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Weighted integrability of \(p\)-adic Fourier transform. (English) Zbl 1456.30081
Summary: We obtain sufficient conditions for functions defined on \(p\)-adic linear space providing the weighted integrability of their Fourier transforms. The Bernstein-Szasz type conditions connected with moduli of smoothness are sharp in a certain sense. As a corollary we deduce recent results of S. S. Platonov. Also we prove Zygmund type tests for integrability of functions having bounded \(s\)-fluctuation and belonging to a Hölder class.
MSC:
30G06 Non-Archimedean function theory
42A38 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
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[1] Vladimirov, V. S.; Volovich, I. V.; Zelenov, E. I., \(p\)-Adic Analysis and Mathematical Physics (1994), Singapore: World Scientific, Singapore · Zbl 0812.46076
[2] Koblitz, N., \(p\)-Adic Numbers, \(p\)-Adic Analysis, and Zeta Functions (1984), N.Y.: Springer-Verlag, N.Y.
[3] Taibleson, M. H., Fourier Analysis on Local Fields (1975), Princeton: Princeton Univ. Press, Princeton · Zbl 0319.42011
[4] Gogoladze, L.; Meskhia, R., On the absolute convergence of trigonometric Fourier series, Proc. Razmazde Math. Inst., 141, 29-40 (2006) · Zbl 1113.42004
[5] Móricz, F., Sufficient conditions for the Lebesgue integrability of Fourier transforms, Anal. Math., 36, 2, 121-129 (2010) · Zbl 1240.42018
[6] Platonov, S. S., Fourier transform of Dini-Lipschitz functions on the field of \(p\)-adic numbers, \(p\)-Adic Numbers Ultrametric Anal. Appl., 11, 4, 307-318 (2019) · Zbl 1440.43009
[7] Golubov, B. I.; Volosivets, S. S., On the integrability and uniform convergence of multiplicative Fourier transform, Georgian Math. J., 16, 3, 533-546 (2009) · Zbl 1188.43003
[8] Golubov, B. I.; Volosivets, S. S., Generalized weighted integrability of the multiplicative Fourier transform, Proceedings of MIPT, 3, 1, 49-56 (2011)
[9] Volosivets, S. S.; Kuznetsova, M. A., Generalized \(p\)-adic Fourier transform and estimates of integral modulus of continuity in terms of this transform, \(p\)-Adic Numbers Ultrametric Anal. Appl., 10, 4, 312-321 (2018) · Zbl 1432.30033
[10] Volosivets, S. S., Hausdorff operators on \(p\)-adic linear spaces and their properties in Hardy, BMO, and Hölder spaces, Math. Notes, 93, 382-391 (2013) · Zbl 1270.42020
[11] Zygmund, A., Trigonometric Series (1959), Cambridge: Cambridge Univ. Press, Cambridge
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