# zbMATH — the first resource for mathematics

Generalized $$p$$-adic Fourier transform and estimates of integral modulus of continuity in terms of this transform. (English) Zbl 1432.30033
The authors introduce a class of complex-valued functions from $$L^q( \mathbb Q_p^n)$$, where $$1<q<\infty$$, $$\mathbb Q_p$$ is the field of $$p$$-adic numbers, for which it is possible to define the Fourier transform belonging to $$L^q( \mathbb Q_p^n)$$. They prove equalities of Parseval type, an inversion formula and a sufficient condition for a function to be represented as this Fourier transform. They also give a sharp estimate of the $$L^2( \mathbb Q_p^n)$$-modulus of continuity in terms of the Fourier transform; the case $$n=1$$ was considered by S. S. Platonov [$$p$$-Adic Numbers Ultrametric Anal. Appl. 9, No. 2, 158–164 (2017; Zbl 1409.11116)]. An $$L^q$$-generalization with $$1<q\le 2$$ is also obtained.

##### MSC:
 30G06 Non-Archimedean function theory 11S80 Other analytic theory (analogues of beta and gamma functions, $$p$$-adic integration, etc.)
Full Text:
##### References:
 [1] V. S. Vladimirov, I. V. Volovich and E.I. Zelenov, p-Adic Analysis and Mathematical Physics (World Scientific, Singapore, 1994). · Zbl 0812.46076 [2] N. Koblitz, p-Adic Numbers, p-Adic Analysis, and Zeta Functions (Springer-Verlag, N.Y., 1984). · Zbl 0364.12015 [3] M. H. Taibleson, Fourier Analysis on Local Fields (Princeton Univ. Press, Princeton, 1975). · Zbl 0319.42011 [4] 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 [5] Platonov, S. S., An analogue of the Titchmarsh theorem for the Fourier transform on the group of $$p$$-adic numbers, p-Adic Numbers Ultrametric Anal. Appl., 9, 158-164, (2017) · Zbl 1409.11116 [6] E. Titchmarsh, Introduction to the Theory of Fourier Integrals (Clarendon Press, Oxford, 1937). · JFM 63.0367.05 [7] B. I. Golubov, A. V. Efimov and V. A. Skvortsov, Walsh Series and Transforms. Theory and Applications (Kluwer Acad. Publ., Dordrecht, 1991). · Zbl 0785.42010 [8] Volosivets, S. S., Generalization of the multiplicative Fourier transform and its properties, Math. Notes, 89, 311-318, (2011) · Zbl 1228.42010 [9] A. N. Kolmogorov and S.V. Fomin, Elements of Function Theory and Functional Analysis (Nauka, Moscow, 1976) [in Russian].
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.