Calculus

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