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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.

30G06 Non-Archimedean function theory
11S80 Other analytic theory (analogues of beta and gamma functions, \(p\)-adic integration, etc.)
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