Interpolation and Approximation with Splines and Fractals
This textbook is intended to supplement the classical theory of uni- and multivariate splines and their approximation and interpolation properties with those of fractals, fractal functions, and fractal surfaces. This synthesis will complement currently required courses dealing with these topics and expose the prospective reader to some new and deep relationships. In addition to providing a classical introduction to the main issues involving approximation and interpolation with uni- and multivariate splines, cardinal and exponential splines, and their connection to wavelets and multiscale analysis, which comprises the first half of the book, the second half will describe fractals, fractal functions and fractal surfaces, and their properties. This also includes the new burgeoning theory of superfractals and superfractal functions. The theory of splines is well-established but the relationship to fractal functions is novel. Throughout the book, connections between these two apparently different areas will be exposed and presented. In this way, more options are given to the prospective reader who will encounter complex approximation and interpolation problems in real-world modeling. Numerous examples, figures, and exercises accompany the material.
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The General Interpolation and Approximation Problem
Interpolation in W s 1
7 other sections not shown
affine fractal function affine fractal surfaces associated attractor B-splines Banach space basis best linear approximation Borel measure box dimension box splines called cardinal spline Ck[a code space code tree coefficients compact metric space complete metric space compute consider continuous function defined Definition denote divided differences element encouraged to verify Example exists exponential spline Figure Fourier transform fractal interpolation functions fractal measure fractal sets function f given graph f Hilbert space homeomorphic IFSs implies interpolation problem interpolation set interval iterated function systems knot sequence knot set Lemma mapping Moreover normed linear space obtain partition polynomial interpolation pre-Hilbert space Proof Exercise properties Proposition random fractals RB operator reader is encouraged real numbers Remark scaling factors Sk(X spline space superfractal superIFS Suppose tensor product Theorem theory truncated power function unique fixed point V-variable fractal vector verify this statement wavelet