V.I. Ivanov. One-dimensional $(k,a)$-generalized Fourier transform ... P. 92-108

We study the two-parametric $(k,a)$-generalized Fourier transform $\mathcal{F}_{k,a}$, $k,a>0$, on the line. For $a\neq 2$ it has deformation properties and, in particular, for a function $f$ from the Schwartz space $\mathcal{S}(\mathbb{R})$, $\mathcal{F}_{k,a}(f)$ may be not infinitely differentiable or rapidly decreasing at infinity. It is proved that the invariant set for the generalized Fourier transform $\mathcal{F}_{k,a}$ and differential-difference operator $|x|^{2-a}\Delta_kf(x)$, where $\Delta_k$ is the Dunkl Laplacian, is the class
\[ \mathcal{S}_{a}(\mathbb{R})=\{f(x)=F_1(|x|^{a/2})+xF_2(|x|^{a/2})\colon F_1,F_2\in\mathcal{S}(\mathbb{R}),\,\, F_1,F_2 - \text{are even}\}. \]
For $a=1/r$, $r\in\mathbb{N}$, we consider two generalized translation operators $\tau^{y}$ and $T^y=(\tau^{y}+\tau^ {-y})/2$. Simple integral representations are proposed for them, which make it possible to prove their $L^{p}$-boundedness as $1\le p\le\infty$ for $\lambda=r(2k-1)>-1/2$. For $\lambda\ge 0$ the generalized translation operator $T^y$ is positive and its norm is equal to one. Two convolutions are defined and Young's theorem is proved for them. For generalized means defined using convolutions, a sufficient $L^{p}$-convergence condition is established. The generalized analogues of the Gauss—Weierstrass, Poisson, and Bochner—Riesz means are studied.

Keywords: $(k,a)$-generalized Fourier transform, generalized translation operator, convolution, generalized means

Received July 10, 2023

Revised August 16, 2023

Accepted August 21, 2023

Funding Agency: This work was carried out within the framework of the state assignment of the Ministry of Education of the Russian Federation, agreement no. 073-03-2023-303/2 dated 02.14.23, the topic of scientific research is “Number-theoretic methods in approximate analysis and their applications in mechanics and physics”.

Valerii Ivanovich Ivanov, Dr. Phys.-Math. Sci., Prof., Tula State University, Tula, 300012 Russia, Tula State Lev Tolstoy Pedagogical University, Tula, 300026 Russia, e-mail: ivaleryi@mail.ru

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Cite this article as: V.I. Ivanov. One-dimensional (k,a)-generalized Fourier transform. Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2023, vol. 29, no. 4, pp. 92–108.