Hermite-Wavelet Transforms of Multivariate Functions on $[0,1]^{d}$

Abstract

For a $d$-dimensional smooth target function $f$ on the cube $[0,1]^{d}$, we propose the Hermite-wavelet transform to overcome boundary effects. In details, we first give a decomposition of $f$ based on its even-order Hermite interpolation on sections of the cube $[0,1]^{d}$: $f=G+r$, where $G$ is a combination of polynomials and the restriction of derivative functions on some part of the boundary of the cube, and $r$ can be extended to a smooth periodic function on $\mathbb{R}^{d}$ after odd extension. Noticing that the restriction of derivative functions on some part of the boundary of the cube has less number of free variables, using similar decomposition again and again, finally the multivariate smooth function $f$ on the cube $[0,1]^{d}$ can be decomposed into a combination of smooth periodic functions and polynomials whose coefficients are completely determined by partial derivatives of $f$ at vertices of the cube $[0,1]^{d}$. After that, we expand all smooth periodic functions into periodic wavelet series. Since these periodic functions have the same smoothness as the target function $f$, the corresponding periodic wavelet coefficients decay fast. Hence the $d$-dimensional smooth target function $f$ on the cube $[0,1]^{d}$ can be reconstructed by values of partial derivatives of $f$ at all vertices of $[0,1]^{d}$ and few periodic wavelet coefficients with small error.