Mathematical Physics Vol 1

6.2 The fundamental convergence theorem for Fourier series

317

By integrating we obtain the coefficients of the Fourier series a 0 2 = k 2 , a n = 4 k n 2 π 2 2cos n π 2 − cos n π − 1 , b n = 0 .

Analyzing the above expressions, we can see that only the terms n = 2 , 6 , 10 , 14 ,... , are different from zero ( a n̸ = 0), so the Fourier series of the initial even-extended function has the form

16 k π 2

x + ... .

2 π ℓ

6 π ℓ

k 2 −

1 2 2

1 6 2

f ( x )=

cos

x +

cos

b) Let us now sketch the odd extension.

Figure 6.6

In this case (odd function), according to (6.18), we obtain

a n = 0 ,

2 ℓ  

x d x 

ℓ/ 2 Z 0

ℓ Z ℓ/ 2

n π ℓ

n π ℓ

2 k ℓ

2 k ℓ

 .

b n =

x sin

x d x +

( ℓ − x ) sin

From here, by partial integration, we obtain

n π 2

8 k n 2 π 2

b n =

sin

.

Finally, the Fourier series for the odd extension takes the following form

8 k π 2

x − ... .

π ℓ

3 π ℓ

5 π ℓ

1 1 2

1 3 2

1 5 2

f ( x )=

sin

x −

sin

x +

sin

6.2.4 Approximation of a function by a trigonometric polynomial. Mean square error

Let f ( x ) be a periodic function, with period 2 π , which can be represented by a Fourier series.