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

The Weinstein wavelet transform Wψφ is developed for functions φ belonging to the space $H(A)$ of Banach-space-valued smooth functions, with respect to a mother wavelet ψ whose Weinstein transformFwψ lies in L²(Rⁿ⁺¹₊).A rigorous formulation of the transform is presented and several fundamental properties are established.An L²-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∂ₜ u(x,t)=ΔⁿW,βu(x,t) on Rⁿ⁺¹₊ issolved using the Weinstein transform technique, yielding an explicit integralrepresentation of the solution in terms of the initial data.