J Austral Math Soc Ser A 48 pp376--383, 1990.

Almost Everywhere Convergence of Lacunary Trigonometric Series With Respect to Riesz Products

J. Peyière

Abstract

Let {l_j}_{j ³ 0} be a sequence of positive integers such that l_j+1/l_j ³ 3 and {a_j}_{j ³ 0} a sequence of complex numbers such that |a_j| £ 1. Let m be the Riesz product Õ_{j ³ 0}[1 + Re(a_je^il_jx)], that is, the weak limit of measures on T the density of which are the partial products. Then of å_{j ³ 0}|a_j|² < ¥, the series å_{j ³ 0}a_j(e^il_jx - ½¾a_j) converges for m-almost every x. The m-a.e. convergence of series å_{j ³ 0}a_je^inl_jx is also investigated as well as the case of Riesz products on a compact commutative group.

1980 AMS Subject Classification (1985 Revision): 42A55, 42A61, 43A25

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Authors

J. Peyrière

Université de Paris-Sud, Unité Associée au CNRS no 757, Mathématiques, bât. 425, 91405 Orsay Cedex, France.