criteria for completeness of orthonormal sets and summability of fourier series

CRITERIA FOR COMPLETENESS OF ORTHONORMAL SETSAND SUMMABILITY OF FOURIER SERIES

BY RALPH PALMER AGNEW

1. Introduction. It is the object of this paper to generalize and systematizemethods that have been found to be useful for determining whether a givenorthonormal set is complete. The methods depend upon use of kernels K(x, y, t)determined by the orthonormal set and a convergence-factor method G of sum-inability. Several necessary and sufficient conditions for completeness are given.These criteria apply to all orthonormal sets of functions n(x), bounded or un-bounded, defined over entire Euclidean spaces or any measurable subsets ofthem. When G is restricted to a wide class containing all familiar methods ofsummabilty (convergence, Cesro, Abel, Euler, Borel, LeRoy, etc.) as well asmany non-regular methods, the criteria are independent of the G used to de-termine the kernel (, y, t). If one method of summability is, in a particularsituation, better than another, it is not because the one kernel has pertinentproperties the other does not have; it is because it is easier to discover thepertinent properties of the one kernel than of the other. That this can be trueis a consequence of the fact that Fourier series of functions in 2, even thoughthey need not be everywhere convergent and may be defined over setsof infinitemeasure, are in many respects as well behaved and as manageable as series con-verging uniformly over sets of finite measure. Several of the fundamental ideasare exhibited by Wiener’s development [7; 51-661 of properties of the Hermiteorthonormal set and proof of completeness of the set. For general discussionsof orthonormal sets, see Ill, [4], and [8]. For extensive references, see I31.

Terminology, notation, and fundamental facts about completeness to be usedlater are given in 2. A general form of the Riesz-Fischer theorem is given in3. Convergence factors n(t), defining a method of summability G not neces-sarily regular, are introduced and used to define (, y, ) heuristically in 4.The kernels (x, y, ) are defined and associated wth the problem of complete-hess in 5. The rSles of step functions and continuous functions are presentedbriefly in 6 and 7. Properties of K(x, y, t), x fixed, are given in 8. Criteriafor completeness involving convergence in mean and summability, respectively,are given n 9 and 10. Use of the simplest of the criteria for completeness isillustrated in 11 by a simple proof of the well-known fact that the Hermiteorthonormal set is complete.

2. Complete sets. Let E be Euclidean space of one or more dimensions or ameasurable subset of one of these spaces. The single symbols x and y will be

Received June 7, 1944; presented to the American Mathematical Society August 12, 1944.

801

802 RALPH PALMER AGNEW

used to denote points of E; in case E is n-dimensional, x may be identified with(xl x2, x,), where xl x2, x= are the coSrdinates of x. All measuresand integrals will be those of Lebesgue. Let L2 denote the space whose elements(or points) are complex-valued functions f(x)., defined almost everywhere andmeasurable over E, such that

(2.1) f If(x) dx

exists. The distance II f- g ll between two elements f and g of 52 is defined by

(2.2) II : g ll If(x) g(x) dx

Two elements of L2 are equal, and we write f g or f(x) g(x) for almost allx in E if and only if f(x) g(x) for all x in E except for a null set (set of measure0). An element (or point) f of L2 is a limit point of a set S in L2 if to each > 0corresponds a point g, of S such that g, f and I] f g, II < . The closureS* of a set S is the union of S and the set of limit points of S.The linear manifold M(S) determined by a set S is the set of linear combina-

tions (with complex constant coefficients) of elements of S. A set S in L2 iscomplete if the closme of its linear manifold is the whole space L2 that is, ifM*(S) L2. If S is the closure of the linear manifold determined by a set inL2, say S M*(S), then M(S) $1 and M*(SI) SI If A C B C L2that is, if A is a subset of B (which may be B itself) and B is a subset of L2, thenobviously M(A) C M(B) and M*(A) C M*(B). If M*(S) S and M*(S’)L2, then

(2.3) M*(S) M*(M*(S)) M*(S’) L2

so M*(S) L.. This proves the following simple lemma which has manyapplications.

LEMMA 2.1. A set S in L2 is complete if the closure M*(S) of its linear manifoldcontains a complete set

Let S denote a set of complex-valued functions

(2.4) 0(x), (x), 2(x),

orthonormal over E; thus the functions (x) are measurable and

(2.5) f ,=(x)*= (x) dx ,=,

where i=, 0 or 1 according as m n or m n. By definition, the orthonormalset S+ is complete if and only if M*(S+) L,. When f L=, we systematically

ORTHONORMAL SETS 8O3

use the symbols fo, fl to denote the Fourier coefficients

(2.6) f. f f(x)*(x) dx (n O, 1, 2,

of f(x). If f L2 and cl c2, cN are constants, the familiar equality

(2.7)

N

[f(x) , cn(x) dxnO

N hr

If(x) c,:h,(x)][f*(X) _, c**(x)] dxn-’O mffiO

shows that

h N

n-O nO_f,,,,,(x) dx ]f() 1 d If. .

n"O n-O

This implies the Bessel inequality

f(2.9) _, f, -- If(x) dx,

and that f M*(S,) if the equality sign holds in (2.9).holds in (2.9), then (2.7) implies that

If the inequality sign

where/t is the positive difference of the members of (2.9), and hence that f is notan element of M*(S,). This proves the following lemma.

LEMMA 2.2. Let f . L2 Then the closure M*(S,) of the linear manifold deter-mined by the orthonormal set S, contains f if and only if the Parseval equality

(2.11) If(x) 12 dx f. [nO

holds for the f.


This implies the well-known fact that M*(S,) L2, that is, S, is complete,if and only if the Parseval equality holds for each f L.. See, for example,[4; 10-12], where several necessary and sufficient conditions for completeness aregiven. Lemma 2.2 and (2.8) imply the following lemma.

LEMMA 2.3. Let f L2 Then f M*(S,) if and only ifN

(2.12) lim If(x)

_f=.(x) [2 dx O,

that is, if and only if the Fourier series off(x) converges in mean to f(x).

This implies that S, is complete if and only if the Fourier series of each f inL2 converges in mean to it. Lemmas 2.1 and 2.2 imply the following moregeneral lemma.

LEMMA 2.4. In order that the orthonormal set S be complete, it is necessary thatthe Parseval equality

f(2.3) If(x) ! dx fn0

E

hold for each. f L2 and sujcient that it hold for each f in a complete set S.

If (2.12) holds and g t L, then the Schwarz inequality shows that

if N

[f(x) : f=,(x)]g*(x) dxn,O

_< If(x) ’ f,,,(x) dx g(x) 12 dx

and hence that, as N --. , the left side converges to 0 and therefore (2.14}holds. This and Lemma 2.4 give the following.

LEMMA 2.5. In order that the orthonormal set S be complete, it is necessary thatthe two-function Parseval equality

(2.14) f(x)g*(x) dx0

hold for each pair f, g offunctions in L and sugcient that it hold for each identical:pair f, f of elements of a complete set S.

We shall use also the following more special lemma.

ORTHONORMAL SETS 805

LEMMA 2.6. If S, is an orthonormal set, complete or incomplete, if f L ifcn is a sequence of constants, if N is an increasing sequence of integers, and if

N(2.15) lim If(x) c.On(x) dx O,

1-’ nO

then the constants cn are the Fourier coeffwients f of f, the Parseval equality (2.11)holds, and (2.12) holds.

To prove this, use (2.7) and the Bessel inequality to obtain, for each1,2,

f(2.16) If(x) c..(x) dx >_ ’ !c.- f,

SinceN -- o as p --. o, this and (2.15) imply that c. f for each n. Thereforethe members of (2.8) converge to 0 as N - o over the sequence N1 N., ....Since If. >- 0, the Parseval equality follows. Finally, (2.12) then follows from(2.8) and Lemma 2.6 is proved.

3. The Riesz-Fischer theorem. In this section, we introduce terminology tobe used later and prove a form of the Riesz-Fischer theorem which is somewhatmore general than that usually stated.

Let T be a set, in a metric or Hausdorff space, having a limit point to notbelonging to T. A function G(x, t) will be said to converge almost everywhereover E to H(x) whenever ---, to sufficiently rapidly if there is a sequence T1, T2,of neighborhoods of to in T such that lim G(x, t) H(X) for almost all x in Ewhenever tn is a sequence such that t T for each n. A sequence convergesessentially uniformly over E if to each > 0 corresponds a set E, of measureE, < such that the sequence converges uniformly over E E,

LEMMA 3.1. If F(x, t) is a complex-valued function defined for almost all x inE when T, if F(x, t) L2 over Efor each T, and if

(3.1) lim f F(x,s)- F(x, t)12dx O,, t-*toE

then there is a function F(x) such that

(3.2) lira F(x, t) F(x)

essentially uniformly (and hence almost everywhere) over E whenever tn ---) to su-ciently rapidly. Moreover, F L2 over E and

(3.3) !i_.m f IF(x)- F(x, t)I dx O.


The basic method of the following proof is that used by Weyl [6] to prove theRiesz-Fischer theorem. Let en be a convergent series of positive terms. Foreach n 1, 2, let T. be a neighborhood of to in T such that

(3.4) f IF(x, s) F(x, t) dx < (s, t T,).

Let the neighborhoods be so chosen that T1 T .-.. Let tn be a sequence,converging to to, such that t Tn for each n. Then, for each n, the set F,, of xfor which the inequality

(3.5) IF(x, t+,) F(x, t) <

fails to hold has measure at most enoutside the set G defined by

Let q be a positive integer.

(3.6)

When x lies

the inequalities (3.5) hold when n >_ q and accordingly the series

(3.7) , F(x, t./l) F(x, t.)

converges unifmznly. Thus the sequence F(x, t) converges uniformly overE G,. Since the measure of G, is dominated by , /1 -t- and thisconverges to 0 as q becomes infinite, it follows that the sequence F(x, t) con-verges essentially uniformly over E. This implies convergence almost every-where over E to a function which we may denote by F(x). Then, for each q,F(x, t.) converges uniformly to F(x) over E GFor each j 1, 2, let E; denote the set of all points x having all of their

coSrdinates dominated by j. From the fact, obtained from (3.4), that, for eachn= 1,2,

f (m >n),(3.8) F(x t,.) F(x, t.) dx < .we conclude that

(3.9) f IF(x, t.,) F(x, t.) [ dx < (m > n).Ei-Oq

With n, j, and q fixed we can use the uniform convergence of F(x, t.,) to F(x)over the bounded set E; G, to obtain

(3.10) fBi-Gq

IF(x) F(x, t)


We may fix n and j and then let q -* o to remove the set Go from (3.10). Then,for each n, we can let j - to obtain

(3.11) f IF(x) F(x, t,) dx <_

Since F(x, t.) is in L over E, it follows from (3.11) that F(x) is in L over E.Moreover, since -- 0,

(3.12) lim fiR(x) 0.

If u is a sequence converging to to, then use of the Minkowski inequality

[f F(x) F(x, u.)

(3.13)

(3.12), and (3.1) shows that F(x,This gives (3.3). The inequality obtained by replacing F(x, u.) by G(x) in(3.13) shows that if F(x, t) converges in.mean to F(x) and to G(x), then F(x)G(x) almost everywhere. Thus the function F(x) is, apart from its values atpoints in null sets, independent of the sequence t. used to deteine it, andLemma 3.1 is proved.The following lemma is a consequence of Lemma 3.1 and the fact that if t

is a sequence converging to to, then one can select a subsequence t convergingto to as rapidly as one wishes.

LEMMA 3.2. If F(x, t) L over E for each T, if F(x, t) converges in mean toF(x) as to, a if there is a sequence t. of points in T convergito to such thatF(x, t) converges almost everywhere over E to G(x) as n , then F(x) G(x)almost everywhere.

If T is the set of positive integers, to , and we write F.(x) for F(x, n),the condition (3.1) takes the familiar fom

(3.14) lim f F(x) F(x) O.E

If F,(x) is the sequence of partial sums of an orthogonal series a,(x) forwhich ]a ]2 < , then (3.14) holds since, when m > n,

f(3.15) F.(x) F.(x) dx a(x) dx a =;kn+l km+l

8O8 RALPH PALMER AGNEW

therefore a..(x) converges in mean to a function in L2 over E. If f L. thenf. 12 < and the Fourier series fn. of f therefore converges in mean to

a function in L2 which we will denote by f*(x) when we need a notation for it.It is (Lemma 2.3) only when f . M*(S,) that f*(x) f(x) almost everywhere.

4. Convergence factors G(t). Let T be a set, in any metric or Hausdorffspace, having a limit point to not belonging to T, and let

(4.1) Go(t), Gl(t), G2(t),

be a sequence of complex-valued functions defined over T and satisfying theconditions

(4.2) , G.(t) < (t T),

(4.3) G.(t) < K (ttT;n 0, 1, 2, ..-),

(4.4) lim G.(t) 1 (n O, 1, 2, ...).t-t

When these functions are used as convergence factors, they determine a methodof summability G by means of which a series u. is summable to L if

(4.5) lim ’ G.,(t)u, L.t-*to nO

The reason for (4.2) will appear later. The service of (4.3) and (4.4) is to givethe conclusion of the following lemma.

LEMMX 4.1. The two conditions (4.3) and (4.4) imply that

(4.6) lim ’ G(t)un _, u.t-*t n.O nO

whenever the series on the right converges absolutely.

To prove this lemma, let un converge absolutely. Suppose > 0. Choose

4.7) (K - 1) ’ un < .Then

(4.8) I: G.(t)un _, u. -- I [G,,(t) 1]u. d- [I G.(t) -t-- 1] u,, I"n,O n-O O

The first term on the right converges to O as -- to, and the last is less than e.

The conclusion follows. While we shall not use the fact, the two conditions (4.3)and (4.4) are essentially necessary to ensure (4.6) whenever u. converges;one can only relax them by replacing T by a neighborhood Vi of to in T. The

an index N so great that


hree conditions (4.2), (4.3), and (4.4) are not sufficient to ensure that the methodof summability G be regular, that is, be such that (4.6) holds whenever Y’ u,converges. For regularity, one needs the additional condition

(4.9) G.+,(t) G.(t) < K

over some neighborhood T1 of to in T; proof and references are given by Moore[2; 13, 35]. The weaker condition

(4.10) _, G.+(t) G.(t)

is implied by (4.2).The kernel K(x, y, t), which will be introduced in the next section, is brought

into the analysis of orthonormnl sets most quickly in the following way. Sincethe terms of the Fourier series of f(x) are

(4.11) f.ch.(x) f f(y.(x)h* (y) dy

the formal computation

G(x, t)= ., G.(t)f.4.(x) _, G.(t).(x*.(y)f(y) dy’0 0

(4.12)

[_, G.(t)4.(x)*.(y)]f(y) dy

gives, for the G transform of the Fourier series of f(x),

(4.13) G(x, t) f g(x, y, Of(Y)dy,

where

(4.14) K(x, y, t) _, G.(t)4.(x)ch* (y).

In case these computations are valid (see Theorem 8.2) the Fourier series_, f,ch,,(x) of f(x) is summable G to .f(x) if and only if

4.15) lim f K(x, y, Of(Y)dy f(x).tto

5. The kernels K(x, y, t). When the functions bo(X), l(x), form anor.honormal set over E, the functions

(5.1) (x, y) 4),,,(x)4)*,(y) (m, n O, 1, 2, ...)


form an orthonormal set over the product set P of points (x, y) for which x t Eand y E; for, by the Fubini theorem, which we shall use repeatedly,

(5.2)

where the right member is 1 when rn p, n q, and is 0 otherwise. We shalluse only the orthonormal subset

(5.3) 4n(x)4>*(y) (n O, 1, 2,...).

Since, by (4.2), the coefficients G(t) in the series

(5..4) G.(t)ch.(x)* (y)

satisfy the condition G,(t) < , it follows from the Riesz-Fischer theoremthat the series converges in mean over P to a function K(x, y, t) belonging toL over P:

N

(5.5) lim K(x, y, t) ., G.(t)cb(x)cb*.(y) dx dy 0P

or

(5.6) K(x, y, t) ,’0 G.(t)4,,(x)ch*(y),0

where the superscript (m) on the indicates that convergence in mean ratherthan convergence is the method by which the "value" of the series is determinedto be K(x, y, t). For each T, the series in (5.6) is the Fourier series of K(x, y, t)for the orthonormal set .(x),* (y). While the hypothesis (4.2) may fail to ensureconvergence of the series in (5.6) to K(x, y, t), some subsequences of the sequenceof partial sums must converge almost everywhere to K(x, y, t). If G(t) <, the series must converge almost everywhere to K(x, y, t); see the remarkfollowing Theorem 8.2.An application of Lemma 2.6 to (5.6) gives the Parseval equality

(5.7) K(x, y, t)Idx dy G.(t) L(t),

where L(t) is defined by the last equality.formation

This implies that the kernel transo

(5.8) g(x, ,) f K(x, y, t)f(y) dy


associates with each $ L2 a transform g(x, t) L2, and the Schwarz inequalityshows that

While L(t) < for each t, it follows from (4.4) that L(t) -- o as --. to. Thekernel K(x, y, t) is Hermitian (such that K*(x, y, t) K(y, x, t)) if and only ifthe functions G,(t) are all real.

Let f(x) and g(x) be functions in L2 over E. Then the function f*(x)g(y)belongs to L2 over P; in fact

(5.10) S s*(=)()I S s(,)Ix S s()I .Hence, as is well known and easily proved by use of (5.5) and the Schwar. in-equality, we can multiply (5.6) by f*(x)g(y) and integrate termwise over P.Replacing the integrals over P by iterated integrals over E gives

(5.11)

and hence

f f*(x) dx f K(x, y, t)g(y)dy

f fG.(t) f*(x)4.(x) dx g(Y)Ch*(Y) dy

f f(5.12) f*(x) dx K(x, y, t)g(y) dy .,where J’, and g, are the Fourier coecients of f(x) and g(x). Since the inequality

(f)(/ )( fg. l) ( If. I)( a. ) If(x) dx g(x)i dx0

shows hegf converges absolugely, i follows from (5.12) and Lemma .1that

(5.13) lim f f K(x, y, t)g(y) dyt--t

This implies that

(5.14) "f*(x)g(x) dx


Thus the limit in the left members of (5.13) and (5.14) exists when f, g Lwhether the orthonormal set be complete or not. From (5.14) and Lemma 2.5,we obtain the following theorem.

THEOREM 5.1. In order that the orthonormal set S be complete, it is necessarythat

(5.15, i_,,m. f --0

for each pair f, g of functions in L2 and suIffcient that

for each f in a complete set S in L=While examples show that the hypothesis

(5.17) lim f f*(x)i(x, t)dx 0 (f , L)t-*to

does not imply the conclusion

(5.18) lim f ti(x, t)I’ O,t-*to

we shll see (Theorem 9.2) that, when L(z, t) is the prticulr function in squarebrackets in (5.16), the hypothesis (5.17) does imply (5.18).

6. Step functions. When n

_1 and a - (a a, a,) and b ------(b, b2, b.) are points such that a < b, k 1, 2, n, we write a ( b.

When a ( b, the set of points x such that a ( x ( b is the interval (a, b). Theinterval is finite when the c06rdinates of a and b are all finite. The characteristicfunction of a finite interval (the function s(x) which is 1 or 0 according as x liesinside or outside the interval) is a simple step function. A linear combination ofsimple step functions is a step function; it is constant over each of a finite setof finite intervals and zero elsewhere. The well-known fact. that, when f Lto each > 0 corresponds a step function f,(x) such that

f lf( ) dx <.

implies that the set S. of step functions is dense in L2 that is, the closureof S is L2 Therefore, since the linear manifold M(S) determined by the setS of simple step functions is S, M*(S) L. Thus the set S of simple step

functions is complete, and Theorem 5.1 implies the following theorem in which fb


denotes the integral over the intersection of the interval a

_x

_b and the set

E over which the functions n(x) are orthonormal.

THEOREM 6.1. The orthonormal set S, is complete if and only if

(6.2) lim 1 K(x, y, t) dy dx 0

for each finite pair a, b of points for which a < b.

7. Continuous functions. Let C denote the set of continuous functions f(x)defined over E, each one of which vanishes outside some finite interval a

_x

_b.

The well-known fact that C is complete is a consequence of the obvious fact thatif g(x) is a step function and > 0, then there is a function f, C such that11 g f, II < . Since C is complete, the following theorem is a corollary ofTheorem 5.1.

THEOREM 7.1. In order that the orthonormal set S be complete, it is necessaryand sucient that

](7.1) lim f*(x) dx f(x) K(x, y, Of(Y) dy 0#-

for each finite pair of points a, b/or which a < b and each continuous/unctionvanishing outside a

_x

_b.

8. Series involving K(x, y, t). We have defined K(x, y, t) as the limit in meanover the product set P of the partial sums of the series

(8.1) Gn(t).(x)* (y).

Examples show that the hypothesis that, as N ---. , K;(x, y, t) converges inmean over the product set P to K(x, y, t) does not imply existence of a singlexo such that K;(Xo y, t) converges in mean to K(xo y, t) over y E; but weshall show that the conclusion holds for almost all Xo when K;(x, y, t), n O,1, is the sequence of partial sums of the series in (8.1). To prove this fact(Theorem 8.1), let be a fixed point in T and use the equation (5.5) definingK(x, y, t) and the Fubini theorem to obtain

(8.2) lim dx K(x, y, t) Gn(t),(x)*(y) I’ dy O.N’-m

Let o -, 1 - be a convergent series whose terms are positive and form adecreasing sequence, and choose an increasing sequence No, N1 of indicessuch that

(8.3)d ddx K(x, y, t)- G(t)(x)*.(y) .dy <


for each p 0 1, .... Then the set E of points x for which the inequality

(8.4) K(x, y, t) G(t)4,.(x)b*(y) dy

fails to hold has measure less than . Hence, when x lies outside the setE,+ E,+ of measure less than + ,+ T (8.4) holds whenp > q and therefore

(8.5) lim K(x, y, t) G(t(x(y) dy O.

The set A of x for which (8.5) fails to hold is a subset of each of the sets FoF and is therefore a set of measure 0. Since K(x, y, t) belongs to L overP, the set A of x for which K(x, y, t) fails (as a function of y) to be in L over Eis also a null set. Let x be a fixed point outside the null union of h and hand set

(8.6) F(y) K(x, y, t), c, G,(t),(x).

Then F(y) is in L over E and (8.5) becomes

(8.7) lira F() e4() d 0.

Use of Lemma .6, applied go ghe orghonoal seg (), gives he followinggheorem.

To8.1. For lmo ll x, he erie (8.1)4 , o K(z, , ).

(8.8) lira K(z, , 0 ()*()()I’ d 0,

or, in oher noio, ]or lmo ll in N

(.) g(x., , ) ’The Parseval equaligy corresponding go (8.4), which hNds for almos all x, is

f(8.10) K(x, y, t)I dy O,(O,,(x)

This implies the fact, which is trivial when the functions ,(x) are uniformlybounded, that, if a ] < , then ]a,(x)] converges almost every-where; for we may define the functions G(t) so that, when has one particularvalue, G(t) a,, n 0, 1, 2, .... Since the terms in the series in (8.10) arenon-negative, we can integrate (8.10) termwise over E to obtain (5.7).

ORTHONORM.kL SETS 815

When x is a point xo for which (8.9) holds, and f L2, we can multiply (8.9)by f(y) and integrate over E, termwise on the right side, and obtain (8.11).

TttEOREM 8.2. Iff $. L then for almost all x in E

(S.ll) K(x, y, Of(Y) dy _, G.(t)f.4.(x)

and

(8.12) K(x, y, Of(Y)dy _,’) G.(t)fcb,,(x).0

The second conclusion (8.12)follows from (8.11); for iff L2 then] fn 12 < o

and, by (4.3), ] Gn(t)fn 12 < o so the series in (8.12) converges in mean to afunction which must be the function to which it converges.Theorem 8.2 shows that, when f L2, the Fourier series of f(x) is summable

G to f(x) if and only if (4.15) holds.It is a corollary of Theorem 8.2, which is trivial when the functions ,(x)

are uniformly bounded, that, if ] an is an absolutely convergent series, thenanOn(x) converges almost everywhere; for if a < o, we can, for one

fixed t, set Gn(t) an ] andf an [1/2/sgn a or 0 according as as 0 or an ’to obtain ] a.,(x) in the right member of (8.11).If we replace f by g in (8.11), multiply the resulting equation by f*(x) and

integrate over E, we obtain again the equation (5.12) which was obtaineddirectly from (5.6) in 5.The preceding work enables us to give a simple proof of the following theorem

which is often as useless as it is elegant when one is seeking to determine whethera given orthonormal set is complete.

THEOREM 8.3. The orthonormal set o(X), (x), is complete if and only ifthe only functions f(x) in 52 for which

(8.13) f K(x, y, t)f(y) dy 0 (t T)

are 0 almost everywhere in E.

Suppose the set bn(x) is not complete. Then there is a function f(x) in Lwhich is not null and which is orthogonal to all of the ’s. Hence fn 0 for eachn and substitution in (8.11) gives (8.13). If, therefore, the only functions inL. satisfying (8.13) are null, the orthonormal set must be complete. Supposethere is a non-null function f(x) in L for which (8.13) holds. Then, for this j’,the left member of (5.16) is positive and Theorem 5.1 implies that the ortho-normal set is incomplete.


9. Convergence in mean and completeness. In this section, we prove thefollowing theorem and use it to obtain further criteria for completeness of ortho-normal sets.

THEOREM 9.1.f*(x) so that

If anCn(x) is an orthogonal series converging in mean over E

N

(9.1) lim f*(x)

_an,,(x) 12 dx O,

N.-.*

then the hypotheses (4.2), (4.3), and (4.4) imply that the G transform

(9.2) G(x, t) Gn(t)a,n(x)

of a,On(x) exists for each T and converges in mean to f*(x) as ---, to, that is,

(9.3) lim f-’-tIf* (x) G(x, t) dx O.

The hypothesis that a(x) converges in mean implies thatHence (4.3) implies that G(t)a < o. Therefore the series in the rightmember of (9.2) converges in mean to a function G(x, t) which is almost everywfiere the function (Theorem 8.2) to which the series converges almost every-where. For each t, the relation

(9.4) f*(x) G(x, t) (’)[1 G,,(t)]a,,(x)

gives the Parseval formula

(9.5) f*(x) G(x, t)I dx "1 1 G,(t) I a0

Let > 0. Then, when N is chosen so great that the last inequality holds,

(9.6)

where K is the constant in (4.3). Hence, when . T,N-1

12(9.7) f*(x) G(x, t)12 dx <_ , 1 On(t)lla, -t- .This inequality and (4.4) imply that the superior limit, as -- to, of the leftmember of (9.7) is less than or equal to . The conclusion (9.3) of Theorem 9.1follows.


THEOREM 9.2. in order that the orthonormal set S, be complete, it is necessarythat

(9.8) lim f (x) f K(x, , t)() ot-*to

hold for each f . L. and sufficient that it hold for each f in a complete set S in L2.

To prove necessity, let the set S, be complete. Suppose f L2. The Fourierseries ’ fCin(x) of f(x) converges in mean over E to f(x). The hypotheses ofTheorem 9.1 are satisfied when a. fn and f*(x) f(x). Moreover, by Theorem8.2 the transform G(x, t) is the inner integral in (9.8). Therefore the conclusion(9.3) of Theorem 9.1 gives the required conclasion (9.8). To prove sufficiency,suppose (9.8) holds for each f in a complete set S. Then, by the Schwarz in-equality, (5.16) holds for each f in S and completeness of S, follows for Theorem5.1.When E is a set of infinite measure (perhaps the interval < x < or

the interval 0 < x < and Theorem 9.2 is being used to establish completenessof an orthonormal set S,, it is convenient to use a complete set S of functionsf(x) (perhaps continuous functions or step functions) each one of which vanishesoutside some finite interval. The set E over which the right integral is takenmay then obviously be replaced by E Eo, where Eo is the set of values of x(or y) for which f(x) 0 (or f(y) 0). The following theorem gives the moresignificant fact that the set E over which the first integral is taken may also bereplaced by E Eo.TEOREM 9.3. In order that the orthonormal set S be complete, it is necessary that

(9.9) iim f y(z) f K(, , ,)y()a a 0t-*to

K-Ko E-Eo

hold for each f L2 and sucient that it hold for each f in a complete set S, the setEo being the set, depending on f(x), of values of x for which f(x) O.

Necessity follows from Theorem 9.2. To prove sufficiency, we observe that,if f S and (9.9) holds, then

(9.10, i,m. f ,.,)[,,)- f (, , ,),()] a=0.E-Eo E-Eo

Adding the sets Eo to the ranges of integration does not change the values ofthese integrals; hence (5.16) holds and Theorem 5.1 implies that the set S iscomplete.The following two theorems are corollaries of Theorem 9.3.


THEOREM 9.4. The orthonormal set S, is complete if and only if

(9.11) lim 1 K(x, y, t) dy dx 0

for each finite pair of points a, b for which a < b.

TIIEOREM 9.5. The orthonormal set S, is complete if and only iff(9.12) lim If(x) j. K(x, y, Of(Y)dy 0

t-*to

/or each finite pair o/points a, b/or which a < b and each/unction f(x) continuousover E and vanishing outside a <_ x <_ b.

10. Summability of Fourier series and completeness. Let G be a method ofsummability determined by convergence factors G.(t) defined in 4. Let

u,,(x) be a series of functions defined over E. If, for each T, the series inthe equation

(10.1) G(x, t) G.(t)u.(x)

converges for almost all x in E, the G transform of u.(x) is said to exist andbe G(x, t). Of the four following definitions the first (which is given here forcomparison with the others) is standard and the others are, in part, naturalextensions of existing terminology.

The series u.(x) is summable G to U(x) for almost allDEFINITION 10.1.xin E i$

(10.2) lim G(x, t) U,(x)

for almost all x in E.DEFINITION 10.2. The series u.(x) is summable G in mean to U,.(x) over E if

(10.3) lim f V2(x) G(x, t) dx O.

DEFINITION 10.3. The series u.(x) is esszntially summable T over E toUa(x) if there is a sequence T. of neighborhoods g to in T such that

(10.4) lim G(x, t.) Ua(x)

for almost all x in E whenever t. is a sequence such that t. to a t. T. for each n.DEFINITION 10.4. The series u.(x) is weakly summable G over E to U(x)

g there is at least one sequence t. such that t. to a

(10.5) lim G(x, t.) U(x).


The following two theorems show that, in so far as application to Fourierseries of functions in L2 is concerned, the three methods of summability inDefinibions 10.2, 10.3, and 10.4 are always equivalent and always effective. Itis still unknown [9; 819] whether the Fourier series of a function f in L2 need beconvergent almost everywhere. If this is so (many persons conjecture that it isnot), then, when applied to Fourier series of functions in L2 all four methodsof summability are always equivalent and always effective provided G is regular.The following two theorems will be used to obtain simple criteria for complete-ness.

THEOREM 10.1. If f . L2 then the Fourier series f..(x) is summable G inmean to f*(x), essentially summable G to f*(x), and weakly summable G to f*(x),where f*(x) is the function in L to which f.(x) converges in mean.

That, under the hypotheses of Theorem 10.1, the series f.(x) is summableG in mean over E to f*(x) is asserted by Theorem 9.1. This result and Lemma3.1 imply that f..(x) is essentially summable and weakly summable to f*(x).

THEOREM i0.2. Iff L2 over E, if A is a measurable subset Of E (which may beE itself), and if the Fourier series . $(x) of f(x) is weakly summable G over Ato F(x), then f(x) is both summable G in mean over A to F(x) and essen-tially summable G over A to F(x).

This is a consequence of Theorem 10.1, since the function F(x) to whichf(x) is weakly summable must be almost everywhere the function f*(x) to

which the series converges in mean.

THEORE 10.3. In order that the orthonormal set S, be complete, it is necessarythat, for each f L

(10.6) lim f g(x, y, t)f(y) dy f(x)

for almost all x,. provided t. is a sequence converging to to sufftciently rapidly.

Let S, be complete. Then, for each f L, the function f* to which f.converges in mean is $. Hence, by Theorem 10.1, f.. is essentially summableG to f. But, by Theorem 8.2, the integral in (10.6) is G(x, t) and (10.6) follows.

THEOREM 10.4. In order that the orthonormal set S, be complete it is sufftcientthat, for each f in a complete set S, there exist at least one sequence t. converging toto such that

(10.7) lim f K(x, y, t.)f(y) dy f(x)

for almost all x in the set E Eo of x for which f(x) O.


Using Theorem 8.2, we see that the hypothesis involving (10.7) implies thatfnCn is weakly summable G over E Eo to f(x). Hence Theorem 10.2 implies

that ’ fnn is summable G in mean over E Eo to f(x). Thus (9.9) holds foreach f in S, and completeness of S, follows from Theorem 9.3.The following two theorems are corollaries of Theorem 10.4.

THEOREM 10.5. In order that the orthonormal set S, be complete, it is sufficientthat, for each finite interval a < x < b and each function f(x) continuous over E andvanishing outside a x b, there exist at least one sequence tn converging to to suchthat (10.7) holds for almost all x in the interval a x < b.

THEOREM 10.6. In order that the orthonormal set S, be complete, it is sucientthat, for each finite interval a x b, there exist at least one sequence t,, convergingto to such that

(10.8) lim K(x, y, tn) dy 1

for almost all x in the interval a < x < b.

11. Completeness of the Hermite orthonormal set. In this section, we illus-trate use of Theorem 10.6 by proving the well-known fact that the Hermitefunctions

d(--1) eiX’--e- (n=O, 1, ...)(11.1) n(x) (2n!t) dx

which are orthonormal over o < x < , form a complete set. For referencesto the literature of Hermite functions, see [3], especially the topic index on page196.

Differentiating the standard formula

e_, 1_ f e_,,e2,,, du(11.2)

n times with respect to x and substituting in (11.1) give

(11.3) ,(x)(2 n!rt)

eta" (-2iu)’e-’e’’ du.

Let G be Abel summability for which T is the interval 0 < < 1, to 1, andG.(t) t’. Using (11.3), one can show in a straightforward way that

(11.4) K(x, y, t)= r-t(1 t2) -1/2 exp [x2 y2 (x Yt-)212 1--

For the details, see Titchmarsh [5; 78]; but note that, in Titchmarsh’s book, the’s denote unnormalized functions and the h’s are our b’s. Writing (11.4) in


the form

(11.5)K(x, y, t)

r-(1 t2)- exp F l+t2( 2xt1 -t

y1 + t/ 2(1 +

and integrating over a _< y < b give, after change of the variable of integration,

(11.6) K(x, y, t)dy v(1 + 2) exp -2(1 + ) x2 e-’" dr,

where a and/ are functions of x, y, and defined by

(11.7) a a1 + 2ii ") l+t 2(1 2)

If a, b, and x are fixed such that a < x < b, then a --* o and B - oo as -- 1.It follows, from (11.6) and the familiar integral formula to which (11.2) reduceswhen x 0, that (10.8) holds. Therefore, by Theorem 10.6, the orthonormalset is complete.

Since the Hermite set is complete, it is of course possible to establish, for thisparticular set, further properties enabling one to establish completeness byverifying one of the more complex criteria for completeness such as those inTheorems 5.1, 6.1, 7.1, 8.3, 9.2, 9.3, 9.4, or 9.5. This is commonly done.

BIBLIOGRAPHY

1. S. KACZMARZ AND H. STEINHAUS, Theorie der Orthogonalreihen, Monografje MatematyczneVI, Warsaw-Lwow, 1935.

2. C. N. MOORE, Summable Series and Convergence Factors, American Mathematical SocietyColloquium Publication, vol. 22, New York, 1938.

3. J. A. SHOHAT, E. HILLE, AND J. L. WALSH, A Bibliography on Orthogonal Polynomials,Bulletin of the National Research Council, no. 103, Washington, 1940.

4. M. H. STONE, Linear Transformations in Hilbert Space and their Applications to Analysis,American Mathematical Society Colloquium Publication, vol. 15, New York, 1932.

5. E. C. TITCHMARSH, Introduction to the Theory of Fourier Integrals, Oxford, 1937.6. H. WEYL, Ueber die Konvergenz yon Reihen, die nach Orthogonal-funktionen fortschreiten,

Mathematische Annalen, vol. 67(1909), pp. 225-245.7. N. WIENER, The Fourier Integral, Cambridge, 1933.8. A. ZYGtUND, Trigonometrical Series, Monografje Matematyczne V, Warsaw-Lwow, 1935.9. A. ZYGMUND, Complex methods in the theory of Fourier series, Bulletin of the American

Mathematical Society, vol. 49(1943), pp. 805-822.

CORNELL UNIVERSITY,

criteria for completeness of orthonormal sets and summability of fourier series

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