Dunkl–Weinstein Multiplier Operators and Applications to Reproducing Kernel Theory

In this paper, we have interested on the Dunkl–Weinstein multiplier operators \(\mathscr {M}_{k,\beta ,a}\) ${\mathcal{M}}_{k,\beta ,a}$ on the Dunkl–Weinstein-type Paley–Wiener space \(\mathscr {H}_s\) ${\mathcal{H}}_s$ . We give for these operators an application to the reproducing kernel theory; and we deduce for them best approximation on the Paley–Wiener space \(\mathscr {H}_s\) ${\mathcal{H}}_s$ . Finally, we give two applications in two particular cases, the first is a Weinstein case \((k=0)\) $(k=0)$ , and the second is the Dunkl case when \(G=\mathbb {Z}_2^d\) $G={\mathbb{Z}}_2^d$ . More precisely, we write the solution of the Tikhonov regularization problem in these cases as a convolution product of two functions.