Description of self-adjoint extensions of differential operators of arbitrary order on an infinite interval in the absolutely indefinite case

1990 ◽  
Vol 48 (3) ◽  
pp. 337-345
Author(s):  
A. M. Khol'kin
Keyword(s):  
Differential Operators ◽  
Arbitrary Order ◽  
Infinite Interval

A family of hyper-Bessel functions and convergent series in them

2014 ◽  
Vol 17 (4) ◽  
Author(s):  
Jordanka Paneva-Konovska
Keyword(s):  
Power Series ◽  
Bessel Function ◽  
Classical Theory ◽  
Bessel Functions ◽  
Differential Operators ◽  
Arbitrary Order ◽  
Convergent Series ◽  
Multi Index ◽  
Fatou Theorem ◽  
Convergence Of Series

AbstractThe Delerue hyper-Bessel functions that appeared as a multi-index generalizations of the Bessel function of the first type, are closely related to the hyper-Bessel differential operators of arbitrary order, introduced by Dimovski. In this work we consider an enumerable family of hyper-Bessel functions and study the convergence of series in such a kind of functions. The obtained results are analogues to the ones in the classical theory of the widely used power series, like Cauchy-Hadamard, Abel and Fatou theorem.

A Friedrichs inequality and an application

1984 ◽  
Vol 97 ◽  
pp. 185-191 ◽  
Author(s):  
Roger T. Lewis
Keyword(s):  
Differential Operators ◽  
Arbitrary Order ◽  
Type Inequality ◽  
Essential Spectra ◽  
Hardy Type Inequality ◽  
Hardy Type ◽  
Partial Differential Operators ◽  
Even Order ◽  
Friedrichs Inequality ◽  
Derivatives Of

SynopsisAn inequality whose origins date to the work of G. H. Hardy is presented. This Hardy-type inequality applies to derivatives of arbitrary order of functions whose domain is a subset of ℝn. The Friedrichs inequality is a corollary. The result is then used to establish lower bounds on the essential spectra of even-order elliptic partial differential operators on unbounded domains.

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 $\mathbb{R}^d$.

Riemann-Liouville and Caputo type multiple Erdélyi-Kober operators

Open Physics ◽  
2013 ◽  
Vol 11 (10) ◽  
Author(s):  
Virginia Kiryakova ◽  
Yuri Luchko
Keyword(s):  
Differential Equations ◽  
Differential Operators ◽  
Arbitrary Order ◽  
Special Functions ◽  
Kernel Functions ◽  
Fractional Integrals ◽  
Human Sciences ◽  
New Applications ◽  
Derivatives Of

AbstractIn this paper some generalized operators of Fractional Calculus (FC) are investigated that are useful in modeling various phenomena and systems in the natural and human sciences, including physics, engineering, chemistry, control theory, etc., by means of fractional order (FO) differential equations. We start, as a background, with an overview of the Riemann-Liouville and Caputo derivatives and the Erdélyi-Kober operators. Then the multiple Erdélyi-Kober fractional integrals and derivatives of R-L type of multi-order (δ 1,…,δ m) are introduced as their generalizations. Further, we define and investigate in detail the Caputotype multiple Erdélyi-Kober derivatives. Several examples and both known and new applications of the FC operators introduced in this paper are discussed. In particular, the hyper-Bessel differential operators of arbitrary order m > 1 are shown as their cases of integer multi-order. The role of the so-called special functions of FC is emphasized both as kernel-functions and solutions of related FO differential equations.

scholarly journals GENERALIZED INTERPOLATION HERMITE – BIRKHOFF POLYNOMIALS FOR ARBITRARY-ORDER PARTIAL DIFFERENTIAL OPERATORS

2018 ◽  
Vol 54 (2) ◽  
pp. 149-163 ◽  
Author(s):  
M. V. Ignatenko ◽  
L. A. Yanovich
Keyword(s):  
Differential Operators ◽  
Arbitrary Order ◽  
Theoretical Research ◽  
Chebyshev System ◽  
Birkhoff Interpolation ◽  
Partial Differential ◽  
Partial Differential Operators ◽  
Special Cases ◽  
Operator Interpolation ◽  
Interpolation Polynomials

This article is devoted to the problem of construction and research of the generalized Hermite – Birkhoff interpolation formulas for arbitrary-order partial differential operators given in the space of continuously differentiable functions of many variables. The construction of operator interpolation polynomials is based both on interpolation polynomials for scalar functions with respect to an arbitrary Chebyshev system, and on the generalized Hermite – Birkhoff interpolation formulas obtained earlier by the authors for general operators in functional spaces. The presented operator formulas contain the Stieltjes integrals and the Gateaux differentials of an interpolated operator. An explicit representation of the error of operator interpolation was obtained. Some special cases of the generalized Hermite – Birkhoff formulas for partial differential operators are considered. The obtained results can be used in theoretical research as the basis for constructing approximate methods for solution of some nonlinear operator-differential equations found in mathematical physics.

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