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Mathematics LibreTexts

4.4: Orthogonality and Normalization

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Consider the series

a02+n=1[ancos(nπxL)+bnsin(nπxL)],LxL.

This is called a trigonometric series. If the series approximates a function f (as will be discussed) it is called a Fourier series and a and b are the Fourier coefficients of f.

In order for all of this to make sense we first study the functions

{1,cos(nπxL),sin(nπxL)},n=1,2,,

and especially their properties under integration. We find that

LL11dx=2L,

LL1cos(nπxL)dx=0

LL1sin(nπxL)dx=0

LLcos(mπxL)cos(nπxL)dx=12LLcos((m+n)πxL)+cos((mn)πxL)dx={0if nmLif n=m,

LLsin(mπxL)sin(nπxL)dx=12LLcos((mn)πxL)cos((m+n)πxL)dx={0if nmLif n=m,

LLcos(mπxL)sin(nπxL)dx=12LLsin((m+n)πxL)+sin((mn)πxL)dx={0if nmLif n=m,

If we consider these integrals as some kind of inner product between functions (like the standard vector inner product) we see that we could call these functions orthogonal. This is indeed standard practice, where for functions the general definition of inner product takes the form

(f,g)=baw(x)f(x)g(x)dx.

If this is zero we say that the functions f and g are orthogonal on the interval [achar3B.pngb] with weight function w. If this function is 1, as is the case for the trigonometric functions, we just say that the functions are orthogonal on [achar3B.pngb].

The norm of a function is now defined as the square root of the inner-product of a function with itself (again, as in the case of vectors),

f=baw(x)f(x)2dx.

If we define a normalised form of f (like a unit vector) as f/f, we have

ff=baw(x)f(x)2dxf2=baw(x)f(x)2dxf=ff=1.

Exercise 4.4.1

What is the normalised form of {1,cos(nπxL),sin(nπxL)}?

Answer

{12L,(1L)cos(nπxL),(1L)sin(nπxL)}

A set of mutually orthogonal functions that are all normalized is called an orthonormal set.


This page titled 4.4: Orthogonality and Normalization is shared under a CC BY-NC-SA 2.0 license and was authored, remixed, and/or curated by Niels Walet via source content that was edited to the style and standards of the LibreTexts platform.

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