tailieunhanh - Advanced Mathematical Methods for Scientists and Engineers Episode 4 Part 4

Tham khảo tài liệu 'advanced mathematical methods for scientists and engineers episode 4 part 4', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | In terms of real functions this is 1 1 - l n -------- cos nx 1 sin nx n m n -tt 2 1 - 1 n ---------- sin nx n in n 1 sign x sin nx . n n n 1 oddn Least Squares Fit to a Function and Completeness Let 1 2 3 . be a set of real square integrable functions that are orthonormal with respect to the weighting function a x on the interval a b . That is 0n 0m ồnm. Let f x be some square integrable function defined on the same interval. We would like to approximate the function f x with a finite orthonormal series. f x 52 an n x n 1 f x may or may not have a uniformly convergent expansion in the orthonormal functions. We would like to choose the an so that we get the best possible approximation to f x . The most common measure of how well a series approximates a function is the least squares measure. The error is defined as the integral of the weighting function times the square of the deviation. E a xd f x 52 n n x I dx Ja n 1 1294 The best fit is found by choosing the an that minimize E. Let cn be the Fourier coefficients of f x . cn ộn ơ f we expand the integral for E. E a a xH f x an n xH dx n 1 f anộn n 1 f an ộn n 1 N N ỵ an n ơ f ỵ ỵ anộn ơ n 1 n 1 N y Onộn f Hf fa f fa f 2 2 an ộn ơ f y y n m ện ơ ệm n 1 n 1 m 1 2 y ancn 1 a2i n 1 n 1 1 ơ f an cn 2 c n n 1 n 1 Each term involving an in non-negative and is minimized for an cn. The Fourier coefficients give the least squares approximation to a function. The least squares fit to f x is thus f x n ơ f ộn x -n 1 .