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86 Polynomial Approximation of Differential Equations
(5.5.3) [Fu](t) := 1
√ u(x) e
2π R
−itx dx ∈ L 2 (R;C),
where i is the imaginary unity and the equality holds in R almost everywhere. Since
u ∈ L 2 (R;C) does not in general imply that u is integrable in R, a little care is necessary
when manipulating expression (5.5.3). With some technical reasoning, one verifies that
the setting is correct after appropriate interpretation (see for instance rudin(1966)).
The function Fu is called Fourier transform of u. The operator F suitably generalizes
the complex discrete Fourier transform (4.3.3).
The following relation is well-known:
(5.5.4) Fu L 2 (R;C) = u L 2 (R;C), ∀u ∈ L 2 (R;C).
In other words, F is an isometry.
Moreover, the inverse operator F −1 : L 2 (R;C) → L 2 (R;C), admits the representation
(5.5.5) [F −1 u](t) = 1
√ 2π
u(x) e
R
itx dx, a.e. ∈ R, ∀u ∈ L 2 (R;C).
Many other properties characterize the transformation F (see rudin(1966)). We just
mention some of those useful in the following sections.
If both the real and the imaginary part of u ∈ L 2 (R;C) have a weak derivative (see
previous section) and u ′ ∈ L 2 (R;C), then the following relation can be established for
almost every t ∈ R:
(5.5.6) [Fu ′ ](t) = −
i t
√ 2π [Fu] (t).
Therefore, setting v(t) := t[Fu](t), t ∈ R, we have
(5.5.7) u ′ ∈ L 2 (R;C) ⇐⇒ v ∈ L 2 (R;C).
Since t ∈ R diverges at infinity at the same rate as ρ(t) := (1 + t 2 ) 1/2 , t ∈ R, (5.5.7)
can be rewritten as
(5.5.8) u ′ ∈ L 2 (R;C) ⇐⇒ ρFu ∈ L 2 (R;C).