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

By Ole Christensen

This concisely written booklet offers an easy advent to a classical quarter of mathematics—approximation theory—in a manner that clearly results in the fashionable box of wavelets. The exposition, pushed through rules instead of technical info and proofs, demonstrates the dynamic nature of arithmetic and the impact of classical disciplines on many components of contemporary arithmetic and functions.

Key positive factors and themes:

* Description of wavelets in phrases instead of mathematical symbols

* basic advent to approximation utilizing polynomials (Weierstrass’ and Taylor’s theorems)

* creation 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, similar to the FBI’s use of wavelets to shop fingerprints

* minimum must haves: common calculus

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

Approximation thought: From Taylor Polynomials to Wavelets could 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 historical past fabric pointing towards complicated literature and study themes in natural and utilized harmonic research and comparable areas.

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Additional resources for Approximation Theory: From Taylor Polynomials to Wavelets

Sample text

A reader who checks the proof will observe that it does not matter that the sum appearing in the integral test starts with n = 1; the same argument works starting with any positive integer-value of n. 1 starting with x = p. 2 Assume that the function f : [1,00[-t [O,oo[ is continuous and decreasing, at least from x = p for some pEN. Then, assuming that f(x)dx has a finite limit as t -t 00, I; N 00 ~ fen) - ~ fen) for all integers N ~ [+1 f(x)dx + feN + 00 5 1) p. 6) with p = N +1. 3 Let a ~ 1, and consider the function f(x) 1 = -, xa x E [1,00[.

2 ~2 -t ~2 a n +1 1 = {n+ an 2n+1 n n n2 for n -t Thus the quotient test shows that the series is convergent. 2 00. o Estimating the sum of an infinite series The definition of an infinite series involves the sum of the series. There exist several ways to check whether a series is convergent or not, but only in relatively rare cases is it possible to find the exact sum of an infinite series. This raises the question how one can find approximative values for the infinite sum. 1 Assume that f : [1,00[-t [O,oo[ is continuous and decreasing.

N=O,l, ... , _ f(x) - n 2: -f(n)(o) , - x , x Ej- p, pl· n. 4 Power series Proof: First, f(O) 33 = ao. 19), f(k)(O) = ak . , kEN. 1. , only one choice of the coefficients an is possible. In Chapter 5 we will see examples of other types of series representations, where there are many possible ways to choose the coefficients. As promised, we now show that not every arbitrarily often differentiable function f : JR -t JR has a power series representation. 12 Consider a function f : JR -t JR with the following properties: (i) f(x) = 1 for all x E [-1,1].