scholarly journals Smoothness and Function Spaces Generated by Homogeneous Multipliers

2012 ◽  
Vol 2012 ◽  
pp. 1-22 ◽  
Author(s):  
Konstantin Runovski ◽  
Hans-Jürgen Schmeisser
Keyword(s):  
Fourier Transform ◽  
Approximation Theory ◽  
Differential Operators ◽  
Function Spaces ◽  
Explicit Representation ◽  
Periodic Functions ◽  
Homogeneous Functions ◽  
Representation Formulas ◽  
The Fourier Transform ◽  
And Function

Differential operators generated by homogeneous functionsψof an arbitrary real orders>0(ψ-derivatives) and related spaces ofs-smooth periodic functions ofdvariables are introduced and systematically studied. The obtained scale is compared with the scales of Besov and Triebel-Lizorkin spaces. Explicit representation formulas forψ-derivatives are obtained in terms of the Fourier transform of their generators. Some applications to approximation theory are discussed.

scholarly journals Some spectral formulas for functions generated by differential and integral operators in Orlicz spaces

2021 ◽  
Vol 13 (2) ◽  
pp. 326-339
Author(s):  
H.H. Bang ◽  
V.N. Huy
Keyword(s):  
Fourier Transform ◽  
Orlicz Space ◽  
Differential Operators ◽  
Orlicz Spaces ◽  
Integral Operators ◽  
Young Function ◽  
The Fourier Transform

In this paper, we investigate the behavior of the sequence of $L^\Phi$-norm of functions, which are generated by differential and integral operators through their spectra (the support of the Fourier transform of a function $f$ is called its spectrum and denoted by sp$(f)$). With $Q$ being a polynomial, we introduce the notion of $Q$-primitives, which will return to the notion of primitives if ${Q}(x)= x$, and study the behavior of the sequence of norm of $Q$-primitives of functions in Orlicz space $L^\Phi(\mathbb R^n)$. We have the following main result: let $\Phi $ be an arbitrary Young function, ${Q}({\bf x} )$ be a polynomial and $(\mathcal{Q}^mf)_{m=0}^\infty \subset L^\Phi(\mathbb R^n)$ satisfies $\mathcal{Q}^0f=f, {Q}(D)\mathcal{Q}^{m+1}f=\mathcal{Q}^mf$ for $m\in\mathbb{Z}_+$. Assume that sp$(f)$ is compact and $sp(\mathcal{Q}^{m}f)= sp(f)$ for all $m\in \mathbb{Z}_+.$ Then $$ \lim\limits_{m\to \infty } \|\mathcal{Q}^m f\|_{\Phi}^{1/m}= \sup\limits_{{\bf x} \in sp(f)} \bigl|1/ {Q}({\bf x}) \bigl|. $$ The corresponding results for functions generated by differential operators and integral operators are also given.

scholarly journals A Sharp Rearrangement Principle in Fourier Space and Symmetry Results for PDEs with Arbitrary Order

2020 ◽  
Author(s):  
Enno Lenzmann ◽  
Jérémy Sok
Keyword(s):  
Fourier Transform ◽  
Phase Retrieval ◽  
Differential Operators ◽  
Arbitrary Order ◽  
Radial Symmetry ◽  
Fourier Space ◽  
Polyharmonic Operator ◽  
Pseudo Differential Operators ◽  
Retrieval Problem ◽  
The Fourier Transform

Abstract We prove sharp inequalities for the symmetric-decreasing rearrangement in Fourier space of functions in $\mathbb{R}^d$. Our main result can be applied to a general class of (pseudo-)differential operators in $\mathbb{R}^d$ of arbitrary order with radial Fourier multipliers. For example, we can take any positive power of the Laplacian $(-\Delta )^s$ with $s> 0$ and, in particular, any polyharmonic operator $(-\Delta )^m$ with integer $m \geqslant 1$. As applications, we prove radial symmetry and real-valuedness (up to trivial symmetries) of optimizers for (1) Gagliardo–Nirenberg inequalities with derivatives of arbitrary order, (2) ground states for bi- and polyharmonic nonlinear Schrödinger equations (NLS), and (3) Adams–Moser–Trudinger type inequalities for $H^{d/2}(\mathbb{R}^d)$ in any dimension $d \geqslant 1$. As a technical key result, we solve a phase retrieval problem for the Fourier transform in $\mathbb{R}^d$. To achieve this, we classify the case of equality in the corresponding Hardy–Littlewood majorant problem for the Fourier transform in $\mathbb{R}^d$.

A note on holomorphic functions and the Fourier-Laplace transform

2017 ◽  
Vol 120 (2) ◽  
pp. 225 ◽  
Author(s):  
Marcus Carlsson ◽  
Jens Wittsten
Keyword(s):  
Fourier Transform ◽  
Laplace Transform ◽  
Half Plane ◽  
Classical Problem ◽  
Holomorphic Functions ◽  
Function Spaces ◽  
Classical Function ◽  
Upper Half Plane ◽  
The Fourier Transform ◽  
Half Axis

We revisit the classical problem of when a given function, which is analytic in the upper half plane $\mathbb{C} _+$, can be written as the Fourier transform of a function or distribution with support on a half axis $(-\infty ,b]$, $b\in \mathbb{R} $. We derive slight improvements of the classical Paley-Wiener-Schwartz Theorem, as well as softer conditions for verifying membership in classical function spaces such as $H^p(\mathbb{C} _+)$.

scholarly journals Весовые неравенства для потенциала Данкля–Рисса

2019 ◽  
Vol 20 (1) ◽  
pp. 131-147
Author(s):  
Дмитрий Викторович Горбачев ◽  
Валерий Иванович Иванов
Keyword(s):  
Approximation Theory ◽  
Function Spaces ◽  
And Function

Для классического потенциала Рисса или дробного интеграла $$I_{\alpha}$$ хорошо известны условия Харди-Литлвуда-Соболева-Стейна-Вейса $$(L^p, L^q)$$ -ограниченности со степенными весами. С помощью преобразования Фурье $$\mathcal{F}$$ потенциал Рисса определяется равенством $$\mathcal{F}(I_{\alpha}f)(y)=|y|^{-\alpha}\mathcal{F}(f)(y)$$. Важным обобщением преобразования Фурье стало преобразование Данкля $$\mathcal{F}_k$$, действующее в лебеговых пространствах с весом Данкля, определяемым с помощью системы корней $$R\subset\mathbb{R}^d$$, ее группы отражений G и неотрицательной функции кратности k на R, инвариантной относительно G. С. Тангавелу и Ю. Шу с помощью равенства $$\mathcal{F}_k(I_{\alpha}f)(y)=|y|^{-\alpha}\mathcal{F}_k(f)(y)$$ определили D-потенциал Рисса. Для D-потенциала Рисса также были доказаны условия ограниченности в лебеговых пространствах с весом Данкля и степенными весами, аналогичные условиям для потенциала Рисса. На конференции "Follow-up Approximation Theory and Function Spaces" в Centre de Recerca Matem`atica (CRM, Barcelona, 2017)  М.Л. Гольдман поставил вопрос об условиях $$(L_p,L_q)$$-ограниченности D-потенциала Рисса с кусочно-степенными весами. Рассмотрение кусочно-степенных весов позволяет выявить влияние на ограниченность D-потенциала Рисса поведения весов в нуле и бесконечности. В настоящей работе на этот вопрос дается полный ответ. В частности,в случае потенциала Рисса получены необходимые и достаточные условия. В качестве вспомогательных результатов доказаны необходимые и достаточные условия ограниченности операторов Харди и Беллмана в лебеговых пространствах с весом Данкля и кусочно-степенными весами.

Spectral theory for a class of pseudodifferential operators

1977 ◽  
Vol 76 (3) ◽  
pp. 197-214
Author(s):  
E. A. Catchpole
Keyword(s):  
Fourier Transform ◽  
Spectral Theory ◽  
Constant Coefficient ◽  
Eigenfunction Expansion ◽  
Differential Operators ◽  
Pseudodifferential Operators ◽  
Multiplication Operator ◽  
Expansion Theorem ◽  
Potential Term ◽  
The Fourier Transform

SynopsisWe investigate the spectral theory for a class of pseudodifferential operators which includes all constant coefficient differential operators, and also operators such as The operators considered are of the form Su(x) = Au(x)+q(x)u(x), where A is an operator which corresponds in the Fourier transform plane to a multiplication operator, and q(x) is a potential term. We prove an eigenfunction expansion theorem for S and derive some results concerning the spectrum of S.

scholarly journals Four Particular Cases of the Fourier Transform

Mathematics ◽  
2018 ◽  
Vol 6 (12) ◽  
pp. 335 ◽  
Author(s):  
Jens Fischer
Keyword(s):  
Fourier Transform ◽  
Fourier Series ◽  
Discrete Time ◽  
Fourier Transforms ◽  
Summation Formula ◽  
Periodic Functions ◽  
Tempered Distributions ◽  
Integral Fourier Transform ◽  
The Fourier Transform ◽  
Poisson’S Summation Formula

In previous studies we used Laurent Schwartz’ theory of distributions to rigorously introduce discretizations and periodizations on tempered distributions. These results are now used in this study to derive a validity statement for four interlinking formulas. They are variants of Poisson’s Summation Formula and connect four commonly defined Fourier transforms to one another, the integral Fourier transform, the Discrete-Time Fourier Transform (DTFT), the Discrete Fourier Transform (DFT) and the integral Fourier transform for periodic functions—used to analyze Fourier series. We prove that under certain conditions, these four Fourier transforms become particular cases of the Fourier transform in the tempered distributions sense. We first derive four interlinking formulas from four definitions of the Fourier transform pure symbolically. Then, using our previous results, we specify three conditions for the validity of these formulas in the tempered distributions sense.

scholarly journals Ray transform on Sobolev spaces of symmetric tensor fields, I: Higher order Reshetnyak formulas

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Venkateswaran P. Krishnan ◽  
Vladimir A. Sharafutdinov
Keyword(s):  
Fourier Transform ◽  
Sobolev Spaces ◽  
Tensor Field ◽  
Differential Operators ◽  
Symmetric Tensor ◽  
Higher Order ◽  
Tensor Fields ◽  
Ray Transform ◽  
The Fourier Transform ◽  
Derivatives Of

<p style='text-indent:20px;'>For an integer <inline-formula><tex-math id="M1">\begin{document}$ r\ge0 $\end{document}</tex-math></inline-formula>, we prove the <inline-formula><tex-math id="M2">\begin{document}$ r^{\mathrm{th}} $\end{document}</tex-math></inline-formula> order Reshetnyak formula for the ray transform of rank <inline-formula><tex-math id="M3">\begin{document}$ m $\end{document}</tex-math></inline-formula> symmetric tensor fields on <inline-formula><tex-math id="M4">\begin{document}$ {{\mathbb R}}^n $\end{document}</tex-math></inline-formula>. Roughly speaking, for a tensor field <inline-formula><tex-math id="M5">\begin{document}$ f $\end{document}</tex-math></inline-formula>, the order <inline-formula><tex-math id="M6">\begin{document}$ r $\end{document}</tex-math></inline-formula> refers to <inline-formula><tex-math id="M7">\begin{document}$ L^2 $\end{document}</tex-math></inline-formula>-integrability of higher order derivatives of the Fourier transform <inline-formula><tex-math id="M8">\begin{document}$ \widehat f $\end{document}</tex-math></inline-formula> over spheres centered at the origin. Certain differential operators <inline-formula><tex-math id="M9">\begin{document}$ A^{(m,r,l)}\ (0\le l\le r) $\end{document}</tex-math></inline-formula> on the sphere <inline-formula><tex-math id="M10">\begin{document}$ {{\mathbb S}}^{n-1} $\end{document}</tex-math></inline-formula> are main ingredients of the formula. The operators are defined by an algorithm that can be applied for any <inline-formula><tex-math id="M11">\begin{document}$ r $\end{document}</tex-math></inline-formula> although the volume of calculations grows fast with <inline-formula><tex-math id="M12">\begin{document}$ r $\end{document}</tex-math></inline-formula>. The algorithm is realized for small values of <inline-formula><tex-math id="M13">\begin{document}$ r $\end{document}</tex-math></inline-formula> and Reshetnyak formulas of orders <inline-formula><tex-math id="M14">\begin{document}$ 0,1,2 $\end{document}</tex-math></inline-formula> are presented in an explicit form.</p>

scholarly journals There Is Only One Fourier Transform

2017 ◽  
Author(s):  
Jens V. Fischer
Keyword(s):  
Fourier Transform ◽  
Discrete Fourier Transform ◽  
Discrete Time ◽  
Fourier Transforms ◽  
Periodic Functions ◽  
Distributional Sense ◽  
Integral Fourier Transform ◽  
The Fourier Transform

Four Fourier transforms are usually defined, the Integral Fourier transform, the Discrete-Time Fourier transform (DTFT), the Discrete Fourier transform (DFT) and the Integral Fourier transform for periodic functions. However, starting from their definitions, we show that all four Fourier transforms can be reduced to actually only one Fourier transform, the Fourier transform in the distributional sense.

Sign in / Sign up

Export Citation Format

Share Document