Multivariate Polysplines: Applications to Numerical and Wavelet AnalysisMultivariate polysplines are a new mathematical technique that has arisen from a synthesis of approximation theory and the theory of partial differential equations. It is an invaluable means to interpolate practical data with smooth functions. Multivariate polysplines have applications in the design of surfaces and "smoothing" that are essential in computer aided geometric design (CAGD and CAD/CAM systems), geophysics, magnetism, geodesy, geography, wavelet analysis and signal and image processing. In many cases involving practical data in these areas, polysplines are proving more effective than wellestablished methods, such as kKriging, radial basis functions, thin plate splines and minimum curvature.

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Multivariate Polysplines: Applications to Numerical and Wavelet Analysis Ognyan Kounchev No preview available  2001 
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analog annulus arbitrary assume ball B(0 belongs biharmonic polysplines boundary conditions boundary operators cardinal Lsplines cardinal polysplines Chapter Chebyshev splines coefficients compact support consider constant convergence Corollary defined Definition differential operator Dirichlet problem divided difference domain elliptic BVP elliptic operator equation estimate Euler polynomials exists following equalities Fourier series Fourier transform Fredholm operator function f given Green formula H¨older spaces harmonic functions Hence holds implies inequality integer interface interpolation conditions interpolation polysplines interval Laplace operator Laplace series Lemma Let the function Let us note logr manifold multivariate nonordered vector norm notation polyharmonic functions polynomial splines polyspline h(x polysplines on annuli polysplines on strips Proposition prove Qz+1 real number Remark representation Riesz basis satisfies Schoenberg Section smoothness Sobolev space solution sphere spherical harmonics TBspline Theorem uniqueness variable wavelet analysis zero
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Page 21  When using the partial variables, we have to use the additional system of equations so that the number of unknowns is equal to the number of equations. The...