Runge-Walsh-wavelet approximation for the Helmholtz equation

Metaharmonic wavelets are introduced for constructing the solution of the H elmholtz equation (reduced wave equation) corresponding to Dirichlet's or N eumann's boundary values on a closed surface Sigma in three-dimensional Euc lidean space R-3. A consistent scale continuous and scale discrete wavelet approach leading to exact reconstruction formulas is considered in more det ail. A scale discrete version of multiresolution is described for potential functions metaharmonic outside the closed surface and satisfying the radia tion condition at infinity. Moreover, we discuss fully discrete wavelet rep resentations of band-limited metaharmonic potentials. Finally, a decomposit ion and reconstruction (pyramid) scheme for economical numerical implementa tion is presented for Runge-wavelet approximation. (C) 1999 Academic Press.