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Almost everywhere convergence of lacunary trigonometric series with respect to Riesz products

Part of: Harmonic analysis in one variable Abstract harmonic analysis

Published online by Cambridge University Press:  09 April 2009

J. PeyriÈre
J. PeyriÈre
Affiliation:
Université de Paris-SudUnité Associée au CNRS no 757 Mathématiques, bât. 425 91405 Orsay Cedex, France
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Abstract

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Let {λj}j≥0 be a sequence of positive integers such that λj+1/λj≥3 and {aj}j≥0 a sequence of complex numbers such that |aj|≤1. Let μ be the Riesz product πj≥0[1+ Re(ajeiλjx)], that is, the weak limit of measures on T the density of which are the partial products. Then if Σj≥0|aj|2≤∞ the series Σj≥0 aj(eiλjx - ½āj) converges for μ-almost every x. The μ-a.e. convergence of series Σ ajeinλjx is also investigated as well as the case of Riesz products on a compact commutative group.

MSC classification

Secondary: 42A55: Lacunary series of trigonometric and other functions; Riesz products 42A61: Probabilistic methods 43A25: Fourier and Fourier-Stieltjes transforms on locally compact and other abelian groups
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 48 , Issue 3 , June 1990 , pp. 376 - 383
Copyright
Copyright © Australian Mathematical Society 1990

References

[1]Brown, G., Moran, W. and Pearce, C. E. M., ‘Riesz products, Hausdorff dimension and normal numbers,’ Math. Proc. Cambridge Philos. Soc. 101 (1987), 529–540.CrossRefGoogle Scholar
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[3]Peyrière, J., ‘Etude de quelques propriétés des produits de Riesz,’ Ann. Inst. Fourier 25 (1975), 127–169.CrossRefGoogle Scholar