Mathematical Physics Vol 1

Chapter 6. Trigonometric Fourier series. Fourier integral

320

As a consequence of this convergence lim k → ∞

a k = 0 , lim k → ∞

b k = 0 ,

i.e.

π Z π Z

lim k → ∞

a k = 0 ⇒ lim k → ∞

f ( x ) cos kx d x = 0 ,

(6.30)

− π

lim k → ∞

b k = 0 ⇒ lim k → ∞

f ( x ) sin kx d x = 0 .

(6.31)

− π

Thus, the Fourier coefficients of a bounded and integrable function tend to zero, when k → ∞ . The relations (6.30) and (6.31) are known in literature as the Riemann theorem . If ∆ N → 0, when N → ∞ , then the inequality (6.29) becomes 1 π π Z − π f 2 ( x ) d x = a 2 0 2 + ∞ ∑ k = 1 a 2 k + b 2 k . (6.32) Let us now demonstrate how the previous relations can be extended to an arbitrary (but finite) period ( − ℓ,ℓ ). Observe a periodic function ϕ ( t ) , with period ( − π , π ), i.e. ϕ ( t )= ϕ ( t + 2 π ) , and another periodic function f ( x ) , with period ( − ℓ,ℓ ), i.e. f ( x )= f ( x + 2 ℓ ) . Let us now introduce the substitution x = ℓ π t , where f ( x )= f ℓ π t = ϕ ( t ) . According to previous substitutions we have ϕ ( t + 2 π )= f ℓ π ( t + 2 π ) = f ℓ π t + 2 ℓ = f ( x + 2 ℓ )= f ( x )= ϕ ( t ) . Further, for trigonometric series, it follows that

∞ ∑ k = 1

a 0 2 a 0 2

ϕ ( t )=

( a k cos kt + b k sin kt )

+

∞ ∑ k = 1

x ,

k π ℓ

k π ℓ

a k cos

x + b k sin

f ( x )=

+

where the appropriate coefficients, say a k , are

π Z

ℓ Z

k π ℓ

1 π

1 ℓ

ϕ ( t ) cos kt d t =

a k =

f ( x ) cos

x d x .

− π

− ℓ

A substitution of integration boundaries took place, given that

ℓ π

for t 1 = − π x 1 =

( − π )= − ℓ

and for t 2 = π x 2 = ℓ π

( π )= ℓ,

while the differential is d t = π ℓ

1 π

1 ℓ

d x , that is,

d t =

d x .