Abstract

The purpose of this paper is to give an extension of Müntz-Szasz theorems to multivariable weighted Banach space. Denote by a sequence of real numbers in . The completeness of monomials in is investigated, where is the weighted Banach spaces which consist of complex continuous functions defined on with exp vanishing at infinity in the uniform norm.

1. Introduction and Notations

The object of this paper is to obtain some completeness criteria for monomials , which is analogous to Müntz-Szasz theorem in one variable.

The following notations will be used. Throughout this paper, points of will be denoted by , where . If , , , then we write . The vectors and are the real and imaginary parts of , respectively, and will be thought of as the set of all with , furthermore; for all , and for all . The set of nonnegative integers will be denoted by . The notations will be used for any multi-index and any . The unit ball of will be denoted by .

By a complete system of elements of a Banach space , we mean ; that is, the completeness is equivalent to the possibility of an arbitrary good approximation of any element of the space by linear combination of elements of this system.

The famous Müntz-Szasz theorem asserts that given a sequence of real numbers the functions are complete if and only if This classical result inspired an intensive research of related questions. Via duality, making use of suitable analytic varieties in the polydisk, in [1], for and , it is shown that there exists a sequence of monomials with for each whose linear span is dense in but not in , where is the Cartesian product of copies of the closed unit interval . The Müntz-Szasz theorem is extended to multivariables and more general results are obtained by replacing by for some function in [2]. For , the so-called Müntz set relative to is defined in [3], which enables one to construct “optimally sparse” lattice points sets for which density holds.

It is a natural goal to consider whether it could give completeness conditions analogous to Müntz-Szasz theorem in the weighted higher-dimensional Banach space on case. The paper is concerned with this problem.

Let be a nonnegative continuous function defined on , henceforth, called a weight, satisfying Given a weight , the weighted Banach space consists of complex continuous functions defined on with vanishing at infinity, normed by

Our space is rooted from [49, 12, 13], in which the exponential polynomial approximation problem is investigated.

Motivated by the Bernstein problem and the Müntz theorem in [10], combining Malliavin’s uniqueness theorem in [11], in his paper [12], Guantie Deng obtained a necessary and sufficient condition for the functions to be dense in . The result which initiated the investigation of Müntz problem on weighted Banach space consists of complex functions continuous on the real axis and is described below.

Theorem 1. Suppose is an even function satisfying (3) and is a convex function on . Let be a sequence of strictly increasing positive integers and let , , . If for each , then is dense in .
Conversely, if the sequence contains all of the odd integers, then, for to be dense in , it is necessary that (6) holds for each .

Deng’s result was generalized to the case where the weighted Banach space consists of complex functions continuous on infinitely many disjoint closed intervals in [7]. The result is described as follows.

Let be a union of infinitely many disjoint closed intervals: where satisfies .

Theorem 2. Suppose is defined by (3) and is a sequence of complex numbers satisfying the following conditions:
Let where is some positive number and If where is the harmonic measure for the domain as seen from and if then the system ( ) is complete in .

Motivated by [49, 12, 13], in this paper, we will investigate the completeness of monomials in , where is a sequence of real numbers in and is a nonnegative continuous function defined in for . Our result can be thought of as a generalization of the results in [7, 8, 13] to multivariable case. It also can be regarded as a generalization of the results in [13]. Our main result depends upon the uniqueness theory of analytic functions on the unit ball . As is well known the zeros of analytic functions in ( ) are never discrete. The multivariable case may be different from a single variable case. That is why it needs to be treated separately (see [9]).

In the sequel, we will use to denote positive constants that may vary in value from one occurrence to the next. The main results of this paper are as follows.

Theorem 3. Let be a nonnegative and nondecreasing function with continuous derivative defined on for some positive constant , satisfying and let be a nonnegative continuous function defined on satisfying where and are fixed positive constants. Let be a sequence of real numbers in . If for some , where is some fixed positive constant, then is complete in .

Theorem 4. Let be a nonnegative continuous function defined on satisfying where is some fixed positive constant. Suppose that is a sequence of real numbers in . If is satisfied for every and arbitrary positive constant , then is incomplete in .

There are obvious ways in which our main result can be generalized: the example of Theorem 1 can be extended to much more general sets by using Lemma 5 in Section 2. We decided not to pursue elaborations; our aim is to present the essence of an interesting qualitative phenomenon, avoiding as far as possible obscuring technicalities.

The remaining part of this paper is organized as follows. In Section 2 we give some notation and we introduce several results used later. In Section 3 we prove our main results.

2. Preliminaries

In this section, we will establish a uniqueness result for functions holomorphic in . The proof of such a result depends on several lemmas.

Following [14], we denote the angles and by and , respectively. Let a function be analytic in and continuous in , and let the relations hold for some . Denote by the most lower bound of all such that Then, the number is called the order of the function . Recall that the canonical Weierstrass factor is defined by The canonical Nevanlinna factor is defined by We define the following modified canonical factor by for , for , and From page 25 of [14], we know that an analytic function of arbitrary finite order admits canonical representation as follows.

Lemma 5. Every function analytic and of a finite order in the right half plane admits the representations where , are complex numbers, , are zeros of , and is a singular boundary function. All integrals and infinite products are absolutely convergent. The following relations hold: where and is an arbitrary positive number.

We can deduce the following lemma by conformal maps.

Lemma 6. Suppose that is analytic in the unit disk , satisfying for some . If, for some fixed , and , where is a sequence of real numbers in , then, .

Proof. We will show that the existence of some satisfying (28) and contradicts Lemma 5.
Suppose that is a nontrivial function analytic in the unit disk, satisfying (28) and . Taking the conformal transformation , it is well known that such transformation maps the unit disk onto the right half plane. Thus, Define then, we get a function which is analytic in the right half plane and satisfies for ; furthermore, we have for sufficient large . Applying Lemma 5, we have for arbitrary ; thus, which is a contradiction to (29).

The following uniqueness lemma is crucial in the establishment of the main result of this paper. It is closely related to results of [15].

Lemma 7. Let be a sequence in , satisfying for all and . Suppose that for some , where is some fixed positive constant. Let be the set of all that have . Let be an analytic function on which satisfies the growth condition for some and . Denote by the zero set of . If , then .

Proof. We will follow the proof of Theorem on pages 135-136 of [15]. Without loss of generality, it is enough to investigate the case where .
Denote, by , . Let be the set of all satisfying It is apparent that is a nonempty set. Thus, it is enough to prove that for fixed .
Define where . Let . Then, and for and . It follows that maps into . Define By (36) and (39), we have Note that Since , the zero sequence satisfies (35) which is (29) in Lemma 6. It follows that for all . In particular, , so that .

We will be concerned with density of polynomials in which is essential in the proof of Theorem 3. We need the following result from [16] (see also similar result in [17]).

Lemma 8. Let be a nonnegative and nondecreasing function with continuous derivative defined on for some positive constant , satisfying
If is a complex measure on such that then the polynomials are dense in .

Proof. Since a real measure has the Jordan decomposition as the difference of positive and negative variation (see page 119 of [18], e.g.), for any complex measure on a -algebra in , there is a measurable function such that for all and such that (see page 124 of [18], e.g.), replacing the positive measure in the proof of Theorems 2.1 and 2.3 in [16]; repeating the proof there word by word, we can see that the same conclusion still holds for the case of complex measures.

We will use the following elementary results on inequality in [19].

Lemma 9. Let for . Then, for .

Lemma 10. Let If for , then, where is some positive constant depending on .

3. Proof of Main Results

In this section, we prove the main results of this paper.

Proof of Theorem 3. If is incomplete in , by the Hahn-Banach theorem there exists a nontrivial bounded linear functional such that and . So by the Riesz representation theorem, there exists a complex measure on satisfying Define then, is holomorphic in and satisfies thus, by (14), we have which yields for all . Denote by the Cayley transform from to ; consider Define the function where is the function defined in (52) and is the inverse of the Cayley transform defined in (53). Thus, we have
Denote, by for . It is obvious that (15) relates to (35) in Lemma 7. Thus, we have ; it follows that for all nonnegative integers . From Lemma 8, it is obvious that , from which the conclusion of Theorem 3 follows.

Proof of Theorem 4. If there exist a real constant and some positive constant such that both (16) and (17) are satisfied, we will show the existence of a nontrivial bounded functional which annihilates .
If (17) holds, we know that there exists analytic functions ( ) which can be represented in the forms in Lemma 5. The function ( verifies for all satisfying (17); furthermore, the following estimate holds for all . It is obvious that For , define where is some positive constant satisfying for the constant defined in (16). Then, we have the following estimate: Suppose that is the analytic function defined in (59); define where and denotes the Lebesgue of . Note first that is in by and (61); furthermore, is continuous on .
Next, we claim that the integral is independent of for arbitrary and complex . Without loss of generality, we prove it for the first coordinate. To see this, let be a rectangular path in the -plane, with one edge on the image axis and one on the line , whose horizontal edges move off to infinity. By Cauchy’s theorem, the integral of the integrand (63) over is 0. From (61) we know that the contribution of the horizontal edges to this integral is also 0. Thus, it follows that (63) is the same for as for , which establishes our claim.
The same can be done for the other coordinates. Hence, we conclude from (62) that for every and . From (61) and (64), we have where . Thus, direct calculation yields From (67) we know that is in . Taking the inverse Fourier transform in (64), we obtain for . Extend to an even function by defining whenever .
It is apparent that, for , we have . Thus, combining Lemmas 9 and 10 with (16), we conclude that there exists some positive constant such that Therefore, from (69) and (67), if (60) holds, choosing satisfying we obtain the bounded linear functional satisfying for , satisfying (17) and

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work was done when Xiangdong Yang visited the IMA (Institute of Mathematics and its Applications) in Minnesota, USA. The author gratefully acknowledges the support from the CSC (Grant no. 5013). The author is also grateful to the IMA for partial financial support during his stay. This work is supported by the National Natural Science Foundation of China (no. 11261024).