Convergence and integrability for some classes of trigonometric series

Živorad Tomovski

Displaying similar documents to “Convergence and integrability for some classes of trigonometric series”

Necessary and sufficient conditions for the $L^{1}$ -convergence of double Fourier series

Péter Kórus (2018)

Czechoslovak Mathematical Journal

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We extend the results of paper of F. Móricz (2010), where necessary conditions were given for the $L^{1}$ -convergence of double Fourier series. We also give necessary and sufficient conditions for the $L^{1}$ -convergence under appropriate assumptions.

Integrability and continuity of functions represented by trigonometric series: coefficients criteria

Mikhail Dyachenko, Sergey Tikhonov (2009)

Studia Mathematica

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We study weighted $L_{p}$ -integrability (1 ≤ p < ∞) of trigonometric series. It is shown how the integrability of a function with weight $x^{- α}$ depends on some regularity conditions on Fourier coefficients. Criteria for the uniform convergence of trigonometric series in terms of their coefficients are also studied.

Absolute convergence of multiple Fourier integrals

Yurii Kolomoitsev, Elijah Liflyand (2013)

Studia Mathematica

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Various new sufficient conditions for representation of a function of several variables as an absolutely convergent Fourier integral are obtained. The results are given in terms of $L_{p}$ integrability of the function and its partial derivatives, each with a different p. These p are subject to certain relations known earlier only for some particular cases. Sharpness and applications of the results obtained are also discussed.

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

Giancarlo Travaglini (1987)

Colloquium Mathematicae

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Norm convergence of Fejér means of two-dimensional Walsh-Fourier series

Ushangi Goginava (2011)

Banach Center Publications

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The main aim of this paper is to prove that there exists a martingale $f \in H_{1 / 2}$ such that the Fejér means of the two-dimensional Walsh-Fourier series of f is not uniformly bounded in the space weak- $L_{1 / 2}$ .

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.

Uniform convergence of N-dimensional Walsh-Fourier series

U. Goginava (2005)

Studia Mathematica

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We establish conditions on the partial moduli of continuity which guarantee uniform convergence of the N-dimensional Walsh-Fourier series of functions f from the class $C_{W} (I^{N}) \cap ⋂_{i = 1}^{N} B V_{i, p (n)}$ , where p(n)↑ ∞ as n → ∞.

The degree of approximation by Hausdorff means of a conjugate Fourier series

Sergiusz Kęska (2016)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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The purpose of this paper is to analyze the degree of approximation of a function $\bar{f}$ that is a conjugate of a function $f$ belonging to the Lipschitz class by Hausdorff means of a conjugate series of the Fourier series.

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...

Regular statistical convergence of double sequences

Ferenc Móricz (2005)

Colloquium Mathematicae

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The concepts of statistical convergence of single and double sequences of complex numbers were introduced in [1] and [7], respectively. In this paper, we introduce the concept indicated in the title. A double sequence $x_{j k} : (j, k) \in ℕ ²$ is said to be regularly statistically convergent if (i) the double sequence $x_{j k}$ is statistically convergent to some ξ ∈ ℂ, (ii) the single sequence $x_{j k} : k \in ℕ$ is statistically convergent to some $ξ_{j} \in ℂ$ for each fixed j ∈ ℕ ∖ ₁, (iii) the single sequence $x_{j k} : j \in ℕ$ is statistically convergent...

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

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...

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)...

A necessary condition for HK-integrability of the Fourier sine transform function

Juan H. Arredondo, Manuel Bernal, Maria G. Morales (2025)

Czechoslovak Mathematical Journal

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The paper is concerned with integrability of the Fourier sine transform function when $f \in {BV}_{0} (ℝ)$ , where ${BV}_{0} (ℝ)$ is the space of bounded variation functions vanishing at infinity. It is shown that for the Fourier sine transform function of $f$ to be integrable in the Henstock-Kurzweil sense, it is necessary that $f / x \in L^{1} (ℝ)$ . We prove that this condition is optimal through the theoretical scope of the Henstock-Kurzweil integration theory.

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.

Universally divergent Fourier series via Landau's extremal functions

Gerd Herzog, Peer Chr. Kunstmann (2015)

Commentationes Mathematicae Universitatis Carolinae

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We prove the existence of functions $f \in A (𝔻)$ , the Fourier series of which being universally divergent on countable subsets of $𝕋 = \partial 𝔻$ . The proof is based on a uniform estimate of the Taylor polynomials of Landau’s extremal functions on $𝕋 ∖ {1}$ .

Right inverses for partial differential operators on Fourier hyperfunctions

Michael Langenbruch (2007)

Studia Mathematica

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We characterize the partial differential operators P(D) admitting a continuous linear right inverse in the space of Fourier hyperfunctions by means of a dual (Ω̅)-type estimate valid for the bounded holomorphic functions on the characteristic variety $V_{P}$ near $ℝ^{d}$ . The estimate can be transferred to plurisubharmonic functions and is equivalent to a uniform (local) Phragmén-Lindelöf-type condition.

The distribution of Fourier coefficients of cusp forms over sparse sequences

Huixue Lao, Ayyadurai Sankaranarayanan (2014)

Acta Arithmetica

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Let $λ_{f} (n)$ be the nth normalized Fourier coefficient of a holomorphic Hecke eigenform $f (z) \in S_{k} (Γ)$ . We establish that $\sum_{n \leq x} λ_{f}^{2} (n^{j}) = c_{j} x + O (x^{1 - 2 / ({(j + 1)}^{2} + 1)})$ for j = 2,3,4, which improves the previous results. For j = 2, we even establish a better result.

On the uniform convergence of double sine series

Péter Kórus, Ferenc Móricz (2009)

Studia Mathematica

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Let a single sine series (*) $\sum_{k = 1}^{\infty} a_{k} s i n k x$ be given with nonnegative coefficients $a_{k}$ . If $a_{k}$ is a “mean value bounded variation sequence” (briefly, MVBVS), then a necessary and sufficient condition for the uniform convergence of series (*) is that $k a_{k} \to 0$ as k → ∞. The class MVBVS includes all sequences monotonically decreasing to zero. These results are due to S. P. Zhou, P. Zhou and D. S. Yu. In this paper we extend them from single to double sine series (**) $\sum_{k = 1}^{\infty} \sum_{l = 1}^{\infty} c_{k l} s i n k x s i n l y$ , even with complex coefficients $c_{k l}$ . We also...

On the Nörlund means of Vilenkin-Fourier series

István Blahota, Lars-Erik Persson, Giorgi Tephnadze (2015)

Czechoslovak Mathematical Journal

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We prove and discuss some new $(H_{p}, L_{p})$ -type inequalities of weighted maximal operators of Vilenkin-Nörlund means with non-increasing coefficients ${q_{k} : k \geq 0}$ . These results are the best possible in a special sense. As applications, some well-known as well as new results are pointed out in the theory of strong convergence of such Vilenkin-Nörlund means. To fulfil our main aims we also prove some new estimates of independent interest for the kernels of these summability results. In the special cases of...

Bases in spaces of analytic germs

Michael Langenbruch (2012)

Annales Polonici Mathematici

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We prove precise decomposition results and logarithmically convex estimates in certain weighted spaces of holomorphic germs near ℝ. These imply that the spaces have a basis and are tamely isomorphic to the dual of a power series space of finite type which can be calculated in many situations. Our results apply to the Gelfand-Shilov spaces $S ¹_{α}$ and $S ₁^{α}$ for α > 0 and to the spaces of Fourier hyperfunctions and of modified Fourier hyperfunctions.

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,...