scholarly journals On discrete pseudo-differential operators and equations

Filomat ◽  
2018 ◽  
Vol 32 (3) ◽  
pp. 975-984 ◽  
Author(s):  
Vladimir Vasilyev
Keyword(s):  
Fourier Transform ◽  
Discrete Fourier Transform ◽  
Differential Operators ◽  
Difference Operators ◽  
Pseudo Differential Operators ◽  
Discrete Spaces

We introduce discrete pseudo-differential operators in appropriate discrete Sobolev-Slobodetskii spaces. Using discrete Fourier transform and factorization concept we study invertibility of such operators in some discrete spaces. Some examples for discrete Calderon-Zygmund operators and difference operators are considered.

scholarly journals More on algebraic properties of the discrete Fourier transform raising and lowering operators

4open ◽  
2019 ◽  
Vol 2 ◽  
pp. 2 ◽  
Author(s):  
Mesuma K. Atakishiyeva ◽  
Natig M. Atakishiyev ◽  
Juan Loreto-Hernández
Keyword(s):  
Fourier Transform ◽  
Discrete Fourier Transform ◽  
Fourier Transforms ◽  
Difference Operators ◽  
Discrete Fourier Transforms ◽  
Raising And Lowering Operators ◽  
Integral Fourier Transform ◽  
Symmetrical Form ◽  
Algebraic Properties ◽  
Lowering Operators

In the present work, we discuss some additional findings concerning algebraic properties of the N-dimensional discrete Fourier transform (DFT) raising and lowering difference operators, recently introduced in [Atakishiyeva MK, Atakishiyev NM (2015), J Phys: Conf Ser 597, 012012; Atakishiyeva MK, Atakishiyev NM (2016), Adv Dyn Syst Appl 11, 81–92]. In particular, we argue that the most authentic symmetrical form of discretization of the integral Fourier transform may be constructed as the discrete Fourier transforms based on the odd points N only, while in the discrete Fourier transforms on the even points N this symmetry is spontaneously broken. This heretofore undetected distinction between odd and even dimensions is shown to be intimately related with the newly revealed algebraic properties of the above-mentioned DFT raising and lowering difference operators and, of course, is very consistent with the well-known formula for the multiplicities of the eigenvalues, associated with the N-dimensional DFT. In addition, we propose a general approach to deriving the eigenvectors of the discrete number operators N(N), that avoids the above-mentioned pitfalls in the structure of each even-dimensional case N = 2L.

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 $\mathbb{R}^d$.

Characterization of Product of Pseudo-Differential Operators Involving Fractional Fourier Transform

2021 ◽  
Vol 88 (1-2) ◽  
pp. 60
Author(s):  
Jitendra Kumar Dubey ◽  
Pradeep Kumar Pandey ◽  
S. K. Upadhyay
Keyword(s):  
Fourier Transform ◽  
Differential Operators ◽  
Fractional Fourier Transform ◽  
Pseudo Differential Operators

Characterizations of product of generalized pseudo-differential operators associated with symbol σ(x,ξ) ∈ S<sup>m</sup> are discussed by exploiting the fractional Fourier transform.

Sign in / Sign up

Export Citation Format

Share Document