ON THE PRODUCTS OF FUNCTIONS REPRESENTED
AS CONVOLUTION TRANSFORMS
I. I. HIRSCHMAN, JR., AND D. V. WIDDER
The authors have in a number of papers,
sidered the convolution transforms
(1)
[l],
[2], and [3],1 con-
/(*) = f G(x- t)da(t)
%J —oo
with kernels of the form
G(t) =(2)
1 rix
I E(s)-1e"ds,
2iriJx~ix
E(s) = eb'fl(l-—)e"at.
k=\ \
aJ
Here b, \ak\î are real constants subject only to the restriction that
E" o-t2< °°• Let Xi and X2 be strictly positive real numbers such
that Xi+X2^1. In the present paper we shall consider results concerning products of functions representable in the form (1) which
are consequences of the identity
(1 - Dor^fMgfax)
+ \ig(\2x)[f(\ix)
We
recall
the
definitions
= \2f(\lX)[gÇh2x)- a-Y(A,*)]
- a-if(\ix)]
+ (1 - Ax-\2)f{\ix)g(\2x).
ai = MaxOj.<0 [ak,
—00],
a2 = Min0t>o
[ak, 00], see [2; l].
Theorem
1. Let G(x)ÇE.class I and let
f(x) = Cg(x - t)da(t),
g(x) = f G(x- t)dß(t)
J —ta
J —00
with a(t) G Î , ß(t) G î . // 0 <XX,0 <X2,Xx+X2g 1, then
f(\ix)g(\2x) = Î* G(x- t)dy(t)
withy(t)E: î .
Received by the editors November 21, 1949 and, in revised form, December 19,
1949.
1 The first number in brackets refers to the references cited at the end of the paper.
The number following the semicolon gives the section number of the reference cited.
97
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98
I. I. HIRSCHMAN AND D. V. WIDDER
Proof. By [3; 36] necessary
have the representation
[February
and sufficient conditions
for h(x) to
G(x - t)db{t)
/CO -00
with ô(0G Î are:
a.
h(x) EC™
(-
b.
h(x) = o(e"ix)
(x —» — oo)
= o(e^x)
»
c.
oo < x < oo),
(x—>+
oo),
j
I (1 - DA,- )h(x) ^ 0
(-
oo < x < oo)
for every selection \A\, A2, • • • , An\ from \ai, a2, • • • }. Our assumptions imply that/(x)
and g(x) fulfill these conditions; it follows
that /(Xix)g(X2x) also satisfies these conditions (a and b trivially,
and c because of (3)). Our desired result follows.
The following theorem may be proved similarly. See [3; 36, 37].
Theorem
2. Let G(x)Eclass II or III and let
/oo
G(x - t)da(t)
(x > n),
-co
s* OO
g(x) = J
G(x - t)dß(t)
(x > r2)
with a(t)G Î , (3(0G Î • If 0<Xi, 0<X2, Xi+X2^l, then
G(x - t)dy(t)
(x > Max [n/\u
r2/\2\)
/CO -00
where y(f) G î •
Using a further
Theorem
representation
theorem
[3; 32] one may establish
3. Let p, q, and r be numbers greater than one such that
r~1 = p-1 + q-i, and let
f{x) = \
G(x-
t)e-cit<t>(t)dt,
J —00
g(x) = f G(x - t)e-c^(t)dt,
J -00
where cti<Ci<a2,
ai<c2<a2,
and
where 4>(t)QLP(— =o, oo), \p(t)
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i95i]
FUNCTIONSREPRESENTED AS CONVOLUTIONTRANSFORMS
GL,(-°°,
99
°°)- H 0<Xi, 0<X2, Xi+X2^l, then
fMg(\tx)
= f G(x- t)e-"x(t)dt
J -00
w&ere e=XiCi-|-X2C2,and where x(¿)GLr( — °°, °°). More precisely we
have
llxcoli,
^ ^xr^^rW^ikwLikwii,.
As an application of Theorem 1 we have the following result concerning the Stieltjes transform. If
A(x)= f (x + t)-Ha(t),
J o+
B(x)= f (x + t)~ldß(t)
J o+
where a(t)E Î, ß(t)G î, and if 0<Xi, 0<X2, Xi+Xa^l, then
A(x^)B(x^)= f (x + t)-Hy(t)
J o+
with7(¿)G
î. Choosing A(x)= B(x)=x~1+t, 0<€<l,wesee
is essential.
that the
condition Xi+X2^l
References
1. I. I. Hirschman, Jr., and D. V. Widder, An inversion and representation theory
for convolution transforms with totally positive kernels, Proc. Nat. Acad. Sei. U.S.A.
vol. 34 (1948) pp. 152-156.
2. -,
The inversion of a general class of convolution transforms, Trans. Amer.
Math. Soc. vol. 66 (1949) pp. 135-201.
3. -,
A representation
theory for a general
class of convolution
Trans. Amer. Math. Soc. vol. 67 (1949) pp. 69-97.
Washington
Harvard
University
University
and
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transforms,