# Orthogonal Polynomials and Generalized Fourier Series

## Orthogonal Polynomials

Two polynomials $${p\left( x \right)}$$ and $${q\left( x \right)}$$ defined on the interval $$\left[ {a,b} \right]$$ are orthogonal if

$\int\limits_a^b {p\left( x \right)q\left( x \right)w\left( x \right)dx} = 0,$

where $${w\left( x \right)}$$ is a nonnegative weight function.

A polynomial sequence $${p_n}\left( x \right),$$ $$n = 0,1,2, \ldots ,$$ where $$n$$ is the degree of $${p_n}\left( x \right),$$ is said to be a sequence of orthogonal polynomials if

$\int\limits_a^b {{p_m}\left( x \right){p_n}\left( x \right)w\left( x \right)dx} = {c_n}{\delta _{mn}},$

where $${c_n}$$ are given constants and $${\delta _{mn}}$$ is the Kronecker delta.

## Generalized Fourier Series

A generalized Fourier series is a series expansion of a function based on a system of orthogonal polynomials. By using this orthogonality, a piecewise continuous function $${f\left( x \right)}$$ can be expressed in the form of generalized Fourier series expansion:

$\sum\limits_{n = 0}^\infty {{c_n}{p_n}\left( x \right)} = \begin{cases} f\left( x \right), \;\text{if}\,f\left( x \right)\,\text{is continuous} \\[0.5em] \frac{{f\left( {x - 0} \right) + f\left( {x + 0} \right)}}{2}, \;\text{at a jump discontinuity} \end{cases}.$

We consider $$4$$ types of orthogonal polynomials: Hermite, Laguerre, Legendre and Chebyshev polynomials.

## Hermite Polynomials

Hermite Polynomials $${H_n}\left( x \right) = {\left( { - 1} \right)^n}{e^{{x^2}}}{\frac{{{d^n}}}{{d{x^n}}}} {e^{ - {x^2}}}$$ are orthogonal on the interval $$\left( { - \infty ,\infty } \right)$$ with respect to the weight function $${e^{ - {x^2}}}:$$

$\int\limits_{ - \infty }^\infty {{e^{ - {x^2}}}{H_m}\left( x \right){H_n}\left( x \right)dx} = \begin{cases} 0, & m \ne n \\ {2^n}n!\sqrt \pi, & m = n \end{cases}.$

An alternative definition uses the weight function $${e^{ - \frac{{{x^2}}}{2}}}.$$ This convention is sometimes preferred in probability theory because $${\frac{1}{{\sqrt {2\pi } }}} {e^{ - \frac{{{x^2}}}{2}}}$$ is the probability density function for the normal distribution.

## Laguerre Polynomials

Laguerre polynomials $${L_n}\left( x \right) = {\frac{{{e^x}}}{{n!}}} {\frac{{{d^n}\left( {{x^n}{e^{ - x}}} \right)}}{{d{x^n}}}},$$ $$n = 0,1,2,3, \ldots$$ are orthogonal on the interval $$\left( {0,\infty } \right)$$ with the weight function $${{e^{ - x}}}:$$

$\int\limits_0^\infty {{e^{ - x}}{L_m}\left( x \right){L_n}\left( x \right)dx} = \begin{cases} 0, & m \ne n \\ 1, & m = n \end{cases}.$

## Legendre Polynomials

Legendre Polynomials $${P_n}\left( x \right) = {\frac{1}{{{{2^n}n!}}} {\frac{{{d^n}{{\left( {{x^2} - 1} \right)}^n}}} {d{x^n}}}},$$ $$n = 0,1,2,3, \ldots$$ are orthogonal on the interval $$\left[ {-1,1} \right]:$$

$\int\limits_{ - 1}^1 {{P_m}\left( x \right){P_n}\left( x \right)dx} = \begin{cases} 0, & m \ne n \\ \frac{2}{{2n + 1}}, & m = n \end{cases}.$

## Chebyshev Polynomials

Chebyshev Polynomials of the first kind $${T_n}\left( x \right) = \cos \left( {n\arccos x} \right)$$ are orthogonal on the interval $$\left[ {-1,1} \right]$$ with the weight function $${\frac{1}{{\sqrt {1 - {x^2}} }}} :$$

$\int\limits_{ - 1}^1 {\frac{{{T_m}\left( x \right){T_n}\left( x \right)}}{{\sqrt {1 - {x^2}} }}dx} = \begin{cases} 0, & m \ne n \\ \pi, & m = n = 0 \\ \frac{\pi }{2}, & m = n \ne 0 \end{cases}.$

## Solved Problems

Click or tap a problem to see the solution.

### Example 1

Show that the set of functions

$1,\cos x,\sin x,\cos 2x,\sin 2x, \ldots ,\cos mx,\sin mx, \ldots$

is orthogonal on the interval $$\left[ { - \pi ,\pi } \right].$$

### Example 2

Find the Fourier-Hermite series expansion of the quadratic function $f\left( x \right) = A{x^2} + Bx + C.$

### Example 1.

Show that the set of functions

$1,\cos x,\sin x,\cos 2x,\sin 2x, \ldots ,\cos mx,\sin mx, \ldots$

is orthogonal on the interval $$\left[ { - \pi ,\pi } \right].$$

Solution.

We evaluate the integrals

${I_1} = \int\limits_{ - \pi }^\pi {\sin mx\sin nxdx} ,\;\; {I_2} = \int\limits_{ - \pi }^\pi {\cos mx\cos nxdx} ,\;\; {I_3} = \int\limits_{ - \pi }^\pi {\sin mx\cos nxdx} .$

The first integral is

${I_1} = \int\limits_{ - \pi }^\pi {\sin mx\sin nxdx} = \frac{1}{2}\int\limits_{ - \pi }^\pi {\left[ {\cos \left( {mx - nx} \right) - \cos \left( {mx + nx} \right)} \right]dx} = \frac{1}{2}\int\limits_{ - \pi }^\pi {\left[ {\cos \left( {m - n} \right)x - \cos \left( {m + n} \right)x} \right]dx} = \frac{1}{2}\left[ {\left. {\Big( {\frac{{\sin \left( {m - n} \right)x}}{{m - n}} - \frac{{\sin \left( {m + n} \right)x}}{{m + n}}} \Big)} \right|_{ - \pi }^\pi } \right].$

For $$m \ne n,$$

${I_1} = \frac{{\sin \left( {m - n} \right)\pi }}{{m - n}} - \frac{{\sin \left( {m + n} \right)\pi }}{{m + n}} = 0.$

For $$m = n$$, we obtain

${I_1} = \int\limits_{ - \pi }^\pi {{{\sin }^2}xdx} = \frac{1}{2}\int\limits_{ - \pi }^\pi {\left( {1 - \cos 2nx} \right)dx} = \frac{1}{2}\left[ {\left. {\left( {x - \frac{{\sin 2nx}}{{2n}}} \right)} \right|_{ - \pi }^\pi } \right] = \frac{1}{2}\left[ {\pi - \frac{{\sin 2n\pi }}{{2n}} - \left( { - \pi } \right) - \frac{{\sin \left( { - 2n\pi } \right)}}{{2n}}} \right] = \pi .$

Thus,

${I_1} = \int\limits_{ - \pi }^\pi {\sin mx\sin nxdx} = \begin{cases} 0, & m \ne n\\ \pi, & m = n\end{cases}.$

Similarly, we can find that

${I_2} = \int\limits_{ - \pi }^\pi {\cos mx\cos nxdx} = \begin{cases} 0, & m \ne n \\ \pi, & m = n \end{cases},$
${I_3} = \int\limits_{ - \pi }^\pi {\sin mx\cos nxdx} = \begin{cases} 0, & m \ne n \\ \pi, & m = n \end{cases}.$

This means that the set of functions

$1,\cos x,\sin x,\cos 2x,\sin 2x, \ldots ,\cos mx,\sin mx, \ldots$

form the orthogonal system on the interval $$\left[ { - \pi ,\pi } \right].$$

### Example 2.

Find the Fourier-Hermite series expansion of the quadratic function $f\left( x \right) = A{x^2} + Bx + C.$

Solution.

We use the explicit expressions for Hermite polynomials:

${H_0}\left( x \right) = 1,\;\; {H_1}\left( x \right) = 2x,\;\; {H_2}\left( x \right) = 4{x^2} - 2.$

Apply the method of undetermined coefficients.

$A{x^2} + Bx + C = {c_0}{H_0}\left( x \right) + {c_1}{H_1}\left( x \right) + {c_2}{H_2}\left( x \right).$

Substituting the Hermite polynomials and equating the coefficients, we obtain

$A{x^2} + Bx + C = {c_0} \cdot 1 + {c_1} \cdot 2x + {c_2} \cdot \left( {4{x^2} - 2} \right),\;\; \Rightarrow A{x^2} + Bx + C = {c_0} + 2{c_1}x + 4{c_2}{x^2} - 2{c_2},\;\; \Rightarrow \left\{ {\begin{array}{*{20}{l}}{4{c^2} = A}\\{2{c_1} = B}\\{{c_0} - 2{c_2} = C}\end{array}} \right.,\;\; \Rightarrow {c_0} = C + \frac{A}{2},\;\; {c_1} = \frac{B}{2},\;\; {c_2} = \frac{A}{4}.$

Hence, the Fourier-Hermite series expansion of the given function is defined by the expression

$f\left( x \right) = A{x^2} + Bx + C = \left( {C + \frac{A}{2}} \right){H_0}\left( x \right) + \frac{B}{2}{H_1}\left( x \right) + \frac{A}{4}{H_2}\left( x \right).$