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A Whittaker-Shannon-Kotel’nikov sampling theorem related to the Dunkl transform on the real line is proved. To this end we state, in terms of Bessel functions, an orthonormal system which is complete in L((−1, 1), |x| dx). This orthonormal system is a generalization of the classical exponential system defining Fourier series.

Let Jμ denote the Bessel function of order μ. The functions xJα+β+2n+1(x 1/2), n = 0, 1, 2, . . . , form an orthogonal system in L2((0,∞), xα+βdx) when α+ β > −1. In this paper we analyze the range of p, α and β for which the Fourier series with respect to this system converges in the Lp((0,∞), xαdx)-norm. Also, we describe the space in which the span of… (More)

We prove two-weight norm inequalities for Cesàro means of generalized Hermite polynomial series and for the supremum of these means. A result about weak boundedness and an almost everywhere convergence result are also obtained.

- Óscar Ciaurri, Luis M. Navas, Francisco J. Ruiz, Juan Luis Varona
- The American Mathematical Monthly
- 2015

- Óscar Ciaurri
- The American Mathematical Monthly
- 2003

Let Jμ denote the Bessel function of order μ. The functions x−α−1Jα+2n+1(x), n = 0, 1, 2, . . . , form an orthogonal system in the space L2((0,∞), x2α+1dx) when α > −1. In this paper we prove that the Fourier series associated to this system is of restricted weak type for the endpoints of the interval of mean convergence, while it is not of weak type if α ≥… (More)

- Ioan Tomescu, Kee-Wai Lau, +14 authors Allen Stenger
- The American Mathematical Monthly
- 2000

- Óscar Ciaurri, Luz Roncal
- Journal of Approximation Theory
- 2013

In this paper, we develop a thorough analysis of the boundedness properties of the maximal operator for the Bochner-Riesz means related to the Fourier-Bessel expansions. For this operator, we study weighted and unweighted inequalities in the spaces Lp((0, 1), x2ν+1 dx). Moreover, weak and restricted weak type inequalities are obtained for the critical… (More)

- Óscar Ciaurri
- Journal of Approximation Theory
- 2004

- Óscar Ciaurri, Carlos Lizama, Luz Roncal, Juan Luis Varona
- Appl. Math. Lett.
- 2015

We relate the fractional powers of the discrete Laplacian with a standard time-fractional derivative in the sense of Liouville by encoding the iterative nature of the discrete operator through a time-fractional memory term.