Splitting the Riesz basis condition for systems of dilated functions through Dirichlet series

Abstract Let π be an automorphic irreducible cuspidal representation of GL m {\operatorname{GL}_{m}} over ℚ {\mathbb{Q}} with unitary central character, and let λ π ⁢ ( n ) {\lambda_{\pi}(n)} be its n-th Dirichlet series coefficient. We study short sums of isotypic trace functions associated to some sheaves modulo primes q of bounded conductor, twisted by multiplicative functions λ π ⁢ ( n ) {\lambda_{\pi}(n)} and μ ⁢ ( n ) ⁢ λ π ⁢ ( n ) {\mu(n)\lambda_{\pi}(n)} . We are able to establish non-trivial bounds for these algebraic twisted sums with intervals of length of at least q 1 / 2 + ε {q^{1/2+\varepsilon}} for an arbitrary fixed ε > 0 {\varepsilon>0} .

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A dominated convergence theorem for Eisenstein series

Annales mathématiques du Québec ◽

10.1007/s40316-021-00157-7 ◽

2021 ◽

Author(s):

Johann Franke

Keyword(s):

Fourier Series ◽

Dirichlet Series ◽

Eisenstein Series ◽

Modular Forms ◽

Convergence Theorem ◽

Rational Functions ◽

New Approach ◽

Series Representations ◽

Dominated Convergence ◽

Generalized Dirichlet

AbstractBased on the new approach to modular forms presented in [6] that uses rational functions, we prove a dominated convergence theorem for certain modular forms in the Eisenstein space. It states that certain rearrangements of the Fourier series will converge very fast near the cusp $$\tau = 0$$ τ = 0 . As an application, we consider L-functions associated to products of Eisenstein series and present natural generalized Dirichlet series representations that converge in an expanded half plane.

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A nonlocal problem for a differential operator of even order with involution

Journal of Applied Analysis ◽

10.1515/jaa-2020-2026 ◽

2020 ◽

Vol 26 (2) ◽

pp. 297-307

Author(s):

Petro I. Kalenyuk ◽

Yaroslav O. Baranetskij ◽

Lubov I. Kolyasa

Keyword(s):

Differential Operator ◽

Differential Equations ◽

Ordinary Differential Equations ◽

Riesz Basis ◽

Existence And Uniqueness ◽

Spectral Properties ◽

Nonlocal Problem ◽

Even Order

AbstractWe study a nonlocal problem for ordinary differential equations of {2n}-order with involution. Spectral properties of the operator of this problem are analyzed and conditions for the existence and uniqueness of its solution are established. It is also proved that the system of eigenfunctions of the analyzed problem forms a Riesz basis.

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