On the theorem of frullani

Theorem 6. Let f: R —► C be Denjoy-Perron integrable. Then the Fourier transform of f has the value (DP) f(t)dt at 0.
Proof. Fix cp € 31. We must prove
_л / r + oo \ / r + OO \
lim(/(Ax), ?(x)) = ((DP) f(t) dt) ( I
\ J — oo / \J— oo /
Let us observe that
(/(Ax), (p(x)) = lf(x),
(y) = lf(t), \ф (y) (t)\ ,
and since
f ^e~²ⁿ‘^XI dx = f
²ⁿ‘^x2‘ dt - $(At),
J— oo \ ^ / J— oo
we have
г+oo
(/(Ax), (x)) = (f(t),
So it remains to prove that

If we put F(x) = (DP) f(t)dt, then lim^ F(x) = lim_;|._₍__oo F(x) - 0. We know that ф is of bounded variation. So, given e > 0, there exists M > 0 such that, for every у > M and A e R, we have

F(t^(At)
[ ^У F(tW(Xt)
J—oo



I^CvWy)| < e, and

|F(-y)p(-Ay)|

Moreover, by Henstock’s lemma (McLeod [18], p. 74), there exists a function 8: R —> У, being У the set of closed intervals of R, verifying that £ €
If a = t₀ < r, < ••• < t_n = b, £_ke [t_k_i,t_k] c 8(£_k), (b, +oo) c a) c J(-oo), then



n
L
k=\

We can also assume that, for every & R, the oscillation of F on b(£) is less than e/(l + V), where V = \

is the total variation of
over R,
and such that <5(+oo) с [M, +oo] and <5(-oo) c [-oo, -M].
Then, if a = t_Q < f < ••• < t_n = b, £_k & [r_A_,, t_k] c S({_k), (b, +oo) c

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