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Sharpness results concerning finite differences in Fourier analysis on the circle group
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.
Publication titleActa Universitatis Szegediensis. Acta Scientiarum Mathematicarum
Department/SchoolSchool of Natural Sciences
PublisherSzegedi Tudomanyegyetem * Bolyai Intezet,University of Szeged, Bolyai Institute
Place of publicationHungary
Rights statement© 2018 Bolyai Institute, University of Szeged.