Orthogonal Functions,  Orthogonal Polynomials, and Orthogonal Wavelets series expansions of function 

? Sergey Moiseev 2009  

Kodofon 

Russia 

smoiseev@kodofon.vrn.ru,  snmoiseev@yandex.ru 

As you read this worksheet, you should execute the commands in sequence as you come to them. 

Introduction 

The worksheet includes all the best known continuous orthogonal series expansions in the closed form. It demonstrates the use of Maple to evaluate expansion of a function by Fourier, Hartley, Fourier-Bessel, Orthogonal Rational Tangent, Rectangular, Haar Wavelet, Walsh, Slant, Piece-Linear-Quadratic, Associated Legandre, Orthogonal Rational, Generalized sinc, Sinc, Sinc Wavelet, Jacobi,  Chebyshev first kind, Chebyshev second kind, Gegenbauer, Generalized Laguerre, Laguerre, Hermite, and classical polynomials orthogonal series. Also the worksheet demonstrates how to create new orthonormal basis of functions by using the Gram-Schmidt orthogonalization process by the example of  Slant, and Piece-Linear-Quadratic orthonormal functions creating. 

Initialization of expansion procedures 

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Gram-Schmidt orthogonalization for L2 space 

The GramSchmidL2(f, x = a..b) calling sequence computes a list of orthonormal functions on the interval from a to b by using the Gram-Schmidt orthogonalization process. 

f - the list of algebraic expressions representing the initial linear independant set of functions .
x - the variable name appearing in . 

a, b - endpoints of interval of the orthogonalization. 

It returns list of orthonormal functions (algebraic expressions). 

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Fourier expansion of a function 

The FourierE(f, x = a..b, n) calling sequence computes the Fourier series expansion 

of the expression f with respect to the variable x on the interval from a to b as folows: 

 

f - the algebraic expression representing the function.
x - the variable name appearing in f. 

a, b - endpoints of the interval on which expansion is taken. 

n - the maximal number of expansion terms that correspond to the number of the expansion coefficients 2n+1. 

It returns list of n Fourier expansions of function f(x) for terms of expansion n=1,2,..,n. 

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Cas (cosin and sine)  functions 

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Hartley expansion of a function 

The HartleyE(f, x = a..b, n) calling sequence computes the Hartley series expansion 

of the expression f with respect to the variable x on the interval from a to b as folows: 

 

f - the algebraic expression representing the function.
x - the variable name appearing in f. 

a, b - endpoints of the interval on which expansion is taken. 

n - the maximal number of expansion terms that correspond to the number of the expansion coefficients 2n+1. 

It returns list of n Hartley expansions of function f(x) for terms of expansion n=1,2,..,n. 

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Fourier-Bessel expansion of a function 

The FourierBesselE(f, x = a..b, n, v) calling sequence computes the Fourier-Bessel series expansion of the expression f with respect to the variable x on the interval from a to b 

as folows: 

where is increasing positive zeroes of the BesselJ(v,x) function.
f - the algebraic expression representing the function. 

x - the variable name appearing in f. 

a, b - endpoints of the interval on which expansion is taken. 

n - the maximal number of expansion terms that correspond to the number of the expansion coefficients n. 

v>-1/2 - the order of BesselJ function.          

It returns list of n Fourier-Bessel expansions of function f(x) for terms of expansion 

n=1,2,..,n. 

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Orthogonal Rational Tangent functions 

The Rattan(p, j, x) calling sequence computes the j-th Orthogonal Rational Tangent function. 

p∈ℝ, p≠0 - the form factor. 

j - the function number. 

x - the argument of the function. 

This functions are orthonormal on the interval . 

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Orthogonal Rational Tangent expansion of a function 

The RattanE(f, x=a..b, n, p) calling sequence computes the Orthogonal Rational Tangent series expansion of the expression f with respect to the variable x on the interval from a to b as folows: 

 

f - the algebraic expression representing the function.
x - the variable name appearing in f. 

a, b - endpoints of the interval on which expansion is taken. 

n - the maximal number of expansion terms that correspond to the number of the expansion coefficients 2n. 

p∈ℝ, p≠0 - the form factor. 

It returns list of n Orthogonal Rational Tangent expansions of function f(x) for terms of expansion n=1,2,..,n. 

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Associated Legendre functions expansion of a function 

The LegendreE(f, x = a..b, n, m) calling sequence computes the Associated Legendre series expansion of the expression f with respect to the variable x on the interval from a to b as folows: 

 

where  

f - the algebraic expression representing the function.
x - the variable name appearing in f. 

a, b - endpoints of the interval on which expansion is taken. 

n - the maximal number of expansion terms that correspond to the number of the expansion coefficients n. 

parameters (order) of the Associated Legandre function.         

It returns list of n Associated Legandre expansions of function f(x) for terms of expansion 

n=1,2,..,n. 

If then the procedure returns Legendre polynomials expansion. 

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Jacobi polynomials expansion of a function 

The JacobiE(f, x = a..b, n, ) calling sequence computes the Jacobi polynomials series expansion of the expression f with respect to the variable x on the interval from a to b  

as folows: 

 

where  

f - the algebraic expression representing the function.
x - the variable name appearing in f. 

a, b - endpoints of the interval on which expansion is taken. 

n - the maximal number of expansion terms that correspond to the number of the expansion coefficients n. 

parameters of the Jacobi polynomial.          

It returns list of n Jacobi polynomials expansions of function f(x) for terms of expansion 

n=1,2,..,n. 

If then the procedure returns Legendre polynomials expansion. 

If then the procedure returns Chebyshev first kind polynomials expansion. 

If then the procedure returns Chebyshev second kind polynomials expansion. 

If then the procedure returns Gegenbauer polynomials expansion. 

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Jacobi functions expansion of a function 

The JacobiFE(f, x = a..b, n, ) calling sequence computes the Jacobi functions series expansion of the expression f with respect to the variable x on the interval from a to b  

as folows: 

 

where  

f - the algebraic expression representing the function.
x - the variable name appearing in f. 

a, b - endpoints of the interval on which expansion is taken. 

n - the maximal number of expansion terms that correspond to the number of the expansion coefficients n. 

parameters of the Jacobi polynomial.          

It returns list of n Jacobi functions expansions of function f(x) for terms of expansion 

n=1,2,..,n. 

If then the procedure returns Legendre polynomials expansion. 

If then the procedure returns Chebyshev first kind functions expansion. 

If then the procedure returns Chebyshev second kind functions expansion. 

If then the procedure returns Gegenbauer functions expansion. 

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Rect function 

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Rect expansion of a function 

The RectE(f, x = a..b, n) calling sequence computes the Rect series expansion 

of the expression f with respect to the variable x on the interval from a to b as folows: 

 

f - the algebraic expression representing the function.
x - the variable name appearing in f. 

a, b - endpoints of the interval on which expansion is taken. 

n - the maximal number of expansion terms that correspond to the number of the expansion coefficients n. 

It returns list of n Rect expansions of function f(x) for terms of expansion n=1,2,..,n. 

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Haar functions  (Haar wavelets) 

The Haar(n, j, x) calling sequence computes the j-th Haar function (wavelet) 

of order n () on interval [0,1). 

n - the order of the Haar function (scale of the wavelet) . 

j - the Haar function number (shift of the wavelet) (). 

x - the argument of Haar function. 

This functions are orthonormal on the interval . 

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Haar expansion of a function 

The HaarE(f, x = a..b, n) calling sequence computes the Haar series expansion 

of the expression f with respect to the variable x on the interval from a to b as folows: 

 

f - the algebraic expression representing the function.
x - the variable name appearing in f. 

a, b - endpoints of the interval on which expansion is taken. 

n - the maximal number of "expansion terms" that correspond to the number of the expansion coefficients . 

It returns list of n Haar expansions of function f(x) for "terms of expansion" n=1,2,..,n. 

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Walsh functions 

The Walsh(n, j, x, opt) calling sequence computes the j-th Walsh function 

of order n () on interval [0,1). The set of the Walsh functions are arranged in optional opt order. 

n - the order of the Walsh function . 

j - the Walsh function number (). 

x - the argument of Walsh function.
opt - optional arranged order of the set of Walsh functions. It is used as follows. 

order=Walsh - sequency (or Walsh) order,
order=Paley - dyadic (or Paley) order,
order=Hadamard - natural (or Hadamard) order . 

By default opt  is equal to order=Walsh. 

This functions are orthonormal on the interval . 

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Walsh expansion of a function 

The WalshE(f, x = a..b, n) calling sequence computes the Walsh series expansion 

of the expression f with respect to the variable x on the interval from a to b as folows: 

 

f - the algebraic expression representing the function.
x - the variable name appearing in f. 

a, b - endpoints of the interval on which expansion is taken. 

n - the maximal number of "expansion terms" that correspond to the number of the expansion coefficients . 

It returns list of n Walsh expansions of function f(x) for "terms of expansion" n=1,2,..,n. 

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List of Slant functions 

The Slant(n) calling sequence computes the list of n Slant functions. 

This functions are orthonormal on the interval . 

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Slant expansion of a function 

The SlantE(f, x = a..b, n) calling sequence computes the Slant series expansion of the expression f with respect to the variable x on the interval from a to b as folows: 

 

f - the algebraic expression representing the function.
x - the variable name appearing in f. 

a, b - endpoints of the interval on which expansion is taken. 

n - the maximal number of expansion terms that correspond to the number of the expansion coefficients . 

It returns list of n Slant expansions of function f(x) for terms of expansion n=1,2,..,n. 

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List of Piece-Linear-Quadratic functions 

The PieceLQ(n) calling sequence computes the list of n Piece-Linear-Quadratic functions. 

This functions are orthonormal on the interval . 

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Piece-Linear-Quadratic expansion of a function 

The PieceLQE(f, x = a..b, n) calling sequence computes the Piece-Linear-Quadratic series expansion of the expression f with respect to the variable x on the interval from a to b as folows: 

 

f - the algebraic expression representing the function.
x - the variable name appearing in f. 

a, b - endpoints of the interval on which expansion is taken. 

n - the maximal number of expansion terms that correspond to the number of the expansion coefficients . 

It returns list of n Piece-Linear-Quadratic expansions of function f(x) for terms of expansion n=1,2,..,n. 

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Generalized Laguerre polynomials expansion of a function 

The LaguerreE(f, x, a, d, n, ) calling sequence computes the Generalized Laguerre polynomials series expansion of the expression f with respect to the variable x on the interval from a to as folows: 

 

f - the algebraic expression representing the function.
x - the variable name appearing in f. 

a - left endpoint of the semi-infinite interval on which expansion is taken. 

d>0 - scale parameter. 

n - the maximal number of expansion terms that correspond to the number of the expansion coefficients n. 

parameter of the Generalized Laguerre polynomial.          

It returns list of Generalized Laguerre polynomials expansions of function f(x) for terms of expansion n=1,2,..,n. 

If then the procedure returns Laguerre polynomials expansion. 

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Generalized Laguerre functions expansion of a function 

The LaguerreFE(f, x, a, d, n, ) calling sequence computes the Generalized Laguerre functions series expansion of the expression f with respect to the variable x on the interval from a to as folows: 

 

f - the algebraic expression representing the function.
x - the variable name appearing in f. 

a - left endpoint of the semi-infinite interval on which expansion is taken. 

d>0 - scale parameter. 

n - the maximal number of expansion terms that correspond to the number of the expansion coefficients n. 

parameter of the Generalized Laguerre polynomial.          

It returns list of Generalized Laguerre functions expansions of function f(x) for terms of expansion n=1,2,..,n. 

If then the procedure returns Laguerre functions expansion. 

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Hermite polynomials expansion of a function 

The HermiteE(f, x, a, d, n) calling sequence computes the Hermite polynomials series expansion of the expression f with respect to the variable x on the interval from to as folows: 

 

f - the algebraic expression representing the function.
x - the variable name appearing in f. 

a - shift parameter. 

d>0 - scale parameter. 

n - the maximal number of expansion terms that correspond to the number of the expansion coefficients n.        

It returns list of Hermite polynomials expansions of function f(x) for terms of expansion n=1,2,..,n. 

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Hermite functions expansion of a function 

The HermiteFE(f, x, a, d, n) calling sequence computes the Hermite functions series expansion of the expression f with respect to the variable x on the interval from to as folows: 

 

f - the algebraic expression representing the function.
x - the variable name appearing in f. 

a - shift parameter. 

d>0 - scale parameter. 

n - the maximal number of expansion terms that correspond to the number of the expansion coefficients n.        

It returns list of Hermite functions expansions of function f(x) for terms of expansion n=1,2,..,n. 

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Orthogonal Rational functions 

The Rat(j, x) calling sequence computes the j-th Orthogonal Rational function. 

j - the function number. 

x - the argument of the function. 

This functions are orthonormal on the interval . 

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Orthogonal Rational expansion of a function 

The RatE(f, x, a, d, n) calling sequence computes the Orthogonal Rational series expansion of the expression f with respect to the variable x on the interval from to as folows: 

 

f - the algebraic expression representing the function.
x - the variable name appearing in f. 

a - shift parameter. 

d>0 - scale parameter. 

n - the maximal number of expansion terms that correspond to the number of the expansion coefficients 2n.        

It returns list of Orthogonal Rational expansions of function f(x) for terms of expansion n=1,2,..,n. 

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Generalized  sinc functions 

The Gsinc(d, a, m, j, x) calling sequence computes the j-th Generalised sinc function. 

j - the function number. 

a - shift parameter. 

d>0 - scale parameter. 

m∈ℕ - rolloff factor (for m=2 and m=3 the Generalised sinc functions are not orthogonal), 

x - the argument of the function. 

If m=1 then the procedure returns sinc function. 

This functions are orthonormal on the interval . 

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Generalized sinc expansion of a function 

The GsincE(f, x, a, d, m, n) calling sequence computes the Generalized sinc series expansion of the expression f with respect to the variable x on the interval from  to as folows: 

 

f - the algebraic expression representing the function.
x - the variable name appearing in f. 

a - shift parameter. 

d>0 - scale parameter. 

m∈ℕrolloff factor (if m=2 or m=3 the expansions is not orthogonal). 

If m=1 then the procedure returns sinc expansion (Whittaker-Shannon-Kotelnikov series expansion). 

n - the maximal number of expansion terms that correspond to the number of the expansion coefficients 2n-1. 

It returns list of n Generalized sinc expansions of function f(x) for terms of expansion n=1,2,..,n. 

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Sinc Wavelets (Shannon wavelets) 

The Sincwave(n, j, x) calling sequence computes the j-th Sinc Wavelet  

of order (scale) n on interval (). 

n - the order of the Sinc Wavelet (scale of the wavelet) 

j - the Sinc Wavelet number (shift of the wavelet). 

x - the argument of Sinc Wavelet. 

This functions are orthonormal on the interval . 

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Sinc Wavelets expansion of a function 

The SincwaveE(f, x , a, d, n) calling sequence computes the Sinc Wavelet series expansion 

of the expression f with respect to the variable x on the interval from to as folows: 

 

f - the algebraic expression representing the function.
x - the variable name appearing in f. 

a - initial shift parameter. 

d>0 - initial scale parameter. 

n - the maximal number of "expansion terms" that correspond to the number of the expansion coefficients . 

It returns list of n Sinc Wavelet expansions of function f(x) for "terms of expansion" n=1,2,..,n. 

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Selection of function and interval of expansion 

                                 Functions for expansion: 

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Enter maximal number of expansion terms:  =  Enter interval for expansion from a to b:            a= b=    Select function for expansion:

                                         Reading of the selected parameters 

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Fourier expansion 

                                      List of the expansions evaluation 

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                                      Animation of expansions 

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Hartley expansion 

                                    List of the expansions evaluation 

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                                      Animation of expansions 

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Fourier-Bessel expansion 

Enter the order of Bessel function :  =

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                                      List of the expansions evaluation 

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                                      Animation of expansions 

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Orthogonal Rational Tangent expansion 

Enter the form factor :  =

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                                      List of the expansions evaluation 

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                                      Animation of expansions 

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Associated Legandre expansion 

Enter the order of Associated Legandre function (m):  =

If then the procedure returns Legendre polynomials expansion.

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                                      List of the expansions evaluation 

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(8.1)

 

                                      Animation of expansions 

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Jacobi expansion 

Enter the parameters       of the polynomials     :  =  :  =

If then the procedure returns Legendre polynomials expansion.   If then the procedure returns Chebyshev first kind polynomials (functions) expansion.   If then the procedure returns Chebyshev second kind polynomials (functions) expansion.   If then the procedure returns Gegenbauer polynomials (functions) expansion.   Otherwise the procedure returns Jacobi polynomials (functions) expansion.

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Jacobi polynomials expansion 

                                      List of the expansions evaluation 

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                                      Animation of expansions 

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Jacobi functions expansion 

                                      List of the expansions evaluation 

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                                      Animation of expansions 

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Rect expansion 

                     List of the expansions evaluation 

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                                      Animation of expansions 

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Haar expansion 

                                      List of the expansions evaluation 

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                                      Animation of expansions 

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Walsh expansion 

                                    List of the expansions evaluation 

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                                      Animation of expansions 

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Slant expansion 

                                    List of the expansions evaluation 

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                                      Animation of expansions 

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Piece-Linear-Quadratic expansion 

                                    List of the expansions evaluation 

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                                      Animation of expansions 

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Generalized Laguerre expansion on  

                                 Functions for expansion: 

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Enter maximal number of expansion terms:          =  Enter interval for expansion from a to :                  a=   Enter scale parameter (d>0):                                       d=   Enter the parameters of the polynomials :  =  ( if then the procedure returns    Laguerre polynomials (functions) expansion)    Select function for expansion:

                                         Reading  of the selected parameters 

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Generalized Laguerre polynomials expansion 

                                       List of the expansions evaluation 

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                                      Animation of expansions 

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Generalized Laguerre functions expansion 

                                       List of the expansions evaluation 

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                                      Animation of expansions 

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Hermite expansion on interval  

                                 Functions for expansion: 

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Enter maximal number of expansion terms:  =  Enter shift parameter:                                            a=   Enter scale parameter (d>0):                                d=     Select function for expansion:

                                         Reading  of the selected parameters 

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Hermite polynomials expansion 

                                     List of the expansions evaluation 

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                                      Animation of expansions 

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Hermite functions expansion 

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                                      Animation of expansions 

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Orthogonal Rational expansion on interval  

                                 Functions for expansion: 

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Enter maximal number of expansion terms:  =  Enter shift parameter:                                            a=   Enter scale parameter (d>0):                                d=     Select function for expansion:

                                         Reading  of the selected parameters 

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                                       List of the expansions evaluation 

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                                      Animation of expansions 

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Generalized sinc expansion on interval  

                                 Functions for expansion: 

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Enter maximal number of expansion terms:  =  Enter shift parameter:                                            a=   Enter scale parameter (d>0):                                d=   Enter rolloff factor (m∈ℕ):                                  m=   If m=1 then the procedure returns sinc expansion (Whittaker-Shannon-Kotelnikov series expansion).  If m=2 or m=3 the expansions is not orthogonal.    Select function for expansion:

                                         Reading  of the selected parameters 

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                                       List of the expansions evaluation 

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                                      Animation of expansions 

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Sinc Wavelets expansion on interval  

                                 Functions for expansion: 

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Enter maximal number of expansion terms:  =  Enter initial shift parameter:                                  a=   Enter initial scale parameter (d>0):                      d=     Select function for expansion:

                                         Reading  of the selected parameters 

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                                       List of the expansions evaluation 

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                                      Animation of expansions 

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