TY - JOUR

T1 - The closed span of an exponential system in the Banach spaces Lp(γ,β) and C[γ,β]

AU - Zikkos, Elias

PY - 2011/9

Y1 - 2011/9

N2 - Let Λ={λn,μn}n=1· be a multiplicity-sequence, that is, a sequence where the λn are complex numbers diverging to infinity, λn≠λk for n≠k, the |λn| are in an increasing order, and each λn appears μn times. We associate to Λ the exponential system EΛ={xkeλnx:k=0,1,2,...,μn-1}n=1·. In the spirit of the Müntz-Szász theorem and assuming that Λ belongs to a certain class of sequences that we denote by Uη, we investigate the closure of span(EΛ) in the Banach spaces Lp(γ,β) and C[γ,β], where -·<γ<β<· and p≥1. We prove that the closed span of EΛ in C[γ,β] is the subspace of functions that admit a Taylor-Dirichlet series representation ∑n=1·(∑j=0μn-1cn,jxj)eλnx,∀xε[γ,β). A similar result holds for the closed span of EΛ in Lp(γ,β) in an almost everywhere sense.

AB - Let Λ={λn,μn}n=1· be a multiplicity-sequence, that is, a sequence where the λn are complex numbers diverging to infinity, λn≠λk for n≠k, the |λn| are in an increasing order, and each λn appears μn times. We associate to Λ the exponential system EΛ={xkeλnx:k=0,1,2,...,μn-1}n=1·. In the spirit of the Müntz-Szász theorem and assuming that Λ belongs to a certain class of sequences that we denote by Uη, we investigate the closure of span(EΛ) in the Banach spaces Lp(γ,β) and C[γ,β], where -·<γ<β<· and p≥1. We prove that the closed span of EΛ in C[γ,β] is the subspace of functions that admit a Taylor-Dirichlet series representation ∑n=1·(∑j=0μn-1cn,jxj)eλnx,∀xε[γ,β). A similar result holds for the closed span of EΛ in Lp(γ,β) in an almost everywhere sense.

KW - Closed span

KW - Müntz-Szász

KW - Taylor-Dirichlet series

UR - http://www.scopus.com/inward/record.url?scp=79960435564&partnerID=8YFLogxK

U2 - 10.1016/j.jat.2011.05.005

DO - 10.1016/j.jat.2011.05.005

M3 - Article

AN - SCOPUS:79960435564

SN - 0021-9045

VL - 163

SP - 1317

EP - 1347

JO - Journal of Approximation Theory

JF - Journal of Approximation Theory

IS - 9

ER -