Incompleteness and closure of a linear span of exponential system in a weighted Banach space

Abstract

In this paper, we obtain a necessary and sufficient condition for the incompleteness of complex exponential polynomials in C a ; where C a is a weighted Banach space of complex continuous functions f on the real axis R with f ðtÞ expðÀaðtÞÞ vanishing at infinity, in the uniform norm with respect to the weight aðtÞ: We also prove that, if the above condition of incompleteness holds, then each function in the closure of complex exponential polynomials can be extended to an entire function represented by a Dirichlet series. r

EXPONENTIAL POLYNOMIAL APPROXIMATION OF WEIGHTED BANACH SPACE ON ℝn

“…In the celebrating work [1], Malliavin's uniqueness theorem in [8] is employed to investigate incompleteness of Span{e λt } in C α where α(t) is a non-negative convex function defined on the real axis ‫,ޒ‬ satisfying (1) for t ∈ ‫,ޒ‬ the closure of Span{e λt } is also characterised by the Dirichlet series. The incomplete theorems in [4] and [11] generalise the work in [1] to the case that the sequence {λ k } ∞ k=1 has an infinite upper density. It is not hard to understand that some classical results of the approximation theory appear to be very difficult and interesting (see [6], for example).…”

“…The set of non-negative integers will be denoted by ‫ޚ‬ + . The notations |z| = (|z 1 n , z, t = z 1 t 1 + · · · + z n t n , e z (t) = exp( z, t ) will be used for any multi-index β, any t ∈ ‫ޒ‬ n and z ∈ ‫ރ‬ n . Let A denote positive constants, it may be different at each occurrence.…”

On Completeness and Minimality of Random Exponential System in a Weighted Banach Space of Functions Continuous on the Real Line*

“…Then using a method similar to [3], we have the following estimates: For every ω ∈ Ω 1 , the function G ω (z) defined by (2.5) is analytic in the closed right half plane C + = {z = x + iy : x ≥ 0} and satisfies:…”

On the completeness of the system $$\{ {t^{{\lambda _n}}}{\log ^{{m_n}}}t\} $$ in C 0(E)

“…Motivated by the Bernstein problem and the Müntz theorem in [3], combining Malliavin's uniqueness theorem in [11], by the approach of Fourier transform, in the papers [5]- [7], a series of intriguing results related to the Berstein polynomial approximation problem were obtained. When Λ = {λ n : n = 1, 2, .…”

“…In this paper, sufficient conditions for M(Λ 1 ) to be complete in C 0 (E) are obtained. In contrast to the method in [5]- [7] which is a combination of Malliavin's uniqueness theorem in [11] and inverse Fourier transformation that cannot be applied in our situation, we will employ the method in [1] and [16]- [18] from which with a combination of Theorem 1.3 our completeness theorem follows.…”