Time sequence and spectral analysis - Earth Dynamics: Deformations and Oscillations of the Rotating Earth - page 155

with coe cients given by the Euler formula

T /2
− T /2 g( t ) e − i 2π T t dt .

G k = (2.247)

The Fourier series representation is periodic in the record length, as is the repres-
entation (2.147) in the DFT pair, seen previously. In the case of the Fourier series
representation of g( t ), we have,

∞

1
T

G k e i 2π T ( t + T )

g( t + T ) =

k =−∞

∞

1
T

G k e i 2π T t + i 2π k

=

k =−∞

∞

1
T

G k e i 2π T t

=

k =−∞

= g( t ). (2.248)

If we now increase the record length without limit, T →∞ and Δ f = 1/ T → 0.

Thus k f becomes the continuous frequency variable, f ,and G k becomes the con-

tinuous function of frequency, G ( f ). We then have that

Δ

∞

G ( f ) e i 2π ft df , g( t ) = (2.249)

−∞

the Fourier integral representation of g( t ), and

∞

g( t ) e − i 2π ft dt , G ( f ) = (2.250)

−∞

the Fourier transform of g( t ).
The Fourier series converges if g( t ) satisfies the Dirichlet conditions : g( t )is
piecewise continuous and has only a finite number of minima and maxima on the
interval − T /2 ≤ t ≤ T /2. The Fourier transform exists if g( t ) is absolutely integ-

rable, or

∞
−∞ | g( t )

| dt < ∞ , (2.251)

for then,

∞ ∞

g( t ) e − i 2π ft dt

g( t ) e − i 2π ft dt ≤

−∞ −∞

∞
−∞ | g( t )

= |· 1 · dt < ∞ . (2.252)