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Averages Involving Fourier Coefficients of Non-Analytic Automorphic Forms

Published online by Cambridge University Press:  20 November 2018

V. Venugopal Rao
V. Venugopal Rao*
Affiliation:
University of Saskatchewan, Regina, Saskatchewan
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Let f(τ) be a complex valued function, defined and analytic in the upper half of the complex τ plane (τ=x+iy, y > 0), such that f(τ+λ) = f(τ) where λ is real and f(-1/τ) = γ(-iτ)k f(τ), k being a complex number. The function (—iτ)k is defined as ek log(-iτ) where log(—iτ) has the real value when — iτ is positive and γ is a complex number with absolute value 1. Such functions have been studied by E. Hecke [4] who calls them functions with signature (λ, k, γ). We further assume that f(τ) = O(|y| -c) as y tends to zero uniformly for all x, c being a positive real number. It then follows that f(τ) has a Fourier expansion of the type f(τ) = a0 + Σ an exp(2πinτ/λ) (n = 1,2,…), the series being convergent absolutely in the upper half plane.

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 13 , Issue 2 , June 1970 , pp. 187 - 198
Copyright
Copyright © Canadian Mathematical Society 1970

Footnotes

(1)

Supported partially by NSF Grant GP-4520.

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