Interpolation and Approximation with Splines and Fractals [Hardcover]

Peter Massopust (Author)

Book Description

Publication Date: January 14, 2010 | ISBN-10: 0195336542 | ISBN-13: 978-0195336542

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.

Editorial Reviews

Review

"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

About 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.

Product Details

• Hardcover: 336 pages
• Publisher: Oxford University Press, USA (January 14, 2010)
• Language: English
• ISBN-10: 0195336542
• ISBN-13: 978-0195336542
• Product Dimensions: 9.2 x 6.1 x 0.9 inches
• Shipping Weight: 1.3 pounds (View shipping rates and policies)
• Average Customer Review: 5.0 out of 5 stars  See all reviews (1 customer review)
• Amazon Best Sellers Rank: #1,522,767 in Books (See Top 100 in Books)

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Customer Reviews

Most Helpful Customer Reviews

5.0 out of 5 stars Topics in analysis, and presented for students, November 21, 2010

By 

Palle E T Jorgensen "Palle Jorgensen" (Iowa City, Iowa United States) - See all my reviews
(VINE VOICE)    (REAL NAME)   

This review is from: Interpolation and Approximation with Splines and Fractals (Hardcover)

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).