Hermite-Wavelet Transforms of Multivariate Functions on [ 0 , 1 ] d $[0,1]^{d}$
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Hermite-Wavelet Transforms of Multivariate Functions on [0, 1]d Zhihua Zhang1 · Palle Jorgensen2
Received: 16 May 2020 / Accepted: 11 September 2020 / Published online: 16 September 2020 © Springer Nature B.V. 2020
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 Rd 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. Keywords Even-order Hermite interpolation · Periodic wavelet · Decomposition of multivariate functions · Hermite-wavelet transform
1 Introduction In recent decades, wavelet algorithms (based on prescribed systems of discrete operations of scaling and translations) have proved extremely successful when compared to alternative This research is supported by European Commission Horizon 2020’s Flagship Project “ePIcenter”, National Key Science Programme No. 2019QZKK0906 and No. 2015CB953602, and Taishan Distinguished Professorship Fund.
B Z. Zhang 1
School of Mathematics, Shandong University, Jinan, 250100, China
2
Department of Mathematics, University of Iowa, Iowa City, USA
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Z. Zhang, P. Jorgensen
harmonic analysis tools. And they have been picked up by engineers at an impressive rate (especially by electrical and computer engineers); and covering such industrial applications data-mining, machine learning, neural networks, to mention just a few. One reason for this success is that wavelet algorithms allow localization. This contrasts Fourier expansions (in one or more variables) based periodic Fourier series. Obviously, when designing orthogonal Fourier expansions, if one makes a local feature-adjustment of a periodic Fourier frequency function, then (because of periodicity) this will messes up features at distances. As a result, by comparison, efficient wavelet expansions can often be achieved with only a small number of significant terms, when contrasted with the cor
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