The Multivariate Müntz-Szasz Problem in Weighted Banach Space on R n

and Applied Analysis 3 where 1 and 2 are fixed positive constants. Let k = (λ1k, λ 2 k , . . . , λ n k )} ∞ k=1 be a sequence of real numbers in R + . If


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 C  will be denoted by  = ( 1 , . . .,   ), where   ∈ C. If   =   +   ,  = ( 1 , . . .,   ),  = ( 1 , . . .,   ), then we write  =  + .The vectors  = R and  = I are the real and imaginary parts of , respectively, and R  will be thought of as the set of all  ∈ C  with I = 0, furthermore; R  + = { = ( 1 ,  2 , . . .,   ) |   > 0 for all 1 ≤  ≤ }, and will be used for any multi-index  and any  ∈ R  .The unit ball of C  will be denoted by B  = { ∈ C  : || < 1}.By a complete system of elements {ℎ  } of a Banach space , we mean Span{ℎ  } = ; 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 0 This classical result inspired an intensive research of related questions.Via duality, making use of suitable analytic varieties in the polydisk, in [1], for 1 ≤  1 <  2 < ∞ and  ≥ 2, it is shown that there exists a sequence of monomials {   } with    ∼  for each  = 1, 2, . . .,  whose linear span is dense in   1 (  ) but not in   2 (  ), where   is the Cartesian product of  copies of the closed unit interval [0, 1].The Müntz-Szasz theorem is extended to multivariables and more general results are obtained by replacing    by ()   for some function () in [2].For Ω ⊂ R  , the so-called Müntz set relative to Ω 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 R  case.The paper is concerned with this problem.
Let () be a nonnegative continuous function defined on R  , henceforth, called a weight, satisfying lim Given a weight (), the weighted Banach space   consists of complex continuous functions  defined on R  with () exp(−()) vanishing at infinity, normed by Our space   is rooted from [4-9, 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 {1,   1 ,   2 , . ..} 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.
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 dist(0,   ) → ∞.Theorem 2. Suppose () is defined by (3) and Λ = {  :  = 1, 2, . ..} is a sequence of complex numbers satisfying the following conditions: the   are all distinct and lim where  0 is some positive number and where (, , C \ ) is the harmonic measure for the domain C \  as seen from  and if then the system {   } ( = 1, 2, . ..) is complete in  0 ().
Motivated by [4-9, 12, 13], in this paper, we will investigate the completeness of monomials {   } in   , where {  = ( 1  ,  2  , . . .,    )} ∞ =1 is a sequence of real numbers in R  + and () is a nonnegative continuous function defined in R  for  ∈ R  .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 [1][2][3].Our main result depends upon the uniqueness theory of analytic functions on the unit ball B  .As is well known the zeros of analytic functions in C  ( ≥ 2) 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 where  1 and  2 are fixed positive constants.Let for some  ∈ {2, . . ., }, where  0 is some fixed positive constant, then {   } is complete in   .
Theorem 4. Let () be a nonnegative continuous function defined on R  satisfying where  is some fixed positive constant.Suppose that is satisfied for every  ∈ {1, 2, . . ., } 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.

Preliminaries
In this section, we will establish a uniqueness result for functions holomorphic in B  .The proof of such a result depends on several lemmas.
Following [14], we denote the angles ( ≤ arg ≤ ) and ( < arg < ) by [, ] and (, ), respectively.Let a function () be analytic in (, ) and continuous in [, ], and let the relations hold for some  > 0. Denote by  the most lower bound of all  * > 0 such that lim sup 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  = 1, 2 . .., for  = 1, . .., 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 R > 0 admits the representations where  = [],  0 , . . .,   are complex numbers,   =      , −/2 <   < /2 are zeros of (), and () is a singular boundary function.All integrals and infinite products are absolutely convergent.The following relations hold: where  = max(, 1) and  is an arbitrary positive number.
We can deduce the following lemma by conformal maps.
The following uniqueness lemma is crucial in the establishment of the main result of this paper.It is closely related to results of [15].for some 0 <  < ∞ and ∞ >  > 1. Denote by () the zero set of ().If () ⊃ , then  ≡ 0.
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 R  such that then the polynomials are dense in  1 (R  , ).
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 ℎ such that |ℎ()| = 1 for all  ∈  and such that  = ℎ|| (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.

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