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Sharpness results concerning finite differences in Fourier analysis on the circle group

Citation

Nillsen, R and Okada, S, Sharpness results concerning finite differences in Fourier analysis on the circle group, Acta Universitatis Szegediensis. Acta Scientiarum Mathematicarum, 84, (3-4) pp. 591-609. ISSN 0001-6969 (2018) [Refereed Article]

Copyright Statement

2018 Bolyai Institute, University of Szeged.

DOI: doi:10.14232/actasm-017-522-y

Abstract

Let G denote the group R or T, let iota denote the identity element of G, and let s is an element of N be given. Then, a difference of order s is a function f is an element of L-2(G) for which there are a is an element of G and g is an element of L-2(G) such that f = (delta(iota)- delta(a))(s) (*)g. Let D-s (L-2(G)) be the vector space of functions that are finite sums of differences of order s. It is known that if f is an element of L-2(R), f is an element of D-s(L-2(R)) if and only if f integral(infinity)(-infinity) vertical bar (f) over cap (x)vertical bar(2)vertical bar x vertical bar(-2s) dx < infinity. Also, if f is an element of L-2(T), f is an element of D-s(L-2(T)) if and only (f) over cap (0) = 0. Consequently, D-s(L-2(G)) is a Hilbert space in a (possibly) weighted L-2-norm. It is known that every function in D-s(L-2(G)) is a sum of 2s + 1 differences of order s. However, there are functions in D-s(L-2(R)) that are not a sum of 2s differences of order s, and we call the latter type of fact a sharpness result. In D-1(L-2(T)), it is known that there are functions that are not a sum of two differences of order one. A main aim here is to obtain new sharpness results in the spaces D-s(L-2(T)) that complement the results known for R, but also to present new results in D-s(L-2(T)) that do not correspond to known results in D-s(L-2(R)). Some results are obtained using connections with Diophantine approximation. The techniques also use combinatorial estimates for potentials arising from points in the unit cube in Euclidean space, and make use of subtraction sets in arithmetic combinatorics.

Item Details

Item Type:Refereed Article
Research Division:Mathematical Sciences
Research Group:Pure mathematics
Research Field:Pure mathematics not elsewhere classified
Objective Division:Expanding Knowledge
Objective Group:Expanding knowledge
Objective Field:Expanding knowledge in the mathematical sciences
UTAS Author:Okada, S (Dr Susumu Okada)
ID Code:152498
Year Published:2018
Deposited By:Mathematics
Deposited On:2022-08-19
Last Modified:2022-09-08
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