Generalized Hankel transform on the line

V. I. Ivanov

Čebyševskij sbornik, Tome 24 (2023) no. 3, pp. 5-25.

Résumé

Since 2012, in harmonic analysis on the line with a power-law weight, the two-parameter $(k,a)$-generalized Fourier transform proposed by S. Ben Saïd, T. Kobayashi, B. Orsted has been intensively studied. It generalizes the Dunkl transform depending on only one parameter $k\ge 0$. Together with an increase in the variety of unitary transforms, the presence of a parameter $a>0$ for $a\neq 2$ leads to the appearance of deformation properties, for example, for functions from the Schwartz space, the generalized Fourier transform may not be infinitely differentiable or rapidly decreasing at infinity. The fast decay is preserved only for the sequence $a=2/n$, $n\in$ $\mathbb{N}$. Some change of variable in this case also improves other properties of the generalized Fourier transform. The generalized Dunkl transform obtained after changing the variable at $a=2/(2r+1)$, $r\in$ $\mathbb{Z}_+$ is devoid of deformation properties and, to a large extent, has already been studied. In this paper, we study the generalized Hankel transform obtained after a change of variable for $a=1/r$, $r\in$ $\mathbb{N}$. An invariant subspace of functions rapidly decreasing at infinity is described for it, and a differential-difference operator is found for which the kernel of the generalized Hankel transform is an eigenfunction. On the basis of a new multiplication theorem for the Bessel functions Boubatra–Negzaoui–Sifi, two generalized translation operators are constructed, and their $L^p$-boundedness and positivity are investigated. A simple proof is given for the multiplication theorem. Two convolutions are defined for which Young's theorems are proved. With the help of convolutions, generalized means are defined, for which sufficient conditions for $L^p$-convergence and convergence almost everywhere are proposed. Generalized analogs of the Gauss-Weierstrass, Poisson and Bochner–Riesz means are investigated.

Détail

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

Bibliographie

[1] Ben Saïd S., Kobayashi T., Orsted B., “Laguerre semigroup and Dunkl operators”, Compos. Math., 148:4 (2012), 1265–1336 | DOI | MR | Zbl

[2] Rösler M., “Dunkl operators. Theory and applications”, Orthogonal Polynomials and Special Functions, Lecture Notes in Math., 1817, Springer-Verlag, 2002, 93–135 | DOI | MR

[3] Gorbachev D., Ivanov V., Tikhonov S., “On the kernel of the $(\kappa,a)$-Generalized Fourier transform”, Forum of Mathematics, Sigma, 11 (2023), e72, 25 pp. | DOI | MR | Zbl

[4] Ivanov V. I., “Undeformed generalized Dunkl transform on the line”, Math. Notes, 114:4 (2023), 509–524 | DOI

[5] Gorbachev D. V., Ivanov V. I., Tikhonov S. Yu., “Positive $Lp$-Bounded Dunkl-Type Generalized Translation Operator and Its Applications”, Constr. Approx., 49:3 (2019), 555–605 | DOI | MR | Zbl

[6] Boubatra M. A., Negzaoui S., Sifi M., “A new product formula involving Bessel functions”, Integral Transforms Spec. Funct., 33:3 (2022), 247–263 | DOI | MR | Zbl

[7] Bateman H., Erdelyi A., Higher Transcendental Functions, v. II, McGraw Hill Book Company, New York, 1953, 414 pp. | MR

[8] Gorbachev D., Ivanov V., Tikhonov S., “Uncertainty principles for eventually constant sign bandlimited functions”, SIAM Journal on Mathematical Analysis, 52:5 (2020), 4751–4782 | DOI | MR | Zbl

[9] Watson G. N., A Treatise on the Theory of Bessel Functions, Cambridge University Press, Cambridge, 1995, 804 pp. | MR | Zbl

[10] Mejjaoli H., “Deformed Stockwell transform and applications on the reproducing kernel theory”, Int. J. Reprod. Kernels, 1:1 (2022), 1–39 | MR

[11] Mejjaoli H., Trimèche K., “Localization Operators and Scalogram Associated with the Deformed Hankel Wavelet Transform”, Mediterr. J. Math., 20:3 (2023), 186 | DOI | MR | Zbl

[12] Platonov S. S., “Bessel harmonic analysis and approximation of functions on the half-line”, Izv. Math., 71:5 (2007), 1001–1048 | DOI | DOI | MR | Zbl

[13] Thangavelu S., Xu Y., “Convolution operator and maximal function for Dunkl transform”, J. d'Analyse. Math., 97 (2005), 25–55 | DOI | MR

[14] Grafacos L., Classical Fourier Analysis, Graduate Texts in Mathematics, 249, Springer Science+Business Media, LLC, New York, 2008, 489 pp. | DOI | MR

[15] Ben Saïd S., Negzaoui S., “Norm inequalities for maximal operators”, Journal of Inequalities and Applications, 2022, 134 | DOI | MR

[16] Chertova D. V., “Jackson's theorems in space with power weight”, Bulletin of Tula State University. Natural Sciences, 2009, no. 3, 100–116 (In Russ.)

[17] Szegö G., Orthogonal polynomials, Amer. Math. Soc., New York, 1959, 440 pp. | MR | Zbl

[18] Gorbachev D. V., Ivanov V. I., Lectures on quadrature formulas and their application in extremal problems, Tula State University, Tula, 2022, 196 pp. (In Russ.)