orthogonality of Chebyshev polynomials

orthogonality of Chebyshev polynomials

By expanding the of

de Moivre identity

cos

⁡

n

⁢

φ

=

(

cos

⁡

φ

+

i

⁢

sin

⁡

φ

)

n

to sum, one obtains as

real part

certain terms containing power products of

cos

⁡

φ

and

sin

⁡

φ

, the latter ones only with even

exponents

.  When these are expressed with cosines (

sin

2

⁡

φ

=

1

-

cos

2

⁡

φ

), the real part becomes a

polynomial

T

n

of degree

n

in the

argument

(

http://planetmath.org/Argument2

)

cos

⁡

φ

:

cos

⁡

n

⁢

φ

=

T

n

⁢

(

cos

⁡

φ

)

(1)

This can be written equivalently (

http://planetmath.org/Equivalent3

)

T

n

⁢

(

x

)

=

cos

⁡

(

n

⁢

arccos

⁡

x

)

.

(2)

It’s a question of

Chebyshev polynomial of first kind

and of

n

(cf.

special cases of hypergeometric function

).

For showing the orthogonality of

T

m

and

T

n

we start from the

integral

∫

0

π

cos

⁡

m

⁢

φ

⁢

cos

⁡

n

⁢

φ

⁢

d

⁢

φ

, which via the substitution

cos

⁡

φ

:=

x

,

d

⁢

x

=

-

sin

⁡

φ

⁢

d

⁢

φ

=

-

1

-

x

2

⁢

d

⁢

φ

changes to

∫

0

π

cos

⁡

m

⁢

φ

⁢

cos

⁡

n

⁢

φ

⁢

d

⁢

φ

=

-

∫

1

-

1

T

m

⁢

(

x

)

⁢

T

n

⁢

(

x

)

⁢

d

⁢

x

1

-

x

2

.

(3)

The left of this equation is evaluated by using the

product formula

in the entry trigonometric identities:

∫

0

π

cos

⁡

m

⁢

φ

⁢

cos

⁡

n

⁢

φ

⁢

d

⁢

φ

=

1

2

⁢

∫

0

π

(

cos

⁡

(

m

-

n

)

⁢

φ

+

cos

⁡

(

m

+

n

)

⁢

φ

)

⁢

𝑑

φ

=

{

0

⁢

for

⁢

m

≠

n

,

π

2

⁢

for

⁢

m

=

n

≠

0

.

By (3), we thus have

∫

-

1

1

T

m

⁢

(

x

)

⁢

T

n

⁢

(

x

)

⁢

d

⁢

x

1

-

x

2

=

{

0

⁢

for

⁢

m

≠

n

,

π

2

⁢

for

⁢

m

=

n

≠

0

,

which means the orthogonality of the polynomials

T

m

⁢

(

x

)

and

T

n

⁢

(

x

)

weighted by

1

1

-

x

2

.

Any

Riemann integrable

real function

f

, defined on

[

-

1

,

1

]

,  may be expanded to the series

f

⁢

(

x

)

=

a

0

2

⁢

T

0

⁢

(

x

)

+

∑

j

=

1

∞

a

j

⁢

T

j

⁢

(

x

)

,

where

a

j

=

2

π

∫

-

1

1

f

(

x

)

T

j

(

x

)

d

⁢

x

1

-

x

2

(

j

=

0

,

1

,

2

,

…

)

This concerns especially the polynomials

f

⁢

(

x

)

:=

x

n

,  for which we obtain

x

n

=

cos

n

⁡

φ

=

cosh

n

⁡

i

⁢

φ

=

2

-

n

⁢

(

e

i

⁢

φ

+

e

-

i

⁢

φ

)

=

2

-

n

⁢

[

(

n

0

)

⁢

(

e

n

⁢

i

⁢

φ

+

e

-

n

⁢

i

⁢

φ

)

+

(

n

1

)

⁢

(

e

(

n

-

2

)

⁢

i

⁢

φ

+

e

-

(

n

-

2

)

⁢

i

⁢

φ

)

+

…

]

=

2

1

-

n

⁢

[

(

n

0

)

⁢

T

n

⁢

(

x

)

+

(

n

1

)

⁢

T

n

-

2

⁢

(

x

)

+

(

n

2

)

⁢

T

n

-

4

⁢

(

x

)

+

…

]

.

(If

n

is even, the last term contains

T

0

⁢

(

x

)

but its coefficient is only a half of the middle number of the Pascal’s triangle row in question.)  Explicitly:

1

=

T

0

x

=

T

1

x

2

=

2

-

1

⁢

(

T

2

+

T

0

)

x

3

=

2

-

2

⁢

(

T

3

+

3

⁢

T

1

)

x

4

=

2

-

3

⁢

(

T

4

+

4

⁢

T

2

+

3

⁢

T

0

)

x

5

=

2

-

4

⁢

(

T

5

+

5

⁢

T

3

+

10

⁢

T

1

)

x

6

=

2

-

5

⁢

(

T

6

+

6

⁢

T

4

+

15

⁢

T

2

+

10

⁢

T

0

)

x

7

=

2

-

6

⁢

(

T

7

+

7

⁢

T

5

+

21

⁢

T

3

+

35

⁢

T

1

)

x

8

=

2

-

7

⁢

(

T

8

+

8

⁢

T

6

+

28

⁢

T

4

+

56

⁢

T

2

+

36

⁢

T

0

)

x

9

=

2

-

8

⁢

(

T

9

+

9

⁢

T

7

+

36

⁢

T

5

+

84

⁢

T

3

+

126

⁢

T

1

)

⁢

⋯

⋯

References

1

Pentti Laasonen:

Matemaattisia erikoisfunktioita

.  Handout No. 261. Teknillisen Korkeakoulun Ylioppilaskunta; Otaniemi, Finland (1969).

Title

orthogonality of Chebyshev polynomials

Canonical name

OrthogonalityOfChebyshevPolynomials

Date of creation

2013-03-22 18:54:42

Last modified on

2013-03-22 18:54:42

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

16

Author

pahio (2872)

Entry type

Derivation

Classification

msc 33C45

Classification

msc 33D45

Classification

msc 42C05

Related topic

OrthogonalPolynomials

Related topic

LaguerrePolynomial

Related topic

ChangeOfVariableInDefiniteIntegral

Related topic

DeterminationOfFourierCoefficients

Related topic

OrthogonalityOfLegendrePolynomials

Related topic

PropertiesOfOrthogonalPolynomials

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