ABSTRACT
Let a function f (x) be integrable on the real axis: f (x) ∈ L1(–∞,∞). Then its Fourier transform F (s) is dened as
F (s) = ∫ ∞
–∞ f (x) eisx dx, F (s) ∈ L1(–∞,∞), (1.1)
and in the case when f (x) is continuous, the following inversion formula is valid:
f (x) = 1 2pi
∫ ∞
–∞ F (s) e-isx ds = 1
2pi lim a→∞
∫ a –a F (s) e-isx ds, x ∈ (–∞,∞). (1.2)
The function f (x) will be called an original and the function F (s) the Fourier image of f (x). The fact that the original and the image are related by formulas (1.1) and (1.2) will be denoted f (x) =⇒ F (s).
