properties of orthogonal polynomials

properties of orthogonal polynomials

A countable

system of

orthogonal polynomials

p

0

⁢

(

x

)

,

p

1

⁢

(

x

)

,

p

2

⁢

(

x

)

,

…

(1)

on an interval

[

a

,

b

]

,  where a

inner product

of two

functions

(

f

,

g

)

:=

∫

a

b

f

⁢

(

x

)

⁢

g

⁢

(

x

)

⁢

W

⁢

(

x

)

⁢

𝑑

x

is defined with respect to a

weighting function

W

⁢

(

x

)

, satisfies the

orthogonality condition

(

http://planetmath.org/OrthogonalVectors

)

(

p

m

,

p

n

)

=

0

always when

m

≠

n

.

One also requires that

deg

⁡

(

p

n

⁢

(

x

)

)

=

n

for all

⁢

n

.

Such a system (1) may be used as basis for the

vector space

of functions defined on

[

a

,

b

]

, i.e. certain such functions

f

may be expanded as a series (

http://planetmath.org/FunctionSeries

)

f

⁢

(

x

)

=

c

0

⁢

p

0

⁢

(

x

)

+

c

1

⁢

p

1

⁢

(

x

)

+

c

2

⁢

p

2

⁢

(

x

)

+

…

where the coefficients

c

n

have the

expression

c

n

=

∫

a

b

f

⁢

(

x

)

⁢

p

n

⁢

(

x

)

⁢

W

⁢

(

x

)

⁢

𝑑

x

.

Other

properties

•

The basis property of the system (1) comprises that any

polynomial

P

⁢

(

x

)

of degree

n

can be uniquely expressed as a finite

linear combination

P

⁢

(

x

)

=

c

0

⁢

p

0

⁢

(

x

)

+

c

1

⁢

p

1

⁢

(

x

)

+

…

+

c

n

⁢

p

n

⁢

(

x

)

.

•

Every member

p

n

⁢

(

x

)

of (1) is

orthogonal

to any polynomial

P

⁢

(

x

)

of degree less than

n

.

•

There is a recurrence relation

p

n

+

1

⁢

(

x

)

=

(

a

n

⁢

x

+

b

n

)

⁢

p

n

⁢

(

x

)

+

c

n

⁢

p

n

-

1

⁢

(

x

)

enabling to determine a .

•

The zeros of

p

n

⁢

(

x

)

are all real and belong to the

open interval

(

a

,

b

)

;  between two of those zeros there are always zeros of

p

n

+

1

⁢

(

x

)

.

•

The Sturm–Liouville

differential equation

Q

⁢

(

x

)

⁢

p

′′

+

L

⁢

(

x

)

⁢

p

′

+

λ

⁢

p

=

0

,

(2)

where

Q

⁢

(

x

)

is a polynomial of at most degree 2 and

L

⁢

(

x

)

a linear polynomial, gives under certain conditions, as

http://planetmath.org/node/8719

solutions

p

a system of orthogonal polynomials

p

0

,

p

1

,

…

corresponding suitable values (

eigenvalues

)

λ

0

,

λ

1

,

…

of the

parametre

λ

.  Those satisfy the

Rodrigues formula

p

n

⁢

(

x

)

=

k

n

W

⁢

(

x

)

⁢

d

n

d

⁢

x

n

⁢

(

W

⁢

(

x

)

⁢

[

Q

⁢

(

x

)

]

n

)

,

where

k

n

is a constant and

W

⁢

(

x

)

:=

1

Q

⁢

(

x

)

⁢

e

∫

L

⁢

(

x

)

Q

⁢

(

x

)

⁢

𝑑

x

.

The classical Chebyshev (

http://planetmath.org/ChebyshevPolynomial

), Hermite (

http://planetmath.org/HermitePolynomials

), Laguerre (

http://planetmath.org/LaguerrePolynomial

), and

Legendre polynomials

all satisfy an equation (2).

[Not ready . . .]

Title

properties of orthogonal polynomials

Canonical

name

PropertiesOfOrthogonalPolynomials

Date of creation

2013-03-22 19:05:34

Last modified on

2013-03-22 19:05:34

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

12

Author

pahio (2872)

Entry type

Topic

Classification

msc 42C05

Classification

msc 33D45

Related topic

HilbertSpace

Related topic

TopicsOnPolynomials

Related topic

IndexOfSpecialFunctions

Related topic

OrthogonalityOfLaguerrePolynomials

Related topic

OrthogonalityOfChebyshevPolynomials

Defines

Rodrigues formula

Generated on Fri Feb 9 21:30:20 2018 by

LaTeXML