Universally divergent Fourier series via Landau's extremal functions

Gerd Herzog; Peer Chr. Kunstmann

Displaying similar documents to “Universally divergent Fourier series via Landau's extremal functions”

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 ${| | φ | |}_{R e H ¹} {\leq c | | f | |}_{ℒ_{λ}}$ . ...

On the order of magnitude of Walsh-Fourier transform

Bhikha Lila Ghodadra, Vanda Fülöp (2020)

Mathematica Bohemica

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For a Lebesgue integrable complex-valued function $f$ defined on $ℝ^{+} : = [0, \infty)$ let $\hat{f}$ be its Walsh-Fourier transform. The Riemann-Lebesgue lemma says that $\hat{f} (y) \to 0$ as $y \to \infty$ . But in general, there is no definite rate at which the Walsh-Fourier transform tends to zero. In fact, the Walsh-Fourier transform of an integrable function can tend to zero as slowly as we wish. Therefore, it is interesting to know for functions of which subclasses of $L^{1} (ℝ^{+})$ there is a definite rate at which the Walsh-Fourier transform tends...

Generalized absolute convergence of single and double Vilenkin-Fourier series and related results

Nayna Govindbhai Kalsariya, Bhikha Lila Ghodadra (2024)

Mathematica Bohemica

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We consider the Vilenkin orthonormal system on a Vilenkin group $G$ and the Vilenkin-Fourier coefficients $\hat{f} (n)$ , $n \in ℕ$ , of functions $f \in L^{p} (G)$ for some $1 < p \leq 2$ . We obtain certain sufficient conditions for the finiteness of the series $\sum_{n = 1}^{\infty} a_{n} {| \hat{f} (n) |}^{r}$ , where ${a_{n}}$ is a given sequence of positive real numbers satisfying a mild assumption and $0 < r < 2$ . We also find analogous conditions for the double Vilenkin-Fourier series. These sufficient conditions are in terms of (either global or local) moduli of continuity of $f$ and give multiplicative...

The Fourier transform in Lebesgue spaces

Erik Talvila (2025)

Czechoslovak Mathematical Journal

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For each $f \in L^{p} (ℝ)$ ( $1 \leq p < \infty$ ) it is shown that the Fourier transform is the distributional derivative of a Hölder continuous function. For each $p$ , a norm is defined so that the space of Fourier transforms is isometrically isomorphic to $L^{p} (ℝ)$ . There is an exchange theorem and inversion in norm.

Extremal plurisubharmonic functions in $C^{N}$

Józef Siciak (1981)

Annales Polonici Mathematici

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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...

The logarithmic capacity in $C^{n}$

S. Kołodziej (1988)

Annales Polonici Mathematici

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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)...

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,...

Siciak’s extremal function via Bernstein and Markov constants for compact sets in $ℂ^{N}$

Leokadia Bialas-Ciez (2012)

Annales Polonici Mathematici

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The paper is concerned with the best constants in the Bernstein and Markov inequalities on a compact set $E \subset ℂ^{N}$ . We give some basic properties of these constants and we prove that two extremal-like functions defined in terms of the Bernstein constants are plurisubharmonic and very close to the Siciak extremal function $Φ_{E}$ . Moreover, we show that one of these extremal-like functions is equal to $Φ_{E}$ if E is a nonpluripolar set with $l i m_{n \to \infty} M ₙ {(E)}^{1 / n} = 1$ where $M ₙ (E) : = s u p {| | | g r a d P | | |}_{E} / {| | P | |}_{E}$ , the supremum is taken over all polynomials P of N variables...

Robin functions and extremal functions

T. Bloom, N. Levenberg, S. Ma'u (2003)

Annales Polonici Mathematici

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Given a compact set $K \subset ℂ^{N}$ , for each positive integer n, let $V^{(n)} (z)$ = $V_{K}^{(n)} (z)$ := sup $1 / (d e g p) V_{p (K)} (p (z))$ : p holomorphic polynomial, 1 ≤ deg p ≤ n. These “extremal-like” functions $V_{K}^{(n)}$ are essentially one-variable in nature and always increase to the “true” several-variable (Siciak) extremal function, $V_{K} (z)$ := max[0, sup1/(deg p) log|p(z)|: p holomorphic polynomial, ${| | p | |}_{K} \leq 1$ ]. Our main result is that if K is regular, then all of the functions $V_{K}^{(n)}$ are continuous; and their associated Robin functions $ϱ_{V_{K}^{(n)}} (z) : = l i m s u p_{| λ | \to \infty} [V_{K}^{(n)} (λ z) - l o g (| λ |)]$ increase to $ϱ_{K} : = ϱ_{V_{K}}$ for all z outside a pluripolar...

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 $G$ .

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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A note on average behaviour of the Fourier coefficients of $j$ th symmetric power $L$ -function over certain sparse sequence of positive integers

Youjun Wang (2024)

Czechoslovak Mathematical Journal

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Let $j \geq 2$ be a given integer. Let $H_{k}^{*}$ be the set of all normalized primitive holomorphic cusp forms of even integral weight $k \geq 2$ for the full modulo group $SL (2, ℤ)$ . For $f \in H_{k}^{*}$ , denote by $λ_{{sym}^{j} f} (n)$ the $n$ th normalized Fourier coefficient of $j$ th symmetric power $L$ -function ( $L (s, {sym}^{j} f)$ ) attached to $f$ . We are interested in the average behaviour of the sum $\sum_{\binom{n = a_{1}^{2} + a_{2}^{2} + a_{3}^{2} + a_{4}^{2} + a_{5}^{2} + a_{6}^{2} \leq x}{(a_{1}, a_{2}, a_{3}, a_{4}, a_{5}, a_{6}) \in ℤ^{6}}} λ_{{sym}^{j} f}^{2} (n),$ where $x$ is sufficiently large, which improves the recent work of A. Sharma and A. Sankaranarayanan (2023).

A variation norm Carleson theorem

Richard Oberlin, Andreas Seeger, Terence Tao, Christoph Thiele, James Wright (2012)

Journal of the European Mathematical Society

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We strengthen the Carleson-Hunt theorem by proving $L^{p}$ estimates for the $r$ -variation of the partial sum operators for Fourier series and integrals, for $r > 𝚖𝚊𝚡 {p^{'}, 2}$ . Four appendices are concerned with transference, a variation norm Menshov-Paley-Zygmund theorem, and applications to nonlinear Fourier transforms and ergodic theory.

On the higher power moments of cusp form coefficients over sums of two squares

Guodong Hua (2022)

Czechoslovak Mathematical Journal

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Let $f$ be a normalized primitive holomorphic cusp form of even integral weight for the full modular group $Γ = SL (2, ℤ)$ . Denote by $λ_{f} (n)$ the $n$ th normalized Fourier coefficient of $f$ . We are interested in the average behaviour of the sum $\sum_{a^{2} + b^{2} \leq x} λ_{f}^{j} (a^{2} + b^{2})$ for $x \geq 1$ , where $a, b \in ℤ$ and $j \geq 9$ is any fixed positive integer. In a similar manner, we also establish analogous results for the normalized coefficients of Dirichlet expansions of associated symmetric power $L$ -functions and Rankin-Selberg $L$ -functions.

Pointwise Fourier inversion of distributions on spheres

Francisco Javier González Vieli (2017)

Czechoslovak Mathematical Journal

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Given a distribution $T$ on the sphere we define, in analogy to the work of Łojasiewicz, the value of $T$ at a point $ξ$ of the sphere and we show that if $T$ has the value $τ$ at $ξ$ , then the Fourier-Laplace series of $T$ at $ξ$ is Abel-summable to $τ$ .

The equidistribution of Fourier coefficients of half integral weight modular forms on the plane

Soufiane Mezroui (2020)

Czechoslovak Mathematical Journal

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Let $f = \sum_{n = 1}^{\infty} a (n) q^{n} \in S_{k + 1 / 2} (N, χ_{0})$ be a nonzero cuspidal Hecke eigenform of weight $k + \frac{1}{2}$ and the trivial nebentypus $χ_{0}$ , where the Fourier coefficients $a (n)$ are real. Bruinier and Kohnen conjectured that the signs of $a (n)$ are equidistributed. This conjecture was proved to be true by Inam, Wiese and Arias-de-Reyna for the subfamilies ${a (t n^{2})}_{n}$ , where $t$ is a squarefree integer such that $a (t) \neq 0$ . Let $q$ and $d$ be natural numbers such that $(d, q) = 1$ . In this work, we show that ${a (t n^{2})}_{n}$ is equidistributed over any arithmetic progression $n \equiv d mod q$ .