Discrete Harmonic Analysis Associated with Jacobi Expansions II: the Riesz Transform

Discrete Harmonic Analysis associated with Jacobi expansions III: the Littlewood-Paley-Stein $g_{k}$-functions and the Laplace type multipliers

“…In fact, the spectrum of J (α,β) is the interval [−1, 1], so that the spectrum of −J (α,β) is [0, 2]. This paper continues in a natural way the study of harmonic analysis associated with J (α,β) of [1] and [3]. In [1] we carried out an exhaustive analysis of the heat semigroup for J (α,β) and in [3] we investigated the Riesz transform.…”

“…This paper continues in a natural way the study of harmonic analysis associated with J (α,β) of [1] and [3]. In [1] we carried out an exhaustive analysis of the heat semigroup for J (α,β) and in [3] we investigated the Riesz transform. The main aim of this paper is to study another classical operator in harmonic analysis, the Littlewood-Paley-Stein g k -function.…”

“…We begin by setting some aspects of our context as in the previous papers [1,3]. For α, β > −1, we take the sequences {a For each sequence {f (n)} n≥0 , we define the operator {J (α,β) f (n)} n≥0 by the relation…”