Interpolation and Approximation with Splines and fractals

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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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Editorial Reviews

From the Publisher
"A very valuable addition to the existing literature on the subject and the exposition takes the reader to very recent topics as Besov or Triebel-Lizorkin spaces, as well as their use in the theory of splines and fractals. Highly recommended for students or researchers working in applied fields who need to refresh their tools with current ones."—Libertas Mathematica

"A useful book, well-written, and covering interesting areas of research in numerical analysis...A comprehensive volume. It is sufficiently introductory to be well readable for the non-expert while at the same time giving many interesting and useful results for the connaisseur."—Zentralblatt Math

"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. The author does a great job in making available to students a set of fundamental topics at the crossroads of numerical analysis, functional analysis, fractals, and numerical harmonic analysis." — Palle E. T. Jorgensen, MathSciNet

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Product Details

  • ISBN-13: 9780195336542
  • Publisher: Oxford University Press, USA
  • Publication date: 1/14/2010
  • Pages: 336
  • Product dimensions: 6.30 (w) x 9.30 (h) x 0.90 (d)

Meet the Author

Peter Massopust holds a Master of Science in physics and a Ph.D. in applied mathematics. Dr. Massopust is best known for his work in fractal geometry, in particular fractal functions and fractal surfaces, and wavelet theory. His current research interests focus on complex splines and wavelets, and their applications to signal and image processing. He is currently a Senior Research Scientist on the Marie Curie Excellence in Research Team MAMEBIA.

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Table of Contents

1. The General Interpolation and Approximation Problem
2. Splines
3. Interpolation in Rs, s > 1
4. Fractals
5. Fractal Functions
6. Fractal Surfaces
7. Superfractals
8. Superfractal Functions

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