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On Multipliers with Unconditionally Converging Fourier Series

Published online by Cambridge University Press:  20 November 2018

Gregory F. Bachelis  and
Louis Pigno
Gregory F. Bachelis
Affiliation:
Kansas State University, Manhattan, Kansas
Louis Pigno
Affiliation:
Kansas State University, Manhattan, Kansas
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Let G be a compact abelian group with dual group Γ. For 1 ≦ p < ∞, 1 ≦ q < ∞, let denote the Banach space of complex-valued functions on Γ which are multipliers of type (p, q) and the subspace of compact multipliers.

Grothendieck [10; 11] has proven that a function in LP(G), 1 ≦ p < 2, has an unconditionally converging Fourier series in LP(G) if and only if it is in L2(G), and Helgason [12] has proven that the derived algebra of LP(G), 1 ≦ p < 2, is L2(G). Using these results we show in § 2 that a multiplier of type (p, g), 1 ≦ p ≦ 2, 1 ≦ q ≦ 2, has an unconditionally converging Fourier series in if and only if it is in (Theorem 2.1), and that, for 1 ≦ p ≦ q ≦ 2, the derived algebra of is (Theorem 2.2). Statements equivalent to the above are also given.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 24 , Issue 3 , June 1972 , pp. 477 - 484
Copyright
Copyright © Canadian Mathematical Society 1972

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