The distribution of Fourier coefficients of cusp forms over sparse sequences

Huixue Lao; Ayyadurai Sankaranarayanan

Displaying similar documents to “The distribution of Fourier coefficients of cusp forms over sparse sequences”

Hybrid sup-norm bounds for Hecke–Maass cusp forms

Nicolas Templier (2015)

Journal of the European Mathematical Society

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Let $f$ be a Hecke–Maass cusp form of eigenvalue $λ$ and square-free level $N$ . Normalize the hyperbolic measure such that $vol (Y_{0} (N)) = 1$ and the form $f$ such that ${∥ f ∥}_{2} = 1$ . It is shown that ${∥ f ∥}_{\infty} ≪_{ϵ} λ^{\frac{5}{24} + ϵ} N^{\frac{1}{3} + ϵ}$ for all $ϵ > 0$ . This generalizes simultaneously the current best bounds in the eigenvalue and level aspects.

A transplantation theorem for ultraspherical polynomials at critical index

J. J. Guadalupe, V. I. Kolyada (2001)

Studia Mathematica

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We investigate the behaviour of Fourier coefficients with respect to the system of ultraspherical polynomials. This leads us to the study of the “boundary” Lorentz space $ℒ_{λ}$ corresponding to the left endpoint of the mean convergence interval. The ultraspherical coefficients $c ₙ^{(λ)} (f)$ of $ℒ_{λ}$ -functions turn out to behave like the Fourier coefficients of functions in the real Hardy space ReH¹. Namely, we prove that for any $f \in ℒ_{λ}$ the series $\sum_{n = 1}^{\infty} c ₙ^{(λ)} (f) c o s n θ$ is the Fourier series of some function φ ∈ ReH¹ with ${| | φ | |}_{R e H ¹} {\leq c | | f | |}_{ℒ_{λ}}$ . ...

Generalized divisor problem for new forms of higher level

Krishnarjun Krishnamoorthy (2022)

Czechoslovak Mathematical Journal

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Suppose that $f$ is a primitive Hecke eigenform or a Mass cusp form for $Γ_{0} (N)$ with normalized eigenvalues $λ_{f} (n)$ and let $X > 1$ be a real number. We consider the sum $𝒮_{k} (X) : = \sum_{n < X} \sum_{n = n_{1}, n_{2}, ..., n_{k}} λ_{f} (n_{1}) λ_{f} (n_{2}) ... λ_{f} (n_{k})$ and show that $𝒮_{k} (X) ≪_{f, ϵ} X^{1 - 3 / (2 (k + 3)) + ϵ}$ for every $k \geq 1$ and $ϵ > 0$ . The same problem was considered for the case $N = 1$ , that is for the full modular group in Lü (2012) and Kanemitsu et al. (2002). We consider the problem in a more general setting and obtain bounds which are better than those obtained by the classical result of Landau (1915) for $k \geq 5$ . Since the result is valid...

On $L^{p}$ integrability and convergence of trigonometric series

Dansheng Yu, Ping Zhou, Songping Zhou (2007)

Studia Mathematica

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We first give a necessary and sufficient condition for $x^{- γ} ϕ (x) \in L^{p}$ , 1 < p < ∞, 1/p - 1 < γ < 1/p, where ϕ(x) is the sum of either $\sum_{k = 1}^{\infty} a_{k} c o s k x$ or $\sum_{k = 1}^{\infty} b_{k} s i n k x$ , under the condition that λₙ (where λₙ is aₙ or bₙ respectively) belongs to the class of so called Mean Value Bounded Variation Sequences (MVBVS). Then we discuss the relations among the Fourier coefficients λₙ and the sum function ϕ(x) under the condition that λₙ ∈ MVBVS, and deduce a sharp estimate for the weighted modulus of continuity of ϕ(x)...

Convergence of greedy approximation II. The trigonometric system

S. V. Konyagin, V. N. Temlyakov (2003)

Studia Mathematica

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We study the following nonlinear method of approximation by trigonometric polynomials. For a periodic function f we take as an approximant a trigonometric polynomial of the form $G ₘ (f) : = \sum_{k \in Λ} f ̂ (k) e^{i (k, x)}$ , where $Λ \subset ℤ^{d}$ is a set of cardinality m containing the indices of the m largest (in absolute value) Fourier coefficients f̂(k) of the function f. Note that Gₘ(f) gives the best m-term approximant in the L₂-norm, and therefore, for each f ∈ L₂, ||f-Gₘ(f)||₂ → 0 as m → ∞. It is known from previous results that in...

Solution of a functional equation on compact groups using Fourier analysis

Abdellatif Chahbi, Brahim Fadli, Samir Kabbaj (2015)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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Let $G$ be a compact group, let $n \in N ∖ {0, 1}$ be a fixed element and let $σ$ be a continuous automorphism on $G$ such that $σ^{n} = I$ . Using the non-abelian Fourier transform, we determine the non-zero continuous solutions $f : G \to C$ of the functional equation $f (x y) + \sum_{k = 1}^{n - 1} f (σ^{k} (y) x) = n f (x) f (y), x, y \in G,$ in terms of unitary characters of $G$ .

Universal forms $\sum a_{i} {(x_{i})}^{n}$ and Waring’s problem

L. Dickson (1936)

Acta Arithmetica

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Boundedness of Fourier integral operators on Fourier Lebesgue spaces and affine fibrations

Fabio Nicola (2010)

Studia Mathematica

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We study Fourier integral operators of Hörmander’s type acting on the spaces $ℱ L^{p} {(ℝ^{d})}_{c o m p}$ , 1 ≤ p ≤ ∞, of compactly supported distributions whose Fourier transform is in $L^{p}$ . We show that the sharp loss of derivatives for such an operator to be bounded on these spaces is related to the rank r of the Hessian of the phase Φ(x,η) with respect to the space variables x. Indeed, we show that operators of order m = -r|1/2-1/p| are bounded on $ℱ L^{p} {(ℝ^{d})}_{c o m p}$ if the mapping $x \mapsto \nabla_{x} Φ (x, η)$ is constant on the fibres, of codimension r,...

Koecher-Maass series of a certain half-integral weight modular form related to the Duke-Imamoḡlu-Ikeda lift

Hidenori Katsurada, Hisa-aki Kawamura (2014)

Acta Arithmetica

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Let k and n be positive even integers. For a cuspidal Hecke eigenform h in the Kohnen plus space of weight k - n/2 + 1/2 for Γ₀(4), let f be the corresponding primitive form of weight 2k-n for SL₂(ℤ) under the Shimura correspondence, and Iₙ(h) the Duke-Imamoḡlu-Ikeda lift of h to the space of cusp forms of weight k for Spₙ(ℤ). Moreover, let $ϕ_{I ₙ (h), 1}$ be the first Fourier-Jacobi coefficient of Iₙ(h), and $σ_{n - 1} (ϕ_{I ₙ (h), 1})$ be the cusp form in the generalized Kohnen plus space of weight k - 1/2 corresponding to...

On the persistence of decorrelation in the theory of wave turbulence

Anne-Sophie de Suzzoni (2013)

Journées Équations aux dérivées partielles

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We study the statistical properties of the solutions of the Kadomstev-Petviashvili equations (KP-I and KP-II) on the torus when the initial datum is a random variable. We give ourselves a random variable $u_{0}$ with values in the Sobolev space $H^{s}$ with $s$ big enough such that its Fourier coefficients are independent from each other. We assume that the laws of these Fourier coefficients are invariant under multiplication by $e^{i θ}$ for all $θ \in ℝ$ . We investigate about the persistence of the decorrelation...

Multifractal analysis of the divergence of Fourier series

Frédéric Bayart, Yanick Heurteaux (2012)

Annales scientifiques de l'École Normale Supérieure

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A famous theorem of Carleson says that, given any function $f \in L^{p} (𝕋)$ , $p \in (1, + \infty)$ , its Fourier series $(S_{n} f (x))$ converges for almost every $x \in 𝕋$ . Beside this property, the series may diverge at some point, without exceeding $O (n^{1 / p})$ . We define the divergence index at $x$ as the infimum of the positive real numbers $β$ such that $S_{n} f (x) = O (n^{β})$ and we are interested in the size of the exceptional sets $E_{β}$ , namely the sets of $x \in 𝕋$ with divergence index equal to $β$ . We show that quasi-all functions in $L^{p} (𝕋)$ have a multifractal behavior with respect to...

Weight reduction for cohomological mod $p$ modular forms over imaginary quadratic fields

Adam Mohamed (2014)

Publications mathématiques de Besançon

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Let $F$ be an imaginary quadratic field and $𝒪$ its ring of integers. Let $𝔫 \subset 𝒪$ be a non-zero ideal and let $p > 5$ be a rational inert prime in $F$ and coprime with $𝔫$ . Let $V$ be an irreducible finite dimensional representation of ${\bar{𝔽}}_{p} [{GL}_{2} (𝔽_{p^{2}})]$ . We establish that a system of Hecke eigenvalues appearing in the cohomology with coefficients in $V$ already lives in the cohomology with coefficients in ${\bar{𝔽}}_{p} \otimes d e t^{e}$ for some $e \geq 0$ ; except possibly in some few cases.

Rearrangement of coefficients of Fourier series on $S U_{2}$

Giancarlo Travaglini (1987)

Colloquium Mathematicae

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Analytical theory of nonlinear oscillations $X$ : some classes of equations $\ddot{x}+g(x)=0$ with no finite Fourier series solution

Chike Obi (1979)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti

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Si dimostra che l'equazione considerata nel titolo ammette soluzioni periodiche rappresentabili da una serie di Fourier con un numero finito di termini solo se $g(x)$ è lineare.

Multiplicatively dependent triples of Tribonacci numbers

Carlos Alexis Ruiz Gómez, Florian Luca (2015)

Acta Arithmetica

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We consider the Tribonacci sequence $T : = {T_{n}}_{n \geq 0}$ given by T₀ = 0, T₁ = T₂ = 1 and $T_{n + 3} = T_{n + 2} + T_{n + 1} + T_{n}$ for all n ≥ 0, and we find all triples of Tribonacci numbers which are multiplicatively dependent.

Fourier approximation and embeddings of Sobolev spaces

D. E. Edmunds, V. B. Moscatelli

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CONTENTSIntroduction............................................................................................................ 51. Preliminaries............................................................................................................. 82. Embedding into $W^{m, p} (Ω)$ into $L^{S} (Ω)$ (n>1).......................................... 103. The case n = 1.......................................................................................................... 284. Embedding $W^{m, p} (Ω)$ into $L^{φ} (Ω)$ ...............................................................

Un ensemble remarquable du point de vue de l’analyse de Fourier sur $Q_{p}$

J. P. Schreiber (1971)

Colloquium Mathematicae

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Steiner forms

Jan Hora (2016)

Commentationes Mathematicae Universitatis Carolinae

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A trilinear alternating form on dimension $n$ can be defined based on a Steiner triple system of order $n$ . We prove some basic properties of these forms and using the radical polynomial we show that for dimensions up to $15$ nonisomorphic Steiner triple systems provide nonequivalent forms over $G F (2)$ . Finally, we prove that Steiner triple systems of order $n$ with different number of subsystems of order $(n - 1) / 2$ yield nonequivalent forms over $G F (2)$ .

The harmonic Cesáro and Copson operators on the spaces $L^{p} (ℝ)$ , 1 ≤ p ≤ 2

Ferenc Móricz (2002)

Studia Mathematica

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The harmonic Cesàro operator is defined for a function f in $L^{p} (ℝ)$ for some 1 ≤ p < ∞ by setting $(f) (x) : = \int_{x}^{\infty} (f (u) / u) d u$ for x > 0 and $(f) (x) : = - \int_{- \infty}^{x} (f (u) / u) d u$ for x < 0; the harmonic Copson operator ℂ* is defined for a function f in $L ¹_{l o c} (ℝ)$ by setting $* (f) (x) : = (1 / x) \int^{x ₀} f (u) d u$ for x ≠ 0. The notation indicates that ℂ and ℂ* are adjoint operators in a certain sense. We present rigorous proofs of the following two commuting relations: (i) If $f \in L^{p} (ℝ)$ for some 1 ≤ p ≤ 2, then ${((f))}^{\land} (t) = * (f ̂) (t)$ a.e., where f̂ denotes the Fourier transform of f. (ii) If $f \in L^{p} (ℝ)$ for some 1 < p ≤ 2, then...

On the 4-norm of an automorphic form

Valentin Blomer (2013)

Journal of the European Mathematical Society

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We prove the optimal upper bound $\sum_{f} {∥f∥}_{4}^{4} ≪ q^{ϵ}$ where $f$ runs over an orthonormal basis of Maass cusp forms of prime level $q$ and bounded spectral parameter.

Modular symbols, Eisenstein series, and congruences

Jay Heumann, Vinayak Vatsal (2014)

Journal de Théorie des Nombres de Bordeaux

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Let $E$ and $f$ be an Eisenstein series and a cusp form, respectively, of the same weight $k \geq 2$ and of the same level $N$ , both eigenfunctions of the Hecke operators, and both normalized so that $a_{1} (f) = a_{1} (E) = 1$ . The main result we prove is that when $E$ and $f$ are congruent mod a prime $𝔭$ (which we take in this paper to be a prime of $\bar{ℚ}$ lying over a rational prime $p > 2$ ), the algebraic parts of the special values $L (E, χ, j)$ and $L (f, χ, j)$ satisfy congruences mod the same prime. More explicitly, we prove that, under certain conditions, ...