Convergence and integrability for some classes of trigonometric series

Živorad Tomovski

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

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.

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)...

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,...

Functions in $L_{p}$ -space and their approximations

P. Chandra (1988)

Matematički Vesnik

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Sidon sets and $I^{0}$ -sets

David Grow (1987)

Colloquium Mathematicae

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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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On general Franklin systems

Gevorkyan Gegham, Kamont Anna

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AbstractWe study general Franklin systems, i.e. systems of orthonormal piecewise linear functions corresponding to quasi-dyadic sequences of partitions of [0,1]. The following problems are treated: unconditionality of the general Franklin basis in $L^{p}$ , 1 < p < ∞, and $H^{p}$ , 1/2 < p ≤ 1; equivalent conditions for the unconditional convergence of the Franklin series in $L^{p}$ for 0< p ≤ 1; relation between Haar and Franklin series with identical coefficients; characterization of the spaces...

On the non-equivalence of rearranged Walsh and trigonometric systems in $L_{p}$

Aicke Hinrichs, Jörg Wenzel (2003)

Studia Mathematica

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We consider the question of whether the trigonometric system can be equivalent to some rearrangement of the Walsh system in $L_{p}$ for some p ≠ 2. We show that this question is closely related to a combinatorial problem. This enables us to prove non-equivalence for a number of rearrangements. Previously this was known for the Walsh-Paley order only.

On the diametral dimension of weighted spaces of analytic germs

Michael Langenbruch (2016)

Studia Mathematica

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We prove precise estimates for the diametral dimension of certain weighted spaces of germs of holomorphic functions defined on strips near ℝ. This implies a full isomorphic classification for these spaces including the Gelfand-Shilov spaces $S ¹_{α}$ and $S ₁^{α}$ for α > 0. Moreover we show that the classical spaces of Fourier hyperfunctions and of modified Fourier hyperfunctions are not isomorphic.

Marcinkiewicz multipliers of higher variation and summability of operator-valued Fourier series

Earl Berkson (2014)

Studia Mathematica

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Let $f \in V_{r} () \cup_{r} ()$ , where, for 1 ≤ r < ∞, $V_{r} ()$ (resp., $_{r} ()$ ) denotes the class of functions (resp., bounded functions) g: → ℂ such that g has bounded r-variation (resp., uniformly bounded r-variations) on (resp., on the dyadic arcs of ). In the author’s recent article [New York J. Math. 17 (2011)] it was shown that if is a super-reflexive space, and E(·): ℝ → () is the spectral decomposition of a trigonometrically well-bounded operator U ∈ (), then over a suitable non-void open interval of r-values,...

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.

A multiplier theorem for Fourier series in several variables

Nakhle Asmar, Florence Newberger, Saleem Watson (2006)

Colloquium Mathematicae

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We define a new type of multiplier operators on $L^{p} (^{N})$ , where $^{N}$ is the N-dimensional torus, and use tangent sequences from probability theory to prove that the operator norms of these multipliers are independent of the dimension N. Our construction is motivated by the conjugate function operator on $L^{p} (^{N})$ , to which the theorem applies as a particular example.

Pointwise convergence of nonconventional averages

I. Assani (2005)

Colloquium Mathematicae

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We answer a question of H. Furstenberg on the pointwise convergence of the averages $1 / N \sum_{n = 1}^{N} U ⁿ (f \cdot R ⁿ (g))$ , where U and R are positive operators. We also study the pointwise convergence of the averages $1 / N \sum_{n = 1}^{N} f (S ⁿ x) g (R ⁿ x)$ when T and S are measure preserving transformations.