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

Juan H. Arredondo; Manuel Bernal; Maria G. Morales

Displaying similar documents to “A necessary condition for HK-integrability of the Fourier sine transform function”

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...

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.

Henstock-Kurzweil integral on $BV$ sets

Jan Malý, Washek Frank Pfeffer (2016)

Mathematica Bohemica

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The generalized Riemann integral of Pfeffer (1991) is defined on all bounded $BV$ subsets of $ℝ^{n}$ , but it is additive only with respect to pairs of disjoint sets whose closures intersect in a set of $σ$ -finite Hausdorff measure of codimension one. Imposing a stronger regularity condition on partitions of $BV$ sets, we define a Riemann-type integral which satisfies the usual additivity condition and extends the integral of Pfeffer. The new integral is lipeomorphism-invariant and closed with respect...

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.

Elementary construction of Hölder functions such that the Kurzweil-Stieltjes integral does not exist

Martin Rmoutil (2025)

Czechoslovak Mathematical Journal

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For any $α, β > 0$ with $α + β < 1$ we provide a simple construction of an $α$ -Hölde function $f : [0, 1] \to ℝ$ and a $β$ -Hölder function $g : [0, 1] \to ℝ$ such that the integral $\int_{0}^{1} f d g$ fails to exist even in the Kurzweil-Stieltjes sense.

Variational Henstock integrability of Banach space valued functions

Luisa Di Piazza, Valeria Marraffa, Kazimierz Musiał (2016)

Mathematica Bohemica

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We study the integrability of Banach space valued strongly measurable functions defined on $[0, 1]$ . In the case of functions $f$ given by $\sum_{n = 1}^{\infty} x_{n} χ_{E_{n}}$ , where $x_{n}$ are points of a Banach space and the sets $E_{n}$ are Lebesgue measurable and pairwise disjoint subsets of $[0, 1]$ , there are well known characterizations for Bochner and Pettis integrability of $f$ . The function $f$ is Bochner integrable if and only if the series $\sum_{n = 1}^{\infty} x_{n} | E_{n} |$ is absolutely convergent. Unconditional convergence of the series is equivalent to Pettis integrability...

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)...

Boundary value problems for the Schrödinger equation involving the Henstock-Kurzweil integral

Salvador Sánchez-Perales, Francisco J. Mendoza-Torres (2020)

Czechoslovak Mathematical Journal

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In the present paper, we investigate the existence of solutions to boundary value problems for the one-dimensional Schrödinger equation $- y^{''} + q y = f$ , where $q$ and $f$ are Henstock-Kurzweil integrable functions on $[a, b]$ . Results presented in this article are generalizations of the classical results for the Lebesgue integral.

On coincidence of Pettis and McShane integrability

Marián J. Fabián (2015)

Czechoslovak Mathematical Journal

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R. Deville and J. Rodríguez proved that, for every Hilbert generated space $X$ , every Pettis integrable function $f : [0, 1] \to X$ is McShane integrable. R. Avilés, G. Plebanek, and J. Rodríguez constructed a weakly compactly generated Banach space $X$ and a scalarly null (hence Pettis integrable) function from $[0, 1]$ into $X$ , which was not McShane integrable. We study here the mechanism behind the McShane integrability of scalarly negligible functions from $[0, 1]$ (mostly) into $C (K)$ spaces. We focus in more detail on...

Linear FDEs in the frame of generalized ODEs: variation-of-constants formula

Rodolfo Collegari, Márcia Federson, Miguel Frasson (2018)

Czechoslovak Mathematical Journal

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We present a variation-of-constants formula for functional differential equations of the form $\dot{y} = ℒ (t) y_{t} + f (y_{t}, t), y_{t_{0}} = ϕ,$ where $ℒ$ is a bounded linear operator and $ϕ$ is a regulated function. Unlike the result by G. Shanholt (1972), where the functions involved are continuous, the novelty here is that the application $t \mapsto f (y_{t}, t)$ is Kurzweil integrable with $t$ in an interval of $ℝ$ , for each regulated function $y$ . This means that $t \mapsto f (y_{t}, t)$ may admit not only many discontinuities, but it can also be highly oscillating and yet, we are...

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...

Compact operators and integral equations in the $ℋ𝒦$ space

Varayu Boonpogkrong (2022)

Czechoslovak Mathematical Journal

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The space $ℋ𝒦$ of Henstock-Kurzweil integrable functions on $[a, b]$ is the uncountable union of Fréchet spaces $ℋ𝒦 (X)$ . In this paper, on each Fréchet space $ℋ𝒦 (X)$ , an $F$ -norm is defined for a continuous linear operator. Hence, many important results in functional analysis, like the Banach-Steinhaus theorem, the open mapping theorem and the closed graph theorem, hold for the $ℋ𝒦 (X)$ space. It is known that every control-convergent sequence in the $ℋ𝒦$ space always belongs to a $ℋ𝒦 (X)$ space for some $X$ . We illustrate how...

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 $G$ .