λ-Convergence of multiple Walsh–Paley series and sets of uniqueness

Let ω0, ω1, … denote the Walsh-Paley functions and let G denote the dyadic group introduced by Fine [3]. Recall that a subset E of G is said to be a set of uniqueness if the zero series is the only Walsh series ∑ akωk which satisfiesA subset E of G which is not a set of uniqueness is called a set of multiplicity.It is known that any subset of G of positive Haar measure is a set of multiplicity [5] and that any countable subset of G is a set of uniqueness [2]. As far as uncountable subsets of Haar measure zero are concerned, both possibilities present themselves. Indeed, among perfect subsets of G of Haar measure zero there are sets of multiplicity [1] and there are sets of uniqueness [5].

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Algebraic Elements and Sets of Uniqueness in the Group of Integers of a p-Series Field

Canadian Mathematical Bulletin ◽

10.4153/cmb-1986-029-7 ◽

1986 ◽

Vol 29 (2) ◽

pp. 177-184 ◽

Cited By ~ 1

Author(s):

Bruce Aubertin

Keyword(s):

Cantor Sets ◽

Pisot Numbers ◽

Algebraic Elements ◽

Sets Of Uniqueness ◽

Field Element

AbstractLet G be the group of integers of a p-series field. A class {E(θ)} of perfect null subsets of G is introduced and classified into M-sets and U-sets according to the arithmetical nature of the field element θ. This is analogous to the well-known classification, involving Pisot numbers, of certain Cantor sets on the circle.

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Sets of Uniqueness for the Group of Integers of a p-Series Field

Canadian Journal of Mathematics ◽

10.4153/cjm-1979-081-7 ◽

1979 ◽

Vol 31 (4) ◽

pp. 858-866 ◽

Cited By ~ 5

Author(s):

William R. Wade

Keyword(s):

Dual Group ◽

Negative Integer ◽

Detailed Account ◽

Basic Properties ◽

Dyadic Group ◽

Image Position ◽

Sets Of Uniqueness ◽

Prime 2

Let G denote the group of integers of a p-series field, where p is a prime ≦ 2. Thus, any element can be represented as a sequence {xi }i = 0∞ with 0 ≦ xi < p for each i ≦ 0. Moreover, the dual group {Ψm}m = 0∞ of G can be described by the following process. If m is a non-negative integer with for each k , and if then(1)where for each integer k ≧ 0 and for each x = {xi} ∈ G the functions Φk are defined by(2)In the case that p = 2, the group G is the dyadic group introduced by Fine [1] and the functions are the Walsh-Paley functions. A detailed account of these groups and basic properties can be found in [4].

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