PROCEEDINGS
of the
AMERICAN MATHEMATICAL
Volume
91, Number
2, June
SOCIETY
1984
A FUNDAMENTAL INEQUALITY IN THE CONVOLUTION
OF L2 FUNCTIONS ON THE HALF LINE
SABUROU SAITOH
ABSTRACT.
inequality
For any positive
for the iterated
integer
convolution
n *f>w
r >=i
q and F¿ G ¿2(0,00),
FLl
|2
2<¡
Jo
ti-2"dt<
we note
the
1 * Pj °f Pj'-
1
2q
r
(2q
l-pj-iordt-
1. Result. For the convolution T7* G of F E Lp(—00,00) and G € Li ( —00,00)
(p > 1), we know the fundamental inequality
^♦GIIp^IIFIUIGHl
(1.1)
See, for example, [4, p. 3]. Note that for F,G E L2( —00,00), in general, F * G £
L2(—00,00).
In this note, we give
THEOREM l. I.
iterated
convolution
We take a positive integer q and Fj E L2(0,00).
Then, for the
Ylj'Li * Pj °f Pj> we obtain the inequality
2q
-i
^Q
roc
(1.2)
3=1
Equality holds here if and only if each Fj is expressible in the form Cje tu for some
constant Cj and for some point u such that Reu > 0 and u is independent of j.
2. Proof of theorem.
the integral transform
(2.1)
For F E L2(0, 00) and, in general, q > 2, wè consider
m
F
e-ztF(t)tq-{1/2)dt.
Jo
Then it is known that f(z) are analytic on Rez > 0, the images f(z) form a Hubert
space Hq admitting the reproducing kernel
(2.2)
Kq(z,ü) = Y(2q)/(z + ü)2q
and
/•OO
(2.3)
\2Hq=jo
\F(t)\2dt.
Received by the editors June 15, 1982.
1980 Mathematics Subject Classification. Primary 26D90; Secondary 30C40.
Key words and phrases. Convolution,
Paley-Wiener
theorem,
Laplace
©1984
transform.
American
0002-9939/84
285
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SABUROU SAITOH
286
See, for example,
[2, pp. 113-114],
In particular,
when q = ¿, we set
/»CXI
(2.4)
fii.z)= I
e-ztF3(t)dt,
Jo
and, therefore
/•OO
(2-5)
\\M1/2=
Jo
m)\2dt.
Note that for an integer q,
(2.6)
Kq(z,û) = (2q-l)\K1/2(z,û)2q.
Hence, from the general theory of Aronszajn
the inequality
2q
(2.7)
I2
[1, pp. 357-362, 350-352], we obtain
,
2<j
1
nu
2q
(2q-l)\11(11^11«!/
as in [3].
On the other hand, we have
2q
(2-8)
ç
Ufj(z) =
3= 1
J°
and since flfii fj{z)^Hqi
29
(2.9)
Y[fj(z)=
/.oo
e~ztF*(t)tq-(l/2)dt
3=1
J°
for some uniquely determined F* E L2(0,oc). From the isometry (2.3) and (2.7),
we thus obtain the desired inequality (1.2).
Next, in order to prove the equality statement, we recall that equality holds in
(2.7) if and only if each fj is expressible in the form CjKx/2(z,u) for some constant
Cj and for some point u such that Reu > 0 and u is independent of j. See [3] for
the case of the unit disc and q = 1. Note that the result [3] also implies the desired
result for, in general, q for the present situation. See again [1, p. 361, Theorem II].
Hence,
/•oo
(2.10)
fj(z) = CjK1/2(z,ü) = cj /
Jo
and we thus see that each Fj is expressible
e-zte~mdt,
in the desired form Cje~tu.
References
1. N. Aronszajn,
Theory of reproducing kernels, Trans.
Amer.
Math.
Soc. 68 (1950),
337-404.
2. J. Burbea, Total positivity of certain reproducing kernels, Pacific J. Math. 67 (1976), 101-130.
3. S. Saitoh,
The Bergman norm and the Szegö norm, Trans.
Amer.
Math.
Soc. 249 (1979),
261-
279.
4. E. M. Stein and G. Weiss, Introduction to Fourier analysis on Euclidean spaces, Princeton
Press, Princeton,
Univ.
N.J., 1971.
Department
of Mathematics,
Faculty
1-5-1, Tenjin-Cho, Kiryu 376, Japan
of Engineering,
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Gunma University,