Download Approximation Theory: From Taylor Polynomials to Wavelets by Ole Christensen PDF

By Ole Christensen

This concisely written booklet offers an straight forward creation to a classical region of mathematics—approximation theory—in a manner that certainly results in the trendy box of wavelets. The exposition, pushed by way of rules instead of technical info and proofs, demonstrates the dynamic nature of arithmetic and the effect 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

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

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

* advent to Fourier analysis

* a number of classical, illustrative examples and constructions

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

* minimum must haves: user-friendly calculus

* workouts that could be utilized in undergraduate and graduate classes on limitless sequence and Fourier series

Approximation idea: From Taylor Polynomials to Wavelets can be a great 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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Additional info for Approximation Theory: From Taylor Polynomials to Wavelets

Example text

7]. 5 General infinite sums of functions As we have seen, a power series consists of infinite sums of terms anx n , n E N. Since not all functions have a representation of this type, it is natural also to consider infinite sums of other types of "simple" functions. More generally, we will consider a family of functions 10,11, h, ... with the same domain of definition I, and attempt to define the function = lo{x) + 11 (x) + h{x) + ... + In(x) + ... = L: In{x). 23) n=O Here it is of course important to know for which values of x the expression for I{x) makes sense: this is the case exactly for the x E I for which the series 01 numbers 2:~=0 In{x) is convergent.

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, ...

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, ... such that f(x) = 2:~=o anxn; • S sends the coefficients ao, al , ... ; • n receives the coefficients ao, al,"'; • n reconstructs the signal by multiplying the coefficients an by xn and forming the infinite series f(x) = 2:~=o anxn. In practice there are a few more steps not described here.

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