scholarly journals Complete transformation of Radon-Kipriyanov. Some properties

2019 ◽  
Vol 489 (2) ◽  
pp. 125-130
Author(s):  
L. N. Lyakhov ◽  
M. G. Lapshina ◽  
S. A. Roshchupkin
Keyword(s):  
Fourier Transform ◽  
Differential Operator ◽  
Differential Operators ◽  
Support Theorem ◽  
Singular Differential Operator ◽  
Wiener Theorem ◽  
Bessel Transform ◽  
Bessel Transforms ◽  
The Fourier Transform ◽  
Complete Transformation

The even Radon-Kipriyanov transform (Kg-transform) is suitable for investigating problems with the Bessel singular differential operator Bi = 2i2+iii,i 0. In this paper, we introduce the odd Radon-Kipriyanov transform and complete Radon-Kipriyanov transform to investigation more general equations containing odd B‑derivativesiBik, k = 0, 1, 2, ... (in particular, gradients of functions). Formulas of K-transforms of singular differential operators are given. Based on the Bessel transforms introduced by B. M. Levitan and the odd Bessel transform introduced by I. A. Kipriyanov and V. V. Katrakhov, a connection was obtained between the complete Radon-Kipriyanov transform with the Fourier transform and the mixed Fourier-Levitan-Kipriyanov-Katrakhov transform. An analogue of Helgasons support theorem and an analogue of the Paley-Wiener theorem are presented.

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.

Semibounded Extensions of Singular Ordinary Differential Operators

1988 ◽  
Vol 31 (4) ◽  
pp. 432-438
Author(s):  
Allan M. Krall
Keyword(s):  
Differential Operator ◽  
Lower Bound ◽  
Boundary Conditions ◽  
Differential Operators ◽  
The Self ◽  
Limit Circle ◽  
Singular Differential Operator ◽  
Ordinary Differential Operators

AbstractThe self-adjoint extensions of the singular differential operator Ly = [(py’)’ + qy]/w, where p < 0, w > 0, q ≧ mw, are characterized under limit-circle conditions. It is shown that as long as the coefficients of certain boundary conditions define points which lie between two lines, the extension they help define has the same lower bound.

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 $s&gt; 0$ 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 weighted version of the Paley–Wiener theorem

1989 ◽  
Vol 105 (2) ◽  
pp. 389-395 ◽  
Author(s):  
T. G. Genchev
Keyword(s):  
Fourier Transform ◽  
Exponential Type ◽  
Entire Functions ◽  
Weighted Version ◽  
Cauchy Theorem ◽  
Wiener Theorem ◽  
Functions Of Exponential Type ◽  
The Fourier Transform

A generalization of the classical theorems of Paley and Wiener[5] and Plancherel and Polya[6] concerning entire functions of exponential type is obtained. The proof relies only on the Cauchy theorem and the Hardy–Littlewood inequality for the Fourier transform (see [8, 9]). Since the functions under consideration are supposed to be defined only in two opposite octants in ℂn, a version of the edge of the wedge theorem [7] is derived as a by-product.

scholarly journals Invariants of optimal integration of rapidly oscillatory functions

2021 ◽  
pp. 121-125
Author(s):  
Valeriy Zadiraka ◽  
Liliya Luts ◽  
Inna Shvidchenko
Keyword(s):  
Fourier Transform ◽  
Wavelet Transform ◽  
Computer Technology ◽  
Quadrature Formulas ◽  
Common Elements ◽  
Optimal Integration ◽  
Different Types ◽  
Bessel Transform ◽  
The Fourier Transform ◽  
Computational Resources

The paper presents some common elements (invariants) of optimal integration of rapidly oscillatory functions for the different types of oscillations, in particular, for calculating the Fourier transform from finite functions, wavelet transform, and Bessel transform. Their brief description is given. The application of the invariants allows to increase the potential of quadrature formulas due to the fullest use of apriori information. Invariants form the basis of computer technology of integration of rapidly oscillatory functions with a given accuracy with limited computational resources.

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.

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