Calculus

Fourier Series

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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:

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

    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 a problem to see the 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