Search results for "FOURIER COEFFICIENTS"
showing 3 items of 3 documents
Exponential sums related to Maass forms
2019
We estimate short exponential sums weighted by the Fourier coefficients of a Maass form. This requires working out a certain transformation formula for non-linear exponential sums, which is of independent interest. We also discuss how the results depend on the growth of the Fourier coefficients in question. As a byproduct of these considerations, we can slightly extend the range of validity of a short exponential sum estimate for holomorphic cusp forms. The short estimates allow us to reduce smoothing errors. In particular, we prove an analogue of an approximate functional equation previously proven for holomorphic cusp form coefficients. As an application of these, we remove the logarithm …
Generalized Henstock integrals in the theory of series in multiplicative systems
2004
Properties of a Henstock type integral defined by means of a differential basis generated by P-adic paths ae studied. It is proved that this integral solves the problem of coefficients reconstruction by using generalized Fourier formulas for a series over multiplivative systems.
Comparison of the P-integral with Burkill's integrals and some applications to trigonometric series
2023
It is proved that the $P_r$-integral [9] which recovers a function from its derivative defined in the space $L^r$, 1 ≤r<∞, is properly included in Burkill’s trigonometric CP-and SCP-integrals. As an application to harmonic analysis, a de La Vallée-Poussin-type theorem for the $P_r$-integral is obtained: convergence nearly everywhere of a trigonometric series to a $P_r$-integrable function f implies that this series is the Pr-Fourier series of f.