Mathematical Physics Vol 1

6.2 The fundamental convergence theorem for Fourier series

319

where the following relations have been used ( α k − a k ) 2 = α 2 k − 2 α k a k + a 2 k ,

2 k − 2 a k α k =( α k − a k ) 2 k − 2 b k β k =( β k − b k )

2 − a 2 k

α

2

= β 2

k − 2 β k b k + b 2 k ,

2 − b 2

( β k − b k )

β

k .

If we now substitute (6.25) into (6.22), we obatin

f 2 d x − "

π Z

N ∑ k = 1

a 2 0 2

1 π

2 k # +

a 2

2 ∆ N =

k + b

(6.26)

+

− π

+ (

( α 0 − a 0 ) 2 2 2 i ) . As the task is to determine the coefficients α k and β k , so that ∆ N is minimal, it can be concluded from (6.26) that the following should be true ( α 0 − a 0 ) 2 2 + N ∑ k = 1 h ( α k − a k ) 2 +( β k − b k ) 2 i = 0 . (6.27) It is clear that this term would constantly increase (the sum of squares), with an increase in N, and in this way the error itself would grow. For that reason, we require that it be equal to zero. From relation (7.70) we obtain the required coefficients α 0 = a 0 , α k = a k , β k = b k . (6.28) From this, we conclude that the best mean square approximation, for an integrable, periodic function f ( x ) , for x ∈ [ − π , π ] , is given by the trigonometric polynomial P N ( x ) , whose coefficients are the Fourier coefficients of the function f ( x ) . Some consequences Note that the error ∆ N = 1 2 π π Z − π [ f ( x ) − P N ( x )] 2 d x is non-negative, as the subintegral function is the square of a real function. Thus, according to (6.26) and (7.70), it follows that + N ∑ k = 1 h ( α k − a k ) 2 +( β k − b k )

f 2 ( x ) d x − "

π Z

N ∑ k = 1

a 2 0 2

1 π

2 k # ≥ 0 ,

a 2

2 ∆ N =

k + b

+

− π

that is

π Z

N ∑ k = 1

a 2 0 2

1 π

2 k , za N = 0 , 1 ,...,

f 2 ( x ) d x ≥

a 2

k + b

(6.29)

+

− π

In literature, this expression (6.29) is known as the Bessel inequality . Note now that the left hand side of the inequality is independent of N . It follows that, when N → ∞ , the right hand side remains bounded, which means that the series of squares of Fourier coefficients a 2 0 2 + ∞ ∑ k = 1 a 2 k + b 2 k is convergent.