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 ![`in`(f[i], `*`(`^`(L(a, b), 2))), i = 1 .. nops(f)](/view.aspx?SI=7256/Expansions_4.gif)
.
x - the variable name appearing in ![f[i]](/view.aspx?SI=7256/Expansions_5.gif)
.
a, b - endpoints of interval of the orthogonalization.
It returns list of 
orthonormal functions (algebraic expressions).
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.
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.
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 ![mu[i]](/view.aspx?SI=7256/Expansions_47.gif)
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.
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.
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 ![d = `*`(`+`(b, `-`(a)), `/`(1, 2)), h[i] = `*`(l[i], `*`(`+`(`*`(2, `*`(i)), 1), `*`(factorial(`+`(i, `-`(m))), `*`(`/`(`+`(`*`(2, `*`(factorial(`+`(i, m))))))))))](/view.aspx?SI=7256/Expansions_83.gif)
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.
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.
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.
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.
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.
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 
.
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.
List of Slant functions
The Slant(n) calling sequence computes the list of n Slant functions.
This functions are orthonormal on the interval 
.
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.
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 
.
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.
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.
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.
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.
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.
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.
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∈ℕ
rolloff 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.
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.
