Integral Transformations - Heat Conduction: Mathematical Models and Analytical Solutions - page 459

Also,

1
β +

F [ I ( x ) f ( x )] =

ω , i

(B.15)

1

lim
β → 0 F

[ I ( x ) f ( x )] =

ω . i

Thus we obtain, by Eq. (B.14) and (B.15), as β → 0,

1

I ( x ) f ( x ) → I ( x ) , F [ I ( x ) f ( x )] →

ω .

(B.16)

i

1
i

This shows that the weak limits of I ,

respectively. Thus, Eq. (B.16) forms a Fourier transformation couple and is denoted
by

( x ) f ( x ) and the image function are I ( x ) and

ω

or F − 1 1
i

1

F [ I ( x )] = = I ( x ) .

i ω ω

Example 2 . Find the generalized Fourier transformation of δ ( x ) .

Solu tio n. It can be shown that the fast-decreasing function sequence
n
π

e − nx 2 is a δ -function sequence such that

n
π

+ ∞ + ∞

e − nx 2

lim
n

ϕ ( x ) d x = ϕ ( 0 )= δ ( x ) ϕ ( x ) d x , ∀ ϕ ∈ K .

→ ∞

− ∞ − ∞

Also,

F n
π

e − nx 2 F n

π

e − nx 2

e − ω 2 = , lim

n → ∞

= 1 .

4 n

Thus, as n → ∞ ,

n
π

F n
π

e − nx 2 1

e − nx 2

→ δ ( x ) , . (B.17)

Therefore the weak limits δ ( x ) and 1 form a Fourier transformation couple such that

1or F − 1

F [ δ ( x )] = [ 1 ]= δ ( x ) . (B.18)

Remark. We can also obtain Eq. (B.18) directly by applying definitions of the δ -

function and the Fourier transformation, i.e.

+ ∞

e − iω x x = 0 =

e − iω x d x 1or F − 1 F [ δ ( x )] = δ ( x ) = [ 1 ]= δ ( x ) .

− ∞