Customer Reviews

Interpolation and Approximation with Splines and Fractals

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The book is aimed at students. It can be used in a course for beginning graduate students, and well motivated undergraduates. It has been tried in courses taught by the author, and it includes a nice selection of pedagogical exercises.

While its focus is two classical topics, interpolation and spline approximation, the tools involved are of a more recent vintage: fractals, iterated function systems, and numerical harmonic analysis. Interpolations and splines are central areas of numerical analysis, but the newer tools involve functional analysis, some operator theory, and scale similarities often used in the study of wavelets.

The beginning of the book (about a third) offers friendly tutorials in these areas. The rest develops the newer topics in a systematic fashion. It should be of use both to students and specialists alike.
There are several attractive features: historical footnotes sprinkled throughout the book, lots of beautifully designed figures, many well chosen and easy to follow examples, and suggested student projects.

Each chapter includes the fundamentals, and traced back to core principles, divided differences in numerical analysis, approximation with polynomials, with Fourier series, with special functions, and with recursive algorithms based on scaling, for example, wavelets, multiresolutions, large but sparse matrices, and iterated function system approximations. The chapters in the second half of the book cover topics prominent in current research: fractals (ch 4), fractal functions (ch 5), fractal surfaces (ch 6), and super fractals (ch 7). The latter involve variable fractal sets and fractal measures, and their dimensions. This in turn involves graph theory, code trees, and random fractal measures.

The author did a great job in making available to students a set of fundamental topics at the cross roads of numerical analysis, functional analysis, fractals, and numerical harmonic analysis. It should also be of interest to students and others from neighboring sciences, computer science, physics, and engineering (image processing).