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1997/1998 Periodic \(L_p\) Functions with \(L_q\) Difference Functions
Tamás Keleti
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Real Anal. Exchange 23(2): 431-440 (1997/1998).


Let \(0\lt p\lt q\lt \infty\). We investigate the following question: For which subsets \(H\) of the circle group \(\mathbb{T}=\mathbb{R}/\mathbb{Z}\) is it true that if \(f\in L_p\) and \(\Delta_h f(x)=f(x+h)-f(x)\in L_q\) for any \(h\in H\), then \(f\in L_q\)? We prove that this is not true for pseudo-Dirichlet sets. Evidence is gathered for the conjecture that the class of counter-examples is precisely the class of \(N\)-sets.


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Tamás Keleti. "Periodic \(L_p\) Functions with \(L_q\) Difference Functions." Real Anal. Exchange 23 (2) 431 - 440, 1997/1998.


Published: 1997/1998
First available in Project Euclid: 14 May 2012

zbMATH: 0943.28002
MathSciNet: MR1640019

Primary: 28A99
Secondary: 39A70 , 42A28 , 43A15 , 43A46

Keywords: {\(L_p\) function} , {\(N\)-set} , {circle group} , {difference function} , {measurable function} , {pseudo-Dirichlet set}

Rights: Copyright © 1999 Michigan State University Press

Vol.23 • No. 2 • 1997/1998
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