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Категория: Математика

Автор: Zikkos Elias

URAL MATHEMATICAL JOURNAL, Vol. 6, No. 1, 2020, pp. 168-175

DOI: 10.15826/umj.2020.1.014

MOMENT PROBLEMS IN WEIGHTED L2 SPACES

ON THE REAL LINE

Elias Zikkos

Khalifa University, PO. Box 127788 Abu Dhabi, United Arab Emirates

elias.zikkos@ku.ac.ae

Abstract: For a class of sets with multiple terms

¡1 — times ¡2 — times ¡1^—times

having density d counting multiplicities, and a doubly-indexed sequence of non-zero complex numbers {dn k : n € N, k = 0, 1,..., — 1} satisfying certain growth conditions, we consider a moment problem of the form

e—2w(t) tk e\\n-t f (t) dt = dn>k, V n € N and k = 0, 1, — 1,

in weighted L2(—œ, œ) spaces. We obtain a solution f which extends analytically as an entire function, admitting a Taylor-Dirichlet series representation

OO ¡ln — 1

f = J2 Cn,kzk)eXnZ, Cnk e C, V z e C.

n = 1 k=0

The proof depends on our previous work where we characterized the closed span of the exponential system {tk: n € N, k = 0, 1, 2,..., — 1} in weighted L2 (—rc, rc) spaces, and also derived a sharp upper bound for the norm of elements of a biorthogonal sequence to the exponential system. The proof also utilizes notions from Non-Harmonic Fourier series such as Bessel and Riesz—Fischer sequences.

(An, ßn}n=1 := (Ai, Ai,..., Ai, A2, A2,..., A2,..., Ak, Ak , ■ ■ ■, Ak , ■ ■ ■ }

1. Introduction

P. Malliavin [5] considered the following in the sense of the classical Bernstein weighted polynomial approximation problem on the real line. Let W(t) be a real-valued continuous function defined on the half-line [0, such that it is log-convex, that is log |W(es)| is a convex function on the real line. Let CW be the weighted Banach space whose elements are the complex-valued continuous functions f defined on [0, to), such that

t^rc W (t)

equipped with the norm

ll/lk = sup|^||:ie[0,oo)

Suppose also that (An}£°=1 is a strictly increasing sequence of positive real numbers diverging to infinity so that liminf(An+1 — An) > 0. Malliavin proved [5, Theorem 8.3] that the span of the

system {tAn is not dense in CW if and only if there exists n € R such that

r+œ

log |W(eCTA)| , . . v-^ 2

———2-- dt < oo, where aA(t) = —

The question of the closure of the non-dense span of the system {tAn was later on addressed by J. M. Anderson and K. G. Binmore [1, Theorem 3]. Provided that the An are positive integers, they proved that any function in the closure extends analytically as an entire function with a gap power series expansion of the form f (z) = ^anzXn.

We note that A. Borichev [2] gave a complete characterization of the closure of polynomials in certain weighted Banach spaces on R, when W is an even log-convex function.

Motivated by the above results, we explored in [7, 8] the properties of a class of exponential systems

Ea := {tkex"t : n € N, k = 0,1, 2,..., — 1},

in certain weighted Banach spaces on the real line. We note that such a system is associated to a set A = {Anwith multiple terms

{An, ^n:= {A^ A1,..., Al, A2, A2,..., A2,..., Ak, Ak,..., Ak,... b * ^ * ^ *

^1 —times ^2—times ^k-times

• {An}~=1 is a strictly increasing sequence of positive real numbers diverging to infinity,

• is a sequence of positive integers, not necessarily bounded.

We say that the set A is a multiplicity sequence.

In [7, 8] we assumed that the multiplicity sequence A belongs to a certain class denoted by U(d, 0). This class and the weighted Banach spaces involved will be recalled in Section 2, while the main results from [7, 8] will be restated in Section 3.

In this paper we continue our investigations by considering a moment problem in a weighted L2 space on the real line. Our result, Theorem 4, is proved in Section 5. Prior to that, we introduce in Section 4 some notions from Non-Harmonic Fourier Series such as Bessel and Riesz-Fischer sequences that will play a decisive role.

The following interesting result is a special case of Theorem 4.

Theorem 1. Let

. . (t2m+2, t > 0, w(t) = < where m € N.

[0, t< 0,

Let {pn}^=1 be the increasing sequence of prime numbers and let = pn+1 — pn for each n € N, that is, is the distance between consecutive primes. Then, for any real number 7 < 2, there exists an entire function f admitting a Taylor-Dirichlet series representation

<X ^n — 1

f (z) = E( E cn,kzk) epnz, Cn,k € C, V z € C,

with the series converging uniformly on compact subsets of C, so that

e—2w(t)tkepntf (t) dt = pnpn, Vn € N and k = 0,1,2,...^ — 1.

2. Notations and definitions from [7, 8] 2.1. Weighted Banach spaces

Definition 1. We denote by Ap,T the class of all non-negative convex functions w(t) defined on the real line that satisfy the following properties:

(i) w(0) = 0 and w(t) > t2, V t > t > 0,

(ii) there is some p > 0 so that w(t) < p|t| V t < 0,

(iii) for all A > 0 there is a positive number t(A) such that w(t + A) > w(t) + t, V t > t(A).

Example 1. Let

then w € Ap,T.

/ x (t2m+2, t > 0,

w(t) = < where m € N,

10, t< 0,

For p > 1 we denote by LW the weighted Banach space of complex-valued measurable functions f defined on R such that

|f (t)e-w(t)|p dt < to,

equipped with the norm

/ /-x \\

LW := (If (t)e-w(t) |p dtj

As usual, LW is a Hilbert space when endowed with the inner product

f(t)g(t)e-&2^ dt.

2.2. The class of multiplicity sequences U(d, 0)

We say that a multiplicity sequence A = (An, }^L1 has finite density d counting multiplicities

lim -= d < oo, where n\\(t) := V^ ßn- (2.1)

i^-x t —<

A„ <t

If = 1 for all n € N the above is equivalent to

---> d as n —> oo.

Definition 2. ^e denote by L(c, d) the class of strictly increasing sequences A = |an having positive real terms an such that A has a finite density d and uniformly separated terms for some c > 0, that is,

--► d as n —>• oo, a,„+i — a,„ > c V n £ N.

0>ra

Suppose now that a sequence A={an}^=1 belongs to the class L(c, d). Then choose two positive numbers a, 5 so that

a < 1 and 5 < min{4,c}.

For each n € N consider the closed segment Tn := {x : |x — an | < a^} C R. Then, choose a point in Tn that we call bn, in an almost arbitrary way, in the sense that

for all n = m either (/) bm = bn or (U) |bm — bn | > 5.

Hence a new sequence B = {bn}^=1 is constructed.

We remark that the condition (/) allows for the presence of multiple terms in B. We may now rewrite B = {bn}^=1 in the form of a multiplicity sequence A = {An}^=1, by grouping together all those terms that have the same modulus.

Definition 3. Fix a nonnegative constant d. We denote by U(d, 0) the class of all the multiplicity sequences A = {An}^=1 constructed in the way described above from sequences A = {an}^=1 which belong to the class L(c, d), for any positive constants a, 5, c, with a < 1 and 5 < min{4, c}.

Remark 1. Clearly L(c, d) is a subclass of U(d, 0).

We now mention two important properties of a sequence A € U(d, 0) [8, Section 2].

(1) A has the same density d counting multiplicities as the original sequence A from which it was constructed, that is, (2.1) holds.

(2) There exists some x > 0 independent of n, so that

^n < xAa V n € N. (2.2)

We also note that since a < 1, then ^n/An — 0 as n — ro, hence for every e > 0 there is n(e) € N so that

< eAn V n > n(e). (2.3)

Remark 2. We use the notation U(d, 0) since A has density d and /An — 0 as n — ro. That is, the second parameter in our notation stands for the relation between the multiplicities and their corresponding frequencies An.

An interesting multiplicity sequence in the U(1,0) class with unbounded multiplicities is the following.

Example 2. Let {pn}^=1 be the increasing sequence of prime numbers, and let = pn+1 — pn for each n € N. Then A = {pn,^n}^=1 belongs to the class U(1,0). It can be constructed in the way described above from the set N of natural numbers which has density 1 (see [7, Example 1.3] and [8, Example 2.1]).

3. Our previous main results and the new one

Assuming that a multiplicity sequence A = {An, ^n}ro=1 belongs to the class U(d, 0), we obtained

in [7] necessary and sufficient conditions in order for the span of Ea to be dense in LW.

Theorem 2 [7, Theorem 1.1]. Let w(t) be a function which belongs to the class Ap>r and suppose that A € U(d, 0) for some d > 0. Then the span of the system Ea is not dense in LW for all p € [1, ro), if and only if there exists n € R such that

An <t

We then characterized in [8] the closure of the non-dense span of Ea. Moreover, in [8] we also derived an upper bound for the norm of the elements of a biorthogonal sequence

rA := {rn,k : n € N, k = 0,1,..., ^n — 1} C L^ to the system Ea in L^, where biorthogonality means

/{ 1, j = n, l = k,

0, j = n, l €{0,1,...,^n — 1}\\{k}, 0, j = n, l €{0,1,...,^j — 1}.

Theorem 3 [8, Theorems 2.1 and 6.1]. Suppose that A € U(d,0) for some d > 0, w(t) € AP;T and (3.1) holds.

Part I. Let f be a function which belongs to the closed span of Ea in LW for some p > 1. Then there is an entire function g(z) which admits a Taylor-Dirichlet series representation

^ ßn — 1

= Ecn>k eAnZ, C„)fc € C, V z € C,

n=1 fc=Q

with the series converging uniformly on compact subsets of C, so that f (x) = g(x) almost everywhere on the real line.

Part II. There is a unique biorthogonal sequence rA to the system Ea in L^ which belongs to its closed span, such that for every e > 0 there is a constant me > 0, independent of n and k, so that

||rn,k||lw < meexp{(—2d + e)AnlogAn}, V n € N, k = 0,1,...,^n — 1. (3.2)

Our aim in this article is to prove the following moment problem result.

Theorem 4. Suppose that A € U(d, 0) for some d > 0, w(t) € AP;T and (3.1) holds. Consider a doubly-indexed sequence of non-zero complex numbers

{dn,k : n € N, k = 0,1,... — 1}

such that

limsup-—-—= 7 < &2d, An = max{|dnifc| : k = 0,1,..., ¡j,n — 1}. (3.3)

n^ro An log An

T/iera i/iere exisis a function f € span (Ea) m L2, i/ioi extends analytically as an entire function, admitting a Taylor-Dirichlet series representation

ro /^n — 1 \\

f (z) = E E Cn,kzk eAnz, Cn,k € C, V z € C,

n=1 \\ k=0 J

with the series converging uniformly on compact subsets of C, so that

e—2w(t)tkeAntf (t) dt = dn,k, V n € N and k = 0,1,2,...^ — 1. (3.4)

We point out that similar moment problems were considered in [8, Theorems 1.2 and 7.1] but the solution obtained is a continuous function on R rather than an entire function.

We also note that Theorem 1 follows by combining Theorem 4 with Example 1, Example 2,

Remark 3. Suppose that A has a positive density d. A sufficient condition for (3.1) to hold (see the proof of [8, Theorem 2.2]) is if w(t) € Ap,T such that

t2 < w(t) < e^, V i > r > 0,

2d

The following results are direct consequences of Theorem 4. Corollary 1. Let w(t) be as in Example 1.

(A) Suppose that {An}X=1 is a sequence in the L(c, d) class for some d > 0 and consider a sequence

ix A™}n=1

of non-zero complex numbers {dn}X=1 such that

v log |dn|

lim sup -—--— < 2d.

n^x Ara log Ara

Then there exists an entire function f admitting a Dirichlet series representation

f (z) = J^ CneAnZ, cn € C, V z € C,

with the series converging uniformly on compact subsets of C, so that

e-2w(t)eAnif (t) dt = dn, V n € N.

(B) There exist entire functions f and g admitting a Dirichlet series representation

f (z) = £ cnenz, g(z) = £ dnenz,

so that for all n € N we have

e-2w(t) entf (t) dt = nn, / e-2w(t)entg(t) dt = n!.

4. Bessel and Riesz—Fischer sequences

The proof of Theorem 4 depends on Theorem 3 and utilizes the following notions from Non-Harmonic Fourier Series.

Let H be a separable Hilbert space endowed with an inner product (•}, and consider two sequences {fn}X=1 and {gn}X=1 in H. We say that [6, Chapter 4, Section 2]:

(i) {fn}X=1 is a Bessel sequence if there exists a constant B > 0 such that

EKf&fn)!2 <B|I/II2 V f G H.

(ii) is a Riesz-Fischer sequence if the moment problem (/, ) = c„ has at least one

solution / € H for every sequence {cn}^=1 in the space 12(N).

Remark 4■ It follows from [3, Proposition 2.3] that if two sequences {/n}^=1 and {gnin H are biorthogonal, that is

(/n, gm) = s .

10, m = n,

and {/n}~=1 is a Bessel sequence, then {gn}^=1 is a Riesz-Fischer sequence.

We give now a sufficient condition in order for {gn}^=1 to be a Riesz-Fischer sequence.

Lemma 1. Let H be a separable Hilbert space and consider two biorthogonal sequences {/^XU

/"Jra=

and {gn}XLi in H. Let cra,m = (/n, /m) and let C = (cra,m) be the Hermitian Gram matrix associated

y™/ra=l

with {/n}~=i. If there is some M > 0 so that

<M for all m = 1,2,3,..., (4.1)

then {/n}X=1 and {gn}X=1 are Bessel and Riesz-Fischer sequences respectively in H.

Proof. Relation (4.1) implies that the Gram matrix C defines a bounded linear operator on the space of sequences 12(N) (see [4, Lemma 3.5.3] and [6, Sec. 4.2, Lemma 1]). It then follows by [4, Lemma 3.5.1] that {/n}X=1 is a Bessel sequence in H. By Remark 4 we conclude that {gn}X=1 is a Riesz-Fischer sequence in H. □

5. Proof of Theorem 4

Clearly span (E\\) in L2, is a separable Hilbert space and let us denote this space by H\\. From Theorem 3 (Part II), let {rn,k} be the biorthogonal sequence to Ea which belongs to its closed span.

Then, define for every n € N and k = 0,1,..., — 1 the following:

£fcgAnt

Un,k{t) &■= ^ndn,krn,k{t) and Vn,k{t) := ^ _-It easily follows that {Ura;k} and {V^} are biorthogonal sequences in Ha.

We now claim that {Ura;k} and {Vn,fc} are Bessel and Riesz-Fischer sequences respectively in Ha. First, since (3.2) and (3.3) hold, if we let e = (2d — y)/2 we get

||Un>k ||L2 < e—eAn, V n € N and k = 0,1,2, — 1.

Then, by the Cauchy-Schwartz inequality we get

KUn,k)| < e—eAn ■ e-eAm, V n,m € N k = 0,1,2,...,^ra — 1 j = 0,1,— 1. (5.1)

Next, let cra)k,mj be the value of (Un,k, ) and let C be the infinite dimensional hermitian matrix with entries the cra,k)TOj&s, that is C is the Gram matrix associated with {Ura,k}. From (2.3) and (5.1) we get

X /Un — 1 X /Um — 1

J2J2J2Y1 |c«>k,mj| <

n=1 k=0 m=1 j=0

It then follows from Lemma 1 that our claim is valid. Thus, the moment problem

f(t)Vn<k(t)e~2w{t) dt = a„,fc V n € N and k = 0,1,2,..., fj,n - 1,

has a solution in Ha whenever ^^|an>k|2 < to. Now, if we let

= t— V n € N and k = 0,1,..., /j,n — 1, An

then the density of A and relation (2.2) imply that

ro Mn — 1 ro

n=1 fc=0 n=1 n

Thus, {an,k} belongs to the space 12(N). Hence, and recalling the definition of Vn,k, there is some function f € Ha so that

ro / tfcp\\nt\\ 1

/ ^ TT^T e dt = lT> V n G N and ^ = 0,1,2,..., Lin - 1.

./ro \\dn,fc ^n/ ^n

Clearly now (3.4) holds.

Finally, since f € Ha it follows from Theorem 3 (Part I) that f extends analytically as an entire function admitting a Taylor-Dirichlet series representation. Our proof is now complete.

REFERENCES

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2. Borichev A. On the closure of polynomials in weighted spaces of functions on the real line. Indiana Univ. Math. J, 2001. Vol. 50. No. 2. P. 829-846. DOI: 10.1512/iumj.2001.50.2044

3. Casazza P., Christensen O., Li S., Lindner A. Riesz-Fischer sequences and lower frame bounds. Z. Anal. Anwend, 2002. Vol. 21. No. 2. P. 305-314. DOI: 10.4171/ZAA/1079

4. Christensen O. An Introduction ta Frames and Riesz Bases. Appl. Numer. Harmon. Anal. Basel: Birkhauser, Springer, 2003. 440 p. DOI: 10.1007/978-0-8176-8224-8

5. Malliavin P. Sur quelques procedes d&extrapolation. Acta Math., 1955. Vol. 93. P. 179-255. DOI: 10.1007/BF02392523

6. Young R. M. An Introduction ta Nanharmanic Fourier Series. Revised first edition. San Diego, CA: Academic Press, Inc., 2001. 234 p.

7. Zikkos E. Completeness of an exponential system in weighted Banach spaces and closure of its linear span. J. Apprax. Theory, 2007. Vol. 146. No. 1. P. 115-148. DOI: 10.1016/j.jat.2006.12.002

8. Zikkos E. The closed span of some exponential system in weighted Banach spaces on the real line and a moment problem. Analysis Math, 2018. Vol. 44. No. 4. P. 605-630. DOI: 10.1007/s10476-018-0311-0

moment problems exponential systems biorthogonal families weighted banach spaces bessel and riesz-fischer sequences

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