Differentiability of arithmetic Fourier series arising from Eisenstein series

On Fourier Coefficients of Eisenstein Series

In this paper we wish to prove that under certain conditions the Fourier coefficients of the Eisenstein series for an arithmetic group acting on a tube domain are all rational numbers. Let G be a connected, simply-connected, semisimple, and almost Q-simple linear algebraic group defined over the rational number field Q. Let R be the real number field. Then GR is connected, and we assume that if...

Eisenstein Series*

group GC defined over Q whose connected component G 0 Q has no rational character. It is also necessary to suppose that the centralizer of a maximal Q split torus of G0C meets every component of GC. The reduction theory of Borel applies, with trivial modifications, to G; it will be convenient to assume that Γ has a fundamental set with only one cusp. Fix a minimal parabolic subgroup P 0 C defin...

Differentiability and Dimension of Some Fractal Fourier Series

Taking into account that the frequencies of Fourier expansions are integers it is not surprising that some arithmetical results play an important role in many theorems and examples in harmonic analysis. This relation is even clearer from the historical point of view and Hardy and Littlewood can be considered at the same time as founders of a substantial part of harmonic analysis and analytic nu...

Asymptotic Relations among Fourier Coefficients of Real-analytic Eisenstein Series

Following Wolpert, we find a set of asymptotic relations among the Fourier coefficients of real-analytic Eisenstein series. The relations are found by evaluating the integral of the product of an Eisenstein series φir with an exponential factor along a horocycle. We evaluate the integral in two ways by exploiting the automorphicity of φir ; the first of these evaluations immediately gives us on...

An Exceptional Arithmetic Group and Its Eisenstein Series