Advanced Mathematical Methods for Scientists and Engineers Episode 4 Part 4 docx

.} be a set of real, square integrable functions that are orthonormal with respect to the weightingfunction σx on the interval [a, b].. .} is a set of real, square integrable functions t

In terms of real functions, this is

= 1π

1

n sin(nx).

25.9 Least Squares Fit to a Function and Completeness

Let {φ1, φ2, φ3, } be a set of real, square integrable functions that are orthonormal with respect to the weightingfunction σ(x) on the interval [a, b] That is,

hφn|σ|φmi = δnm.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) may or may not have a uniformly convergent expansion in the orthonormal functions

We would like to choose the αnso 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 theweighting function times the square of the deviation

The “best” fit is found by choosing the αn that minimize E Let cn be the Fourier coefficients of f (x).

cn= hφn|σ|f i

we expand the integral for E

E(α) =

Z b a

f

+

... λn and the eigenfunctions as φn for n ∈ Z+ For the moment we assume that

λ = is not an eigenvalue and that the eigenfunctions are real-valued We expand the function... class="text_page_counter">Trang 14< /span>

= 0

for all elements... class="text_page_counter">Trang 24< /span>

Most of the differential equations that we study in this book are second order, formally self-adjoint, with real-valuedcoefficient