Mathematical Physics Vol 1

Chapter 6. Trigonometric Fourier series. Fourier integral

318

Let us approximate the observed function by a trigonometric polynomial of N -th order

N ∑ k = 1

α 0 2

( α k cos kx + β k sin kx ) ≡ P N ( x ) .

f ( x ) ≈

(6.20)

+

The coefficients of this polynomial, α k i β k , are currently indeterminate, and can be expressed in several ways. However, it is clear that we are interested in the form for which the best approximation is obtained, i.e. with the smallest approximation error, for a fixed value of N . To this end, we first define the approximation error, which can also be done in several ways, but it is most natural to define it by the expression | f ( x ) − P N ( x ) | = ∆ N , for x ∈ [ − ℓ,ℓ ] . However, in this task, it is more convenient to define the error by an integral, in the form

π Z

1 2 π

2 d x

∆ N =

[ f ( x ) − P N ( x )]

(6.21)

.

− π

Defined in this way, ∆ N is called the mean square error . The task is to determine, for a fixed N , the coefficients α k i β k of the polynomial (6.20), so that ∆ N is minimal. Observe first the subintegral function. Given that ( f − P N ) 2 = f 2 − 2 fP N + P 2 N , it follows from (6.21) that

π Z

π Z − π

1 2 π

1 2 π

2 N d x .

2 d x

f 2 − 2 fP

∆ N =

[ f ( x ) − P N ( x )]

N + P

(6.22)

=

− π

Further, as by the initial assumption, the function f ( x ) can be represented by a convergent Fourier series, it follows that π Z − π f P N d x =

" a 0 2

( a k cos kx + b k sin kx ) # · "

( α k cos kx + β k sin kx ) # d x =

π Z

∞ ∑ k = 1

N ∑ k = 1

α 0 2

=

+

+

− π

= π "

( a k α k + b k β k ) # .

N ∑ k = 1

a 0 α 0 2

(6.23)

+

We have used here R cos kx d x = 0, R sin kx d x = 0 and R sin kx cos kx d x = 0. In a similar way, we also obtain π Z − π P 2 N d x = π " α 2 0 2 + N ∑ k = 1 α 2 k + β 2 k # .

(6.24)

The last two relations (6.23) and (6.24) yield π Z − π P 2 N − 2 fP N d x = π ( ( α 0 − a 0 ) 2 2 + N ∑ k = 1 h

2 i )

( α k − a k ) 2

+( β k − b k )

−

− π "

N ∑ k = 1

a 2 0 2

2 k # ,

a 2

k + b

(6.25)

+