Approximation Theory: From Taylor Polynomials to Wavelets by Ole Christensen

By Ole Christensen

This concisely written booklet supplies an uncomplicated advent to a classical sector of mathematics—approximation theory—in a fashion that clearly results in the fashionable box of wavelets. The exposition, pushed through principles instead of technical information and proofs, demonstrates the dynamic nature of arithmetic and the impact of classical disciplines on many components of contemporary arithmetic and purposes.

Key beneficial properties and subject matters:

* Description of wavelets in phrases instead of mathematical symbols

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

* creation to endless sequence, with emphasis on approximation-theoretic aspects

* creation to Fourier analysis

* quite a few classical, illustrative examples and constructions

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

* minimum necessities: uncomplicated calculus

* routines which may be utilized in undergraduate and graduate classes on limitless sequence and Fourier series

Approximation idea: From Taylor Polynomials to Wavelets should be an exceptional textbook or self-study reference for college students and teachers in natural and utilized arithmetic, mathematical physics, and engineering. Readers will locate motivation and heritage fabric pointing towards complicated literature and examine issues in natural and utilized harmonic research and similar areas.

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Additional info for Approximation Theory: From Taylor Polynomials to Wavelets

Sample text

2) only introduces Fourier series on the formal level: so far we have not discussed whether the series is convergent. 2. However, already now it is clear why we restrict attention to 211'-periodic functions: all the trigonometric functions appearing in the Fourier series are 211'-periodic, so if we want any pointwise relationship between f and the Fourier series, this assumption is necessary. , f(x) l a +21f f(x)dx, Va E llt = f( -x) for all x, then r f(x)dx 21 f(x)dx, Va> i-a a = i: O. , f(x) = - f( -x) for all x, then f(x)dx = 0, Va> O.

AN with a certain precision, for example, 50 digits. So in reality S will send some numbers and n receives coefficients corresponding to the signal N j(x) = L iinxn. n=O The fact that the numbers an can only be sent with a certain precision is well known in engineering; one speaks about quantification. (x) ::: f(x) - it does not help if S sends a picture of an elephant and n receives something that looks like a giraffe! Sometimes it is unsuitable just to send 46 2 Infinite Series the first N + 1 coefficients ao, ...

1vergent. 1. 2, which shows that for any NEN, 1 N-1 Ln Ln 00 n=1 2 - n=1 [00 1 1 < iN+! 55. 645. 2 only applies to series with positive terms. •. + (_I)n~lbn + ... 8) n=1 where bn 2': 0 for all n E N, or bn ~ 0 for all n E N. The following criterion is useful in order to check convergence for an alternating series. 4 An alternating series for which the sequence {bn}~1 decreases monotonically to zero, is convergent, and for all N E N, 00 N n=l n=1 L) -It-Ibn - 2:) -It-Ibn ~ bN+1. 4, the series f)-It-l~ n=l is convergent.

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