Discrete harmonic analysis associated with Jacobi expansions I: The heat semigroup

A weighted transplantation theorem for Jacobi coefficients

“…In this way, we obtain that Therefore, it is enough to prove that the kernels e,e K γ,δ α,β , e,o K γ,δ α,β , o,e K γ,δ α,β , and o,o K γ,δ α,β satisfy the properties (a) and (b). These facts are consequence of the following propositions (see [2] for similar estimates in other setting).…”

Discrete diffusion semigroups associated with Jacobi-Dunkl and exceptional Jacobi polynomials

“…In the third section, by a general theorem proved in [4], we extend the maximal operator of the diffusion semigroup associated with exceptional Jacobi polynomials to weighted l p spaces. In the last section we deduce a similar theorem for discrete Jacobi-Dunkl semigroup from the corresponding result of [2].…”

“…If the Radon-Nikodym derivative of µ is positive on (−1, 1), by [18,Theorem 4.5.7] (see also [20]) then the recurrence coefficients fulfil the asymptotics lim n→∞ a n = 1 2 and lim n→∞ b n = 0, (2.17) and so −2A = 2(J − I) can be decomposed to the sum of a symmetric and a compact operator, where the symmetric part is just ∆ d ; in other words the rows of −2A tends to the rows of ∆ d . In these cases {W t } t≥0 is called a discrete heat semigroup again, see [4] and [2] in ultraspherical and Jacobi cases, respectively. Similarly to the previous one, a recurrence relation with a d-diagonal matrix generates a difference operator on the right-hand side (possibly more complicated than ∆ d ), and by the previous computations the corresponding diffusion semigroup generates solution to the initial-value problem (2.16).…”

“…Besides the continuous operator semigroup, lately the study of discrete diffusion semigroups has come to the forefront of interest, see e.g. [9,1,2,4].…”

The convergence of discrete Fourier-Jacobi series

“…holds. This problem is the discrete counterpart of (1) and it belongs to the study of the discrete harmonic analysis for Jacobi series developed in [1,2,3] by the authors. In those papers the starting point is a discrete Laplacian defined by the three-term recurrence relation for the Jacobi polynomials.…”