Integrability of the Fourier transform: functions of bounded variation

0.11 MB·English

by  E. Liflyand

INTEGRABILITY OF THE FOURIER TRANSFORM: FUNCTIONS OF
BOUNDED VARIATION
E. LIFLYAND
Abstract. CertainrelationsbetweentheFouriertransformofafunctionofboundedvariationand
2 the Hilbert transform of its derivative are revealed. The widest subspaces of the space of functions
1 of bounded variation are indicated in which the cosine and sine Fourier transforms are integrable.
0
2

n
a
J 1. Introduction

6
We aregoingto comparethe Fouriertransformof afunction ofbounded variationandthe Hilbert
2
transformof a related function. For this, let us start with some known results. The first one is given

] in [7, Thm.2] (see also [2]). We define the following T-transform of a function g : R = [0,∞) → C:
A
C
. t/2 g(t+s)−g(t−s)
h
Tg(t) = ds,
t Z s
a 0
m
where the integral is understood in the improper (principal value) sense, that is, as lim .
[ δ→0+ δ
R

1 Theorem 1. Let f : R → C be locally absolutely continuous, of bounded variation and lim f(t) =
v +
t→∞
8 0. Let also Tf′ ∈ L1(R ). Then the cosine Fourier transform of f
+
5
5
5 ∞
. (1) f (x) = f(t)cosxtdt
1 c Z
0 0
b
2 is Lebesgue integrable on R , with
+
1
:
v
i (2) kf k . kf′k +kTf′k ,
X c L1(R+) L1(R+) L1(R+)
r and for the sine Fourier transbform, we have, with x > 0,
a
∞ 1 π
(3) f (x) = f(t)sinxtdt = f +F(x),
s Z x (cid:16)2x(cid:17)
0
b
where
(4) kFk . kf′k +kTf′k .
L1(R+) L1(R+) L1(R+)
Here and in what follows we use the notation “.” and “&” as abbreviations for “≤ C” and
“≥ C”, with C being an absolute positive constant.
Let us now turn to the Hilbert transform of an integrable function g
1991 Mathematics Subject Classification. Primary 42A38; Secondary 42A50.
Key words and phrases. Fourier transform, integrability, Hilbert transform, Hardy space.
1
2 E. LIFLYAND
1 g(t)
(5) Hg(x) = dt,
π Z t−x
R
where the integral is also understood in the improper (principal value) sense, now as lim .
|t−x|>δ
δ→0+
R
It is not necessarily integrable, and when it is, we say that g is in the (real) Hardy space H1(R). If
g ∈ H1(R), then
(6) g(t)dt = 0.
Z
R
It was apparently first mentioned in [6].
An odd function always satisfies (6). However, not every odd integrable function belongs to
H1(R), for a counterexample see, e.g., [8]. When in the definition of the Hilbert transform (5) the
function g is odd, we will denote this transform by H , and it is equal to
0
2 ∞ tg(t)
(7) H g(x) = dt.
0 π Z t2 −x2
0
If it is integrable, we will denote the corresponding Hardy space by H1(R).
0
Since
H g(x) = Tg(x)+Γ(x),
0
where Γ is such that
∞ ∞
|Γ(x)|dx . |g(t)|dt,
Z Z
0 0
the right-hand sides of (2) and (4) can be treated as kf′k . This has been observed in [7] and
H01(R+)
later on in [2].
The space of integrable functions g with integrable Tg, or just H1(R ), is one of the widest
0 +
spaces the belonging of the derivative f′ to which ensures the integrability of the cosine Fourier
transform of f. However, the possibility of existence (or non-existence) of a wider space of such
type is of considerable interest. Let us show that such a space does exist, moreover, it is the widest
possible, at least provides a necessary and sufficient condition for the integrability of the cosine
Fourier transform. In fact, it has in essence been introduced (for different purposes) in [4] as
|g(x)|
(8) Q = {g : g ∈ L1(R), dx < ∞}.
Z |x|
R
b
With the obvious norm
|g(x)|
kgk + dx
L1(R) Z |x|
R
b
it is a Banach space and ideal in L1(R). What we will actually use is the space Q of the odd
0
functions from Q
INTEGRABILITY OF THE FOURIER TRANSFORM: FUNCTIONS OF BOUNDED VARIATION 3
∞ |g (x)|
(9) Q = {g : g ∈ L1(R),g(−t) = −g(t), s dx < ∞};
0 Z x
0 b
such functions naturally satisfy (6).
Theorem 2. Let f : R → C be locally absolutely continuous, of bounded variation and lim f(t) =
+
t→∞
0. Then the cosine Fourier transform of f given by (1) is Lebesgue integrable on R if and only if
+
f′ ∈ Q .
0
The situationismoredelicate withthesine Fouriertransform, where asort ofasymptotic relation
can be obtained. In what follows we shall denote
g (x)
(10) T (x) = s .
g x
b
Theorem 3. Let f : R → C be locally absolutely continuous, of bounded variation and lim f(t) =
+
t→∞
0. Then for the sine Fourier transform of f given in (3) there holds for any x > 0
∞ 1 π
(11) fs(x) = Z f(t)sinxtdt = xf (cid:16)2x(cid:17)+H0Tf′(x)+G(x),
0
b
where
(12) kGk . kf′k .
L1(R+) L1(R+)
This theorem makes it natural to consider a Hardy type space H1(R ) which consists of Q
Q + 0
functions g with integrable H T .
0 g
Corollary 4. Let a function f satisfy the assumptions of Theorem 3 and such that f′ ∈ H1(R ).
Q +
Then
1 π
(13) f (x) = f +G(x),
s x (cid:16)2x(cid:17)
b
where
(14) kGk . kf′k .
L1(R+) HQ1(R+)
Technically, this is an obvious corollary of Theorem 3. We shall discuss it in Section 3.
2. Proofs
Proof of Theorem 2. The assumptions of the theorem give a possibility to integrate by parts. This
yields
1 ∞ 1
f (x) = − f′(t)sinxtdt = − f′(x).
c x Z x s
0
Integrating both sides overbR completes the proof. b (cid:3)
+
4 E. LIFLYAND
Proof of Theorem 3. Let us start with integration by parts in
π π
2x 1−cosxt π 1 2x
f(t)sinxtdt = f(t) |2x + f′(t)[cosxt−1]dt
Z x 0 x Z
0 0
π
1 π 1 2x
= f + f′(t)[cosxt−1]dt.
x (cid:16)2x(cid:17) x Z
0
π
The last value is bounded by 2x t|f′(t)|dt, and
0
R
∞ 2πx π ∞
(15) t|f′(t)|dtdx = |f′(t)|dt.
Z Z 2 Z
0 0 0
Let us now consider
∞
(16) I = I(x) = f(t)sinxtdt.
Z
π
2x
We will start with the following statement.
Lemma 1. There holds
2 ∞ ∞ cosv 2 ∞ xt sinv
H0Tf′(x) = xπ Z f′(t)sinxt Z v dvdt+ xπ Z f′(t)cosxt Z v dvdt.
0 xt 0 0
Proof of Lemma 1. We have
2 ∞ 1
(17) H0Tf′(x) = π Z fs′(u) u2 −x2 du
0
2 ∞ b ∞ 1
(18) = f′(t) sinutdudt.
π Z Z u2 −x2
0 0
Denoting, as usual,
∞ cost
Ci(u) = − dt
Z t
u
and
u sint π ∞ sint
Si(u) = dt = − dt,
Z t 2 Z t
0 u
we will make use of the formula (see [1, Ch.II, (18)])
∞ 1 1
sinyxdx = [sinayCi(ay)−cosaySi(ay)],
Z a2 −x2 a
0
where the integrals is understood in the principal value sense and a,y > 0. We apply this formula
to the inner integral on the right-hand side of (17), with a = x and y = t. Using the expressions
for Ci and Si, we complete the proof of the lemma. (cid:3)
With this in hand, we are going to prove that
INTEGRABILITY OF THE FOURIER TRANSFORM: FUNCTIONS OF BOUNDED VARIATION 5
(19) I(x) = H0Tf′(x)+G(x),
where
(20) kGk . kf′k .
L1(R+) L1(R+)
We denote the two summands in the expression obtained in the lemma by I and I . For both, we
1 2
make use of the fact that
∞ cosv 1
dv = O( ).
Z v xt
xt
The same true when cosv is replaced by sinv. When t ≥ 1 we have
x
∞ 1 ∞ 1 ∞ ∞ 1 ∞
(21) |f′(t)| dtdx = |f′(t)|t dx = |f′(t)|dt.
Z x Z xt Z Z x2 Z
0 1 0 1 0
x t
When t ≤ 1 we split the inner integral into two. First,
x
∞ cosv
dv = O(1),
Z v
1
and using sinxt ≤ t, we arrive at the estimate similar to (15). Further, we have
(cid:12) x (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12) 1 cosv 1
dv = O(ln ).
Z (cid:12) v (cid:12) xt
xt(cid:12) (cid:12)
(cid:12) (cid:12)
By this, integrating in x over (0,∞), w(cid:12)e hav(cid:12)e to estimate
∞ 1/t 1 ∞
|f′(t)|t ln dxdt = |f′(t)|dt.
Z Z xt Z
0 0 0
Here we use that
1/t 1 1
ln dx = .
Z xt t
0
In conclusion, I can be treated as G.
1
Let us proceed to I . Using that
2
1 xt sinv
dv = O(t),
x(cid:12)Z v (cid:12)
(cid:12) 0 (cid:12)
(cid:12) (cid:12)
we arrive for t ≤ π at (15). Let now(cid:12)t ≤ π . We h(cid:12)ave
2x 2x
xt sinv π ∞ sinv
dv = − dv.
Z v 2 Z v
0 xt
Now
6 E. LIFLYAND
2 ∞ π
f′(t)cosxtdt = I.
xπ Z 2
0
For the integral ∞ sinv dv, the estimates are exactly like those in (21).
xt v
Combining (19R) and estimates before Lemma 1, we complete the proof of the theorem. (cid:3)
3. Discussion
Discussion and comments are in order. At first sight, Theorem 2 does not seem to be a result at
all, at most a technical reformulation of (1). This could be so but not after the appearance of the
analysis of Q in [4]. Indeed, the well-known extension of Hardy’s inequality (see, e.g., [3, (7.24)])
|g(x)|
(22) dx . kgk
Z |x| H1(R)
R
b
implies
(23) H1(R) ⊆ Q ⊆ L1(R),
0
where the latter is the subspace of g in L1(R) which satisfy the cancelation property (6).
It is worth noting that (23) immediately proves (2) from Theorem 1. The initial proof in [7] is
essentially more complicated.
It is doubtful that Q (or Q ) may be defined in terms of f itself rather than its Fourier transform,
0
therefore it is of interest to find certain proper subspaces of Q wider than H1 belonging to which
0
is easily verifiable. We mention the paper [9] in which a family of subspaces between H1 and L1
is introduced and duality properties of that family are studied. However, it is not clear how to
compare that family with Q .
0
Back to Theorem 3, let us analyze (19). On the one hand, we have
∞ ∞
Z |I(x)|dx = Z |H0Tf′(x)|dx+O(kf′kH01(R+)).
0 0
On the other hand, it is proved in [7] that
∞
|I(x)|dx = O(kf′k ).
Z H01(R+)
0
This leads to
Proposition 1. If g is an integrable odd function, then
kH T k . kgk .
0 g L1(R+) H01(R+)
The above proof of Proposition 1 looks ”artificial”. A direct proof, preferable simple enough will
be very desirable. In any case, this implies an updated chain of embeddings
(24) H1(R ) ⊆ H1(R ) ⊆ Q ⊆ L1(R ).
0 + Q + 0 0 +
INTEGRABILITY OF THE FOURIER TRANSFORM: FUNCTIONS OF BOUNDED VARIATION 7
It is very interesting to figure out which of these embeddings are proper. Correspondingly, inter-
mediate spaces are of interest, both theoretical and practical.
References
[1] G. Bateman, A. Erd´elyi, Tables of Integral Transforms, I, McGraw Hill Book Company, New York, 1954.
[2] S. Fridli, Hardy Spaces Generated by an Integrability Condition, J. Approx. Theory, 113 (2001), 91–109.
[3] J. Garcia-Cuerva and J.L. Rubio de Francia, Weighted Norm Inequalities and Related Topics, North-Holland,
1985.
[4] R.L. Johnson and C.R. Warner, The convolution algebra H1(R), J. Function Spaces Appl. 8 (2010), 167–179.
[5] F.W. King, Hilbert transforms, Vol.1, Enc. Math/ Appl., Cambridge Univ. Press, Cambridge, 2009.
[6] H. Kober, A note on Hilberts operator, Bull. Amer. Math. Soc., 48:1 (1942), 421–426.
[7] E. Liflyand, Fourier transforms of functions from certain classes, Anal. Math. 19 (1993), 151–168.
[8] E. Liflyand and S. Tikhonov, Weighted Paley-Wiener theorem on the Hilbert transform, C.R. Acad. Sci. Paris,
Ser.I 348 (2010), 1253–1258.
[9] C. Sweezy, Subspaces of L1(Rd), Proc. Amer. Math. Soc., 132 (2004), 3599–3606.
Department of Mathematics, Bar-Ilan University, 52900 Ramat-Gan, Israel
E-mail address: [email protected]