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Approximation Theory: From Taylor Polynomials to Wavelets by Ole Christensen

By Ole Christensen

This concisely written e-book provides an basic creation to a classical quarter of mathematics—approximation theory—in a manner that obviously results in the trendy box of wavelets. The exposition, pushed through principles instead of technical info and proofs, demonstrates the dynamic nature of arithmetic and the effect of classical disciplines on many parts of contemporary arithmetic and purposes.

Key positive aspects and themes:

* Description of wavelets in phrases instead of mathematical symbols

* user-friendly advent to approximation utilizing polynomials (Weierstrass’ and Taylor’s theorems)

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

* creation to Fourier analysis

* a number of classical, illustrative examples and constructions

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

* minimum necessities: hassle-free calculus

* workouts which may 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 students and teachers in natural and utilized arithmetic, mathematical physics, and engineering. Readers will locate motivation and heritage fabric pointing towards complicated literature and learn themes in natural and utilized harmonic research and similar areas.

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Extra info for Approximation Theory: From Taylor Polynomials to Wavelets

Sample text

We obtain oscillations with very high frequencies. The consequence is that the sum function f oscillates so much around each point x that it is not differentiable. 7. 6 Uniform convergence 41 the domain of f, the partial sums of the series converge to f (x). However, as we will explain now, this type of convergence is not sufficient in all situations. 23) is to write a complicated signal f as an infinite sum of simpler functions. , by a certain partial sum. In other words, we need to be able to find N E N such that is sufficiently small, simultaneously for all x in the considered interval.

7 Signal transmission 45 then all information about the function f is stored in the coefficients {an}~o. 27) to find out which signal we are dealing with. Let us explain how this can be used in signal transmission. Assume for example that a sender S wishes to send a picture to a receiver n. For simplification, we assume that the picture is given as a graph of a function f, and that f can be represented as a power series. Then the transmission can be done in the following way: • S finds the coefficients ao, al, ...

26) goes to zero as N ---t 00. D We end this section with a few important results concerning continuity and differentiability of infinite series of functions; the first result below gives conditions for the associated sum function being well defined and continuous. 5 Assume that the functions 11,12, ... •. such that (i) Ifn(x)1 ::; kn, '

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