Select Page

Fourier Series

Orthogonal Polynomials and Generalized Fourier Series

Page 1
Problem 1
Page 2
Problems 2-5

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:

${\sum\limits_{n = 0}^\infty {{c_n}{p_n}\left( x \right)} \text{ = }}\kern0pt {\begin{cases} f\left( x \right), \;\text{if}\,f\left( x \right)\,\text{is continuous} \\ \frac{{f\left( {x – 0} \right) + f\left( {x + 0} \right)}}{2}, \;\text{at a jump discontinuity} \end{cases}}$

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

Hermite Polynomials

Hermite Polynomials $${H_n}\left( x \right) =$$ $${\left( { – 1} \right)^n}{e^{{x^2}}}{\large\frac{{{d^n}}}{{d{x^n}}}\normalsize} {e^{ – {x^2}}}$$ 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 $${\large\frac{1}{{\sqrt {2\pi } }}\normalsize} {e^{ – \frac{{{x^2}}}{2}}}$$ is the probability density function for the normal distribution.

Laguerre Polynomials

Laguerre polynomials $${L_n}\left( x \right) =$$ $${\large\frac{{{e^x}}}{{n!}}\normalsize} {\large\frac{{{d^n}\left( {{x^n}{e^{ – x}}} \right)}}{{d{x^n}}}\normalsize},$$ $$n = 0,1,2,3, \ldots$$ 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 $${P_n}\left( x \right) =$$ $${\large\frac{1}{{{{2^n}n!}}\normalsize} {\large\frac{{{d^n}{{\left( {{x^2} – 1} \right)}^n}}} {d{x^n}}}\normalsize},$$ $$n = 0,1,2,3, \ldots$$ 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 $${\large\frac{1}{{\sqrt {1 – {x^2}} }}\normalsize} :$$

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

Solved Problems

Click on problem description to see solution.

✓Example 1

Show that the set of functions

${1,\cos x,\sin x,\cos 2x,\sin 2x, \ldots ,\;}\kern-0.3pt{\cos mx,\sin mx, \ldots }$

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 3

Find the Fourier-Laguerre series expansion of the power function $$f\left( x \right) = {x^p},$$ $$p \ge 1.$$

✓Example 4

Find the Fourier-Legendre series expansion of the step function

${f\left( x \right) }= {\begin{cases} 0, & -1 \lt x \lt 0 \\ 1, & 0 \lt x \lt 1 \end{cases}.}$

✓Example 5

Find the Fourier-Chebyshev series expansion of the function $$f\left( x \right) = {x^3}$$ on the interval $$\left[ { – 1,1} \right].$$

Example 1.

Show that the set of functions

${1,\cos x,\sin x,\cos 2x,\sin 2x, \ldots ,\;}\kern-0.3pt{\cos mx,\sin mx, \ldots }$

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

Solution.

We evaluate the integrals

${{I_1} = \int\limits_{ – \pi }^\pi {\sin mx\sin nxdx} ,\;\;\;}\kern-0.3pt {{I_2} = \int\limits_{ – \pi }^\pi {\cos mx\cos nxdx} ,\;\;\;}\kern-0.3pt {{I_3} = \int\limits_{ – \pi }^\pi {\sin mx\cos nxdx} .}$

The first integral is

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

For $$m \ne n,$$

${{I_1} \text{ = }}\kern0pt {\frac{{\sin \left( {m – n} \right)\pi }}{{m – n}} – \frac{{\sin \left( {m + n} \right)\pi }}{{m + n}} }={ 0.}$

For $$m = n$$, we obtain

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

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

${1,\cos x,\sin x,\cos 2x,\sin 2x, \ldots ,\;}\kern-0.3pt{\cos mx,\sin mx, \ldots }$

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

Page 1
Problem 1
Page 2
Problems 2-5