Behavior of the sequence of norms of primitives of a function
type: Item
instance of: scholarly article
title: Behavior of the sequence of norms of primitives of a function
zbMATH Open document ID: 1204.46017
author: Ha Huy Bang
author: Vu Nhat Huy
published in: Journal of Approximation Theory
publication date: 2010-06-25T00:00:00Z
review text: The behavior of the sequence of the \(L_p\)-norm of primitives of a function is characterized by its spectrum, being the smallest closed set outside of which the Fourier transform \(\hat{f}\) vanishes. More precisely, let \(1 \leq p \leq \infty\) and \((I^nf)_{n=0}^{\infty} \subset L_p(\mathbb{R})\) be the sequence of primitives of \(f\), then it is proved that \( \lim_{n \rightarrow \infty}\|I^n f\|_p^{1/n} = \sigma^{-1}\), where \(\sigma = \inf\{ |\xi| : \xi \in \text{supp}\hat{f} \}\).
reviewed by: Nele De Schepper
Mathematics Subject Classification ID: 46E30
Mathematics Subject Classification ID: 46F05
zbMATH DE Number: 5726582
zbMATH Keywords: \(L_p\)-spaces
zbMATH Keywords: generalized functions
MaRDI profile type: Publication
full work available at URL: j.jat.2009.12.011
OpenAlex ID: W2081164669
cites work: On real Paley--Wiener theorems for certain integral transforms.
cites work: On the inversion of the Laplace and Abel transforms for causal symmetric spaces
cites work: Q3023593
cites work: Real Paley-Wiener theorems for the inverse Fourier transform on a Riemannian symmetric space.
cites work: A Property of Infinitely Differentiable Functions
cites work: Functions with Bounded Spectrum
cites work: Properties of functions in Orlicz spaces that depend on the geometry of their spectra
cites work: Investigation of the properties of functions in the space \(N_{\Phi}\) depending on the geometry of their spectrum.
cites work: The sequence of Luxemburg norms of derivatives
cites work: A remark on the Kolmogorov-Stein inequality
cites work: Q4780280
cites work: Functions of exponential type
cites work: On the supports of functions
cites work: Q4438433
cites work: Spectrum of signals
cites work: Generalization of a theorem of Boas to a class of integral transforms
cites work: Paley-Wiener-type theorems for a class of integral transforms
cites work: Q4552839
DOI: 10.1016/J.JAT.2009.12.011
description: scientific article; zbMATH DE number 5726582