The Weinstein Wavelet Transform on Banach-Space-Valued Function Space

The Weinstein wavelet transform $\mathcal{W}_{\psi}\phi$ is developed for functions $\phi$ belonging to the space $H(A)$ of Banach-space-valued smooth functions, with respect to a mother wavelet $\psi$ whose Weinstein transform$\mathcal{F}_{w}\psi$ lies in $L^{2}(\mathbb{R}^{n+1}_{+})$.A rigorous formulation of the transform is presented and several fundamental properties are established.An $L^{2}$-estimate for the product of two Weinstein wavelet transforms is obtained by examining the relation between the Weinstein wavelet transform and the Weinstein-Hausdorff operator within the context of Weinstein convolution theory.As an application, the degenerate Weinstein heat equation$\partial_t u(x,t)=\Delta^{n}_{W,\beta}u(x,t)$ on $\mathbb{R}^{n+1}_{+}$ issolved using the Weinstein transform technique, yielding an explicit integralrepresentation of the solution in terms of the initial data.