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Ole Christensen's Approximation Theory: From Taylor Polynomials to Wavelets PDF

By Ole Christensen

ISBN-10: 0817636005

ISBN-13: 9780817636005

ISBN-10: 0817644482

ISBN-13: 9780817644482

This concisely written e-book supplies an easy creation to a classical sector of mathematics—approximation theory—in a fashion that obviously ends up in the fashionable box of wavelets. The exposition, pushed via principles instead of technical information and proofs, demonstrates the dynamic nature of arithmetic and the effect of classical disciplines on many components of recent arithmetic and purposes.

Key positive factors and subject matters:

* Description of wavelets in phrases instead of mathematical symbols

* undemanding creation to approximation utilizing polynomials (Weierstrass’ and Taylor’s theorems)

* advent to countless sequence, with emphasis on approximation-theoretic aspects

* creation to Fourier analysis

* a variety of classical, illustrative examples and constructions

* dialogue of the position of wavelets in electronic sign processing and information compression, resembling the FBI’s use of wavelets to shop fingerprints

* minimum necessities: hassle-free calculus

* routines that could be utilized in undergraduate and graduate classes on endless sequence and Fourier series

Approximation thought: From Taylor Polynomials to Wavelets might be a very good textbook or self-study reference for college kids and teachers in natural and utilized arithmetic, mathematical physics, and engineering. Readers will locate motivation and history fabric pointing towards complicated literature and learn themes in natural and utilized harmonic research and comparable areas.

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Extra info for Approximation Theory: From Taylor Polynomials to Wavelets

Sample text

If f is an even function, then x I-t f (x) cos nx is even and x I-t f (x) sin nx is odd; if f is odd, then x I-t f(x)cosnx is odd and x I-t f(x)sinnx is even. 1 If f is an even junction, then bn = an = -211f f(x) cosnx dx, n 11' 0 ° for all n, and = 0,1,2, ... If f is odd, then an = 0 for all n, and bn = -211f f(x)sinnx dx, n = 1,2, ... 1 says that the Fourier series for an even function consists of cosine-functions, while for an odd function it consists of sinefunctions. Let us find the Fourier series for a few 211'-periodic functions.

If f is an even function, then x I-t f (x) cos nx is even and x I-t f (x) sin nx is odd; if f is odd, then x I-t f(x)cosnx is odd and x I-t f(x)sinnx is even. 1 If f is an even junction, then bn = an = -211f f(x) cosnx dx, n 11' 0 ° for all n, and = 0,1,2, ... If f is odd, then an = 0 for all n, and bn = -211f f(x)sinnx dx, n = 1,2, ... 1 says that the Fourier series for an even function consists of cosine-functions, while for an odd function it consists of sinefunctions. Let us find the Fourier series for a few 211'-periodic functions.

7, which shows the partial sum x3 Ss(x) = x - 3! ; this is clearly not a 21l"-periodic function. This illustrates that very contraintuitive things might happen for infinite series; we will meet several other 0 such instances later. 17) derived for the exponential function and the sine function, all have the form L:~=o anx n for some coefficients an. In general, a series of the form L:~=o anx n is called a power series, and a function which can be written in the form 00 f(x) = I>n xn n=O for some coefficients an, is said to have a power series representation.

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Approximation Theory: From Taylor Polynomials to Wavelets by Ole Christensen


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