orthogonal polynomials

orthogonal polynomials

1

Orthogonal Polynomials

Polynomials

of degree

n

are

analytic functions

that can be written in the form

p

n

⁢

(

x

)

=

a

0

+

a

1

⁢

x

+

a

2

⁢

x

2

+

⋯

+

a

n

⁢

x

n

They can be differentiated and integrated for any value of

x

, and are fully determined by the

n

+

1

coefficients

a

0

⁢

…

⁢

a

n

. For this simplicity they are frequently used to approximate more complicated or unknown

functions

. In

approximations

, the necessary degree

n

of the polynomial is not normally defined by criteria other than the quality of the approximation.

Using polynomials as defined above tends to lead into numerical difficulties when determining the

a

i

, even for small values of

n

. It is therefore customary to stabilize results numerically by using

orthogonal polynomials

over an

interval

[

a

,

b

]

, defined with respect to a

positive

weighting function

W

⁢

(

x

)

>

0

by

∫

a

b

p

n

⁢

(

x

)

⁢

p

m

⁢

(

x

)

⁢

W

⁢

(

x

)

⁢

𝑑

x

=

0

⁢

for

⁢

n

≠

m

Orthogonal polynomials are obtained in the following way: define the scalar product.

(

f

,

g

)

=

∫

a

b

f

⁢

(

x

)

⁢

g

⁢

(

x

)

⁢

W

⁢

(

x

)

⁢

𝑑

x

between the functions

f

and

g

, where

W

⁢

(

x

)

is a weight factor. Starting with the polynomials

p

0

⁢

(

x

)

=

1

,

p

1

⁢

(

x

)

=

x

,

p

2

⁢

(

x

)

=

x

2

, etc., from the

Gram-Schmidt decomposition

one obtains a sequence of orthogonal polynomials

q

0

⁢

(

x

)

,

q

1

⁢

(

x

)

,

…

, such that

(

q

m

,

q

n

)

=

N

n

⁢

δ

m

⁢

n

. The normalization factors

N

n

are arbitrary. When all

N

i

are equal to one, the polynomials are called

orthonormal

.

Some important orthogonal polynomials are:

a

b

W

⁢

(

x

)

name

-1

1

1

Legendre polynomials

-1

1

(

1

-

x

2

)

-

1

/

2

Chebyshev polynomials

-

∞

∞

e

-

x

2

Hermite polynomials

Orthogonal polynomials of successive orders can be expressed by a

recurrence relation

p

n

=

(

A

n

+

B

n

⁢

x

)

⁢

p

n

-

1

+

C

n

⁢

p

n

-

2

This relation can be used to compute a finite series

a

0

⁢

p

0

+

a

1

⁢

p

1

+

⋯

+

a

n

⁢

p

n

with arbitrary coefficients

a

i

, without computing explicitly every polynomial

p

j

(Horner’s Rule).

Chebyshev polynomials

T

n

⁢

(

x

)

are also

orthogonal

with respect to discrete values

x

i

:

∑

i

T

n

⁢

(

x

i

)

⁢

T

m

⁢

(

x

i

)

=

0

⁢

for

⁢

n

<

m

≤

M

where the

x

i

depend on

M

.

For more information, see

[

2

,

3

]

.

References

1

Originally from The Data Analysis Briefbook
(

http://rkb.home.cern.ch/rkb/titleA.html

http://rkb.home.cern.ch/rkb/titleA.html)

2

M. Abramowitz and I.A. Stegun (Eds.), Handbook of Mathematical Functions, National Bureau of Standards, Dover, New York, 1974.

3

W.H. Press, S.A. Teukolsky, W.T. Vetterling, and B.P. Flannery, Numerical Recipes in C, Second edition, Cambridge University Press, 1995. (The same book exists for the Fortran language). There is also an Internet version which you can work from.

Title

orthogonal polynomials

Canonical

name

OrthogonalPolynomials

Date of creation

2013-03-22 12:06:20

Last modified on

2013-03-22 12:06:20

Owner

akrowne (2)

Last modified by

akrowne (2)

Numerical id

11

Author

akrowne (2)

Entry type

Definition

Classification

msc 42C05

Classification

msc 33D45

Related topic

OrthogonalMatrices

Related topic

Orthonormal

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