Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS

Remote Access
Green Open Access
Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)


Divided differences and systems of nonharmonic Fourier series

Author: David Ullrich
Journal: Proc. Amer. Math. Soc. 80 (1980), 47-57
MSC: Primary 42C30; Secondary 46B15
MathSciNet review: 574507
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Suppose that $ {\omega _{n,0}},{\omega _{n,1}}, \ldots ,{\omega _{n,k}}$ are distinct complex numbers with $ \vert n - {\omega _{n,j}}\vert \leqslant \delta $ for all $ n \in {\mathbf{Z}},j = 0,1, \ldots ,k$. We show that if $ \delta > 0$ is small enough then, given complex numbers $ {c_{n,j}}(n \in {\mathbf{Z}},j = 0,1, \ldots ,k)$ there exists $ f \in {L^2}( - (k + 1)\pi ,(k + 1)\pi )$ with

$\displaystyle \int_{ - (k + 1)\pi }^{(k + 1)\pi } {f(t){e^{ - it{\omega _{n,j}}}}} dt = {c_{n,j}}\quad {\text{for}}\;n \in {\mathbf{Z}},j = 0,1, \ldots ,k$

if and only if certain ``divided differences'' involving the $ {c_{n,j}}$'s and the $ {\omega _{n,j}}$'s are square summable. This extends a classical theorem of Paley and Wiener, which is equivalent to the case $ k = 0$ above.

References [Enhancements On Off] (What's this?)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 42C30, 46B15

Retrieve articles in all journals with MSC: 42C30, 46B15

Additional Information

PII: S 0002-9939(1980)0574507-8
Keywords: Nonharmonic Fourier series, Riesz basis, divided difference
Article copyright: © Copyright 1980 American Mathematical Society

Comments: Email Webmaster

© Copyright , American Mathematical Society
Contact Us · Sitemap · Privacy Statement

Connect with us Facebook Twitter Google+ LinkedIn Instagram RSS feeds Blogs YouTube Podcasts Wikipedia