minimality property of Fourier coefficients

minimality property of Fourier coefficients

Let

f

be a

Riemann integrable

periodic real function with period

2

⁢

π

and

n

a

positive integer

.  Among all “trigonometric polynomials”

φ

⁢

(

x

)

:=

α

0

2

+

∑

j

=

1

n

(

α

j

⁢

cos

⁡

j

⁢

x

+

β

j

⁢

sin

⁡

j

⁢

x

)

,

the

polynomial

with the coefficients

α

j

and

β

j

being the

Fourier coefficients

α

j

=

a

j

:=

1

π

⁢

∫

-

π

π

f

⁢

(

x

)

⁢

cos

⁡

j

⁢

x

⁢

d

⁢

x

and

β

j

=

b

j

:=

1

π

⁢

∫

-

π

π

f

⁢

(

x

)

⁢

sin

⁡

j

⁢

x

⁢

d

⁢

x

for the

Fourier series

of

f

produces the minimal value of the

mean square deviation

1

2

⁢

π

⁢

∫

-

π

π

[

f

⁢

(

x

)

-

φ

⁢

(

x

)

]

2

⁢

𝑑

x

.

Proof.

For any fixed number

n

, it’s a question of giving the least value to the

definite integral

m

:=

1

2

⁢

π

⁢

∫

-

π

π

[

f

⁢

(

x

)

-

α

0

2

-

∑

j

=

1

n

(

α

j

⁢

cos

⁡

j

⁢

x

+

β

j

⁢

sin

⁡

j

⁢

x

)

]

2

⁢

𝑑

x

(

≧

0

)

.

(1)

Expanding

m

and integrating termwise yields

m

=

1

2

⁢

π

⁢

∫

-

π

π

(

f

⁢

(

x

)

)

2

⁢

𝑑

x

-

α

0

2

⁢

π

⁢

∫

-

π

π

f

⁢

(

x

)

⁢

𝑑

x

-

1

π

⁢

∑

j

=

1

n

α

j

⁢

∫

-

π

π

f

⁢

(

x

)

⁢

cos

⁡

j

⁢

x

⁢

d

⁢

x

-

1

π

⁢

∑

j

=

1

n

β

j

⁢

∫

-

π

π

f

⁢

(

x

)

⁢

sin

⁡

j

⁢

x

⁢

d

⁢

x

+

1

2

⁢

π

⁢

α

0

2

4

⁢

∫

-

π

π

𝑑

x

+

1

2

⁢

π

⁢

∑

j

=

1

n

α

j

2

⁢

∫

-

π

π

cos

2

⁡

j

⁢

x

⁢

d

⁢

x

+

1

2

⁢

π

⁢

∑

j

=

1

n

β

j

2

⁢

∫

-

π

π

sin

2

⁡

j

⁢

x

⁢

d

⁢

x

+

α

0

2

⁢

π

⁢

∑

j

=

1

n

α

j

⁢

∫

-

π

π

cos

⁡

j

⁢

x

⁢

d

⁢

x

+

α

0

2

⁢

π

⁢

∑

j

=

1

n

β

j

⁢

∫

-

π

π

sin

⁡

j

⁢

x

⁢

d

⁢

x

+

1

π

⁢

∑

j

=

1

n

∑

k

=

1

n

α

j

⁢

β

k

⁢

∫

-

π

π

cos

⁡

j

⁢

x

⁢

sin

⁡

k

⁢

x

⁢

d

⁢

x

+

1

π

⁢

∑

j

=

1

n

∑

k

≠

j

α

j

⁢

α

k

⁢

∫

-

π

π

cos

⁡

j

⁢

x

⁢

cos

⁡

k

⁢

x

⁢

d

⁢

x

+

1

π

⁢

∑

j

=

1

n

∑

k

≠

j

β

j

⁢

β

k

⁢

∫

-

π

π

sin

⁡

j

⁢

x

⁢

sin

⁡

k

⁢

x

⁢

d

⁢

x

.

Here, we have the Fourier coefficients

1

π

⁢

∫

-

π

π

f

⁢

(

x

)

⁢

𝑑

x

=

a

0

,

1

π

⁢

∫

-

π

π

f

⁢

(

x

)

⁢

cos

⁡

j

⁢

x

⁢

d

⁢

x

=

a

j

,

1

π

⁢

∫

-

π

π

f

⁢

(

x

)

⁢

sin

⁡

j

⁢

x

⁢

d

⁢

x

=

b

j

.

Furthermore,

∫

-

π

π

cos

2

⁡

j

⁢

x

⁢

d

⁢

x

=

∫

-

π

π

sin

2

⁡

j

⁢

x

⁢

d

⁢

x

=

π

,

∫

-

π

π

cos

⁡

j

⁢

x

⁢

sin

⁡

k

⁢

x

⁢

d

⁢

x

=

0

and

∫

-

π

π

cos

⁡

j

⁢

x

⁢

cos

⁡

k

⁢

x

⁢

d

⁢

x

=

∫

-

π

π

sin

⁡

j

⁢

x

⁢

sin

⁡

k

⁢

x

⁢

d

⁢

x

=

0

for

⁢

k

≠

j

.

Using all these we can write

m

=

1

2

⁢

π

⁢

∫

-

π

π

(

f

⁢

(

x

)

)

2

⁢

𝑑

x

-

α

0

⁢

a

0

2

-

∑

i

=

1

n

(

α

i

⁢

a

i

+

β

i

⁢

b

i

)

+

a

0

2

4

+

1

2

⁢

∑

i

=

1

n

(

α

i

2

+

β

i

2

)

.

Adding and subtracting still the sum

a

0

2

4

+

1

2

⁢

∑

i

=

1

n

(

a

i

2

+

b

i

2

)

yields finally the form

m

=

1

2

⁢

π

⁢

∫

-

π

π

(

f

⁢

(

x

)

)

2

⁢

𝑑

x

-

a

0

2

4

-

1

2

⁢

∑

i

=

1

n

(

a

i

2

+

b

i

2

)

+

1

4

⁢

(

α

0

-

a

0

)

2

+

1

2

⁢

∑

i

=

1

n

[

(

α

i

-

a

i

)

2

+

(

β

i

-

b

i

)

2

]

.

The three first addends of this sum do not depend on the choice of the quantities

α

i

and

β

i

.  The other addends are non-negative, and their sum is minimal, equal 0, when

α

i

=

a

i

,

β

i

=

b

i

∀

i

.

Accordingly, the mean square deviation

m

, i.e. (1), is minimal when one uses the Fourier coefficients. Q.E.D.

References

1

N. Piskunov:

Diferentsiaal- ja integraalarvutus kõrgematele tehnilistele õppeasutustele

.  Kirjastus Valgus, Tallinn (1966).

Title

minimality property of Fourier coefficients

Canonical name

MinimalityPropertyOfFourierCoefficients

Date of creation

2013-03-22 18:22:00

Last modified on

2013-03-22 18:22:00

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

13

Author

pahio (2872)

Entry type

Theorem

Classification

msc 42A16

Classification

msc 42A10

Related topic

CommonFourierSeries

Related topic

UniquenessOfFourierExpansion

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