Sets of Uniqueness on the Group 2&#969

Abstract

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Sets of uniqueness in compact, 0-dimensional metric groups

“…Taking the infimum over a > f(xo) gives the result. 0 The next result generalizes Theorem 3 of [8] (see also [9]), which covers the Walsh series case. …”

“…Call elements of QM(X) quasi-measures. If 8 is a quasi-measure and f is a trigonometric polynomial, we let (8,1) denote the value of 8 at f. One way of defining a quasi-measure is to specify (8, ~u) for U E U~o Cn. The matching condition (1.1 )…”

A Unified Approach to Uniqueness of Walsh Series and Haar Series

“…THEOREM 1. Let f be integrable on G, ni,n2)... be a subsequence of nonnegative integers, and S be a Walsh series which satisfies (3) limsup |52n(x)| < oo Since each Walsh function takes on only the values ±1, Schwartz' inequality implies that any Walsh series whose coefficients tend to zero necessarily satisfies the C-S condition. Therefore, Theorem 1 also contains the following result: It should be noted that following Skvorcov [6], the techniques of the next section can be modified to handle the case when / is Denjoy integrable in the narrow sense.…”

Recent developments in the theory of Walsh series

“…Crittenden and Shapiro [1965] proved that a Borel set E is a LI-set for the class of Walsh series which satisfy Skvorcov [1968] showed that this result remains true if "integrable" is replaced by "Perron integrable" and "Walsh-Fourier series" is replaced by "Perron-Walsh-Fourier series". In [1980b] [1974].…”