An Analogy of the Carleson–Hunt Theorem with Respect to Vilenkin Systems

• Published in: The Journal of fourier analysis and applications Vol. 28; no. 3
• Main Authors: Persson, L.-E., Schipp, F., Tephnadze, G., Weisz, F.
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.06.2022; Springer; Springer Nature B.V
• Subjects: Abstract Harmonic Analysis; Almost everywhere convergence; Approximations and Expansions; Carleson-Hunt theorem; Fourier Analysis; Fourier series; Kolmogorov theorem; Martingales; Matematik; Mathematical Methods in Physics; Mathematics; Mathematics and Statistics; Operator Theory and Fourier Analysis; Partial Differential Equations; Signal,Image and Speech Processing; Theorems; Vilenkin group; Vilenkin system; Vilenkin-Fourier series; 42C10; Kolmogorov theorem; 42B25; Vilenkin group; Vilenkin–Fourier series; Almost everywhere convergence; Carleson–Hunt theorem; Fourier analysis; Vilenkin system
• ISSN: 1069-5869; 1531-5851; 1531-5851
• DOI: 10.1007/s00041-022-09938-2

In this paper we discuss and prove an analogy of the Carleson–Hunt theorem with respect to Vilenkin systems. In particular, we use the theory of martingales and give a new and shorter proof of the almost everywhere convergence of Vilenkin–Fourier series of f ∈ L p ( G m ) for p > 1 in case the Vilenkin system is bounded. Moreover, we also prove sharpness by stating an analogy of the Kolmogorov theorem for p = 1 and construct a function f ∈ L 1 ( G m ) such that the partial sums with respect to Vilenkin systems diverge everywhere.