ωj+1.

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Φ(j+1)l(x) =1

ωd(d− 1)∑n∈z+

h(n)(2n+ d− 1)×G(d−12

)n (η

(j+1)l . x)

< F,Φ0 > Φ0 +∞∑i=0

∑l∈Ωi+1

Set

Λj,ω(F ) =< F,Φ0 > Φ0+

j−1∑i=0

∑l∈Ωi+1

+∑l∈ω

where ω is a subset of Ωj+1.

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Φ(j+1)l(x) =1

ωd(d− 1)∑n∈z+

h(n)(2n+ d− 1)×G(d−12

)n (η

(j+1)l . x)

< F,Φ0 > Φ0 +∞∑i=0

∑l∈Ωi+1

Set

Λj,ω(F ) =< F,Φ0 > Φ0+

j−1∑i=0

∑l∈Ωi+1

+∑l∈ω

where ω is a subset of Ωj+1.Askari-Hemmat ([email protected]) Polynomial Wavelet Type Expansions on the Spline Jan., 06, 2014 12 / 22

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.Main Theorem..

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For any F ∈ C(Sd)limj→∞

∥F − Λj,ω(F )∥ = 0 (4)

First we will prove that the operators Λj,ω : C(Sd) −→ C(Sd), are

uniformly bounded. We show that (4) holds on the set of sphericalpolynomials and our claim is a consequence of Banach-Steinhaustheorem. To prove the main theorem we need the following lemma:

.Lemma..

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For any F ∈ C(Sd)

< F,Φ0 > Φ0+

j−1∑i=0

∑l∈Ωi+1

=∑l∈Ωj

a(j)l < F,Φjl > Φjl. (5)

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.Main Theorem..

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For any F ∈ C(Sd)limj→∞

∥F − Λj,ω(F )∥ = 0 (4)

First we will prove that the operators Λj,ω : C(Sd) −→ C(Sd), are

uniformly bounded. We show that (4) holds on the set of sphericalpolynomials and our claim is a consequence of Banach-Steinhaustheorem. To prove the main theorem we need the following lemma:

.Lemma..

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For any F ∈ C(Sd)

< F,Φ0 > Φ0+

j−1∑i=0

∑l∈Ωi+1

=∑l∈Ωj

a(j)l < F,Φjl > Φjl. (5)

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Construct a polynomial basis for C(Sd).

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The following theorem due to Krein-Milman-Rutman plays the mainrole:Let {Qk}∞k=0 be a basis for a Banach space H and let αk ∈ H∗,k = 0, 1, ..., be coefficient functionals for this basis. If Pk ∈ Hand ∥Pk −Qk∥ ≤ 2

−k−2

∥αk∥ =: λk for all k = 0, 1, ..., then the

sequence {Pk}∞k=0 is a basis for H.

If we can find a basis, say {Qn}, for C(Sd) and set λk = 2−k−2

∥αk∥ where

{αk}∞k=1 is the sequence of corresponding coefficient functional.So to construct a polynomial basis for C(Sd) it’s enough to constructa basis for it.

We start with finding an initial basis {Qk} for C(Sd)

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−k−2

sequence {Pk}∞k=0 is a basis for H.If we can find a basis, say {Qn}, for C(Sd) and set λk = 2

−k−2

∥αk∥ where

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−k−2

sequence {Pk}∞k=0 is a basis for H.If we can find a basis, say {Qn}, for C(Sd) and set λk = 2

−k−2

∥αk∥ where

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.Step 1...

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Counstruction a basis for the space C([0, 1]d) and

C0([0, 1]d) :=

{f ∈ C([0, 1]d) : f(x) = 0, ∀x ∈ ∂([0, 1]d)

}.

Let {fn}∞n=0 be the Faber-Cshaude basis for C[0, 1] defined by

f0(x) = 1, x ∈ [0, 1]

f1(x) = x, x ∈ [0, 1],

for n = 2k + i, k = 0, 1, ..., i = 1, 2, ..., 2k, fn is linear and continuous on[ i−12k

, 2i−12k+1 ] and on [2i−12k+1

, i2k],

fn(x) =

{0 if x /∈ ( i−1

2k, i2k)

1 if x = 2i−12k+1

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.Step 1...

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The tensor product of d systems fn is a basis for C([0, 1]d), say

B = {Fk}∞k=1. Elements of B are functions of the form

F (x) = fn1(x)fn2(x) · · · fnd(x), xj ∈ [0, 1], nj ∈ Z+, j = 1, · · · , d.

Denote by B′ the subset of B that contains of all the functions F suchthat nj ̸= 0, nj ̸= 1, j = 1, · · · , d. Then B′ is a basis foe C0([0, 1]d).

.Step 2...

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Construction of basis for the space C(Bd) and

C0(Bd) :=

{f ∈ C(Bd) : f(x) = 0, ∀x ∈ ∂Bd

}.

By a change of variable, we can replace [0, 1]d by [−1, 1]d, preserving thesame notation B, B′ for the corresponding basis.

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.Step 1...

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The tensor product of d systems fn is a basis for C([0, 1]d), say

B = {Fk}∞k=1. Elements of B are functions of the form

F (x) = fn1(x)fn2(x) · · · fnd(x), xj ∈ [0, 1], nj ∈ Z+, j = 1, · · · , d.

Denote by B′ the subset of B that contains of all the functions F suchthat nj ̸= 0, nj ̸= 1, j = 1, · · · , d. Then B′ is a basis foe C0([0, 1]d).

.Step 2...

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Construction of basis for the space C(Bd) and

C0(Bd) :=

{f ∈ C(Bd) : f(x) = 0, ∀x ∈ ∂Bd

}.

By a change of variable, we can replace [0, 1]d by [−1, 1]d, preserving thesame notation B, B′ for the corresponding basis.Askari-Hemmat ([email protected]) Polynomial Wavelet Type Expansions on the Spline Jan., 06, 2014 16 / 22

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.Step 2...

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Define a mapϕ : Bd −→ [−1, 1]d

ϕ(x) = (ρr(θ), θ)

where x = (ρ, θ), 0 ≤ ρ ≤ 1, θ ∈ Sd−1 and r(θ) is the length of thesegment {

y = (t, θ) : t ≥ 0, y ∈ [−1, 1]d}.

Then θ is bijective and the functions Gk := Fk(ϕ), k = 1, 2, ..., constitutea basis for C(Bd). Denote this basis by B′.A basis B′′ = {G

(0)k }

∞k=1 for C0(B

d) can be constructed similarly fromB′.

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.Step 3...

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Construction of a basis for C(Sd).

Let C(o)(Sd) and C(e)(Sd) be respectively the set of functions f ∈ C(Sd)

such that

f(x1, x2, ..., xd, xd+1) = f(x1, x2, ..., xd,−xd+1)

and the set of functions f ∈ C(Sd) such that

f(x1, x2, ..., xd, xd+1) = −f(x1, x2, ..., xd,−xd+1).

Each f ∈ C(Sd) can be represented in the form f = f (e) + f (o), wheref (e) ∈ C(e)(Sd), f (o) ∈ C(o)(Sd). It is obvious that if {H(e)k }

∞k=1 and

{H(o)k }∞k=1 are bases for C

(e)(Sd) and C(o)(Sd), then the system

{H(e)k ,H(o)k }

∞k=1 is a basis for C(S

d).

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.Step 3...

......

Construction of a basis for C(Sd).

Let C(o)(Sd) and C(e)(Sd) be respectively the set of functions f ∈ C(Sd)

such that

f(x1, x2, ..., xd, xd+1) = f(x1, x2, ..., xd,−xd+1)

and the set of functions f ∈ C(Sd) such that

f(x1, x2, ..., xd, xd+1) = −f(x1, x2, ..., xd,−xd+1).

Each f ∈ C(Sd) can be represented in the form f = f (e) + f (o), wheref (e) ∈ C(e)(Sd), f (o) ∈ C(o)(Sd). It is obvious that if {H(e)k }

∞k=1 and

{H(o)k }∞k=1 are bases for C

(e)(Sd) and C(o)(Sd), then the system

{H(e)k ,H(o)k }

∞k=1 is a basis for C(S

d).

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. . . . . .

.Step 3...

......

So it remaines to find bases for C(e)(Sd) and C(o)(Sd).

For x = (x1, x2, ..., xd+1) ∈ Sd, set

H(e)k (x) = Gk(x1, x2, ..., xd)

H(o)k (x) =

{G

(o)k (x1, x2, ..., xd) if xd+1 ≥ 0

−G(o)k (x1, x2, ..., xd) if xd+1 ≤ 0

{H(e)k }∞k=1 and {H

(o)k }

∞k=1 are the required bases.

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. . . . . .

REFERENCES

.

......

1. C. Foias and I. Singer, Some remarks on strongly independent sequences and bases inBanach spaces, Rev. Math. Pure et Appl. Acad. R.P.R., 6 (1961), no. 3, 589–594.2. Al. A. Privalov, On the growth of degrees of polynomial bases and approximation oftrigonometric projections, Mat. Zametki [Math Notes], 42 (1987) no. 2, 207–214.3. R. A. Lorentz and A. A. Saakyan (Sahakian), Orthogonal trigonometric Schauderbases of optimal degree for C(0, 2π), J. Fourier Anal. Appl., 1 (1994), no. 1, 103–112.4. M. A. Skopina, Orthogonal polynomial Schauder bases for C[−1, 1] of optimaldegree, Mat. Sb. [Russian Acad. Sci. Sb. Math.], 192 (2001), no. 3, 115–136.5. W. Freeden and M. Schreiner, Orthogonal and non-orthogonal multiresolutionanalysis, scale discrete and exact fully discrete wawelet transform on the sphere,Constructive Approximation, 14 (1998), 493–515.6. Yu. Farkov, B-spline wavelets on the sphere, in: Self-Similar Systems, Proceedings ofthe International Workshop (July 30–August 7, 1998, Dubna, Russia), JINR, E5-99-38,Dubna, 1999, pp. 79–82.7. M. Skopina, Polynomial expansions of continuous functions on the sphere and on thedisk, in: Preprint no. 2001:5, University of South Carolina, Department of Mathematics,2001.8. C. Müller, Spherical Harmonics, Lecture Notes in Math., vol. 17, Springer-Verlag,Berlin, 1966.

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. . . . . .

REFERENCES

.

......

9. E. M. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces,Princeton Univ. Press, Princeton, NJ, 1971.10. H. N. Mhaskar, F. J. Narcowich, and J. D. Ward, Spherical Marcinkiewicz–Zygmundinequalities and positive quadrature, Math. Comp., 70 (2001), no. 235, 1113–1130.11. H. N. Mhaskar, F. J. Narcowich, J. D. Ward, and J. Prestin, Polynomial frames onthe sphere, Adv. Comput. Math., 13 (2000), 387–403.12. R. Askary, Orthogonal Polynomials and Spherical Functions, SIAM, Phildelphia,1975.13. G. Szegö, Orthogonal Polynomials, Amer. Math. Soc., New York, 1959.14. B. S. Kashin and A. A. Saakyan, Orthogonal Series, [in Russian], Izd, AFTs,Moscow, 1999.

15. Wang Kunyang and Li Luoking, Harmonic Analysis and Approximation on the Unit

Sphere, Graduate Series in Mathematics, 2000.

polynomial wavelet type expansions on the...

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