Integral Transforms and Special Functions
Vol. 21, No. 1, January 2010, 75–83
Generalized Cauchy transformation with applications
to boundary values in generalized function spaces
Vesna Manova Erakovi´ c
a
, Stevan Pilipovi´ c
b
* and Vasko Reˇ ckovski
c
a
University of Skopje, Skopje, Macedonia;
b
University of Novi Sad, Novi Sad, Serbia;
c
University of Ohrid, Ohrid, Macedonia
(Received 7 April 2009 )
Generalized Cauchy transformation, actually the generalized Cauchy formula for functions, distributions
and ultradistributions with appropriate growth rate, is given and used for the boundary value representation
in the sense of generalized functions.
Keywords: Cauchy formula for functions; distributions of finite order; ultradistributions; Cauchy
boundary value representation
AMS Subject Classification: 44A15; 46F12; 46F20
1. Introduction
The aim of this article is to give:
(1) generalized Cauchy formula, also called generalized Cauchy transformation, for a function
which is bounded by a real analytic function of appropriate growth and to apply this result to:
(2) boundary value characterization of distributions of the form
F(x) =
m
i =0
(P
i
(x)f
i
(x))
(i)
,
where, for every i ∈{0, 1,...,m},f
i
∈ L
1
(R) and P
i
is a real analytic function on the real
line R different from zero on R of suitable growth rate and to
(3) boundary value characterization of tempered ultradistributions of the form
F(x) = P
0
d
dx
(P (x)f (x)),
where P
0
is an ultradifferential operator of class (M
p
), respectively of class {M
p
}, while P(x)
is a subexponential function of class (M
p
), respectively of class {M
p
}.
*Corresponding author. Email: pilipovic@im.ns.ac.yu
ISSN 1065-2469 print/ISSN 1476-8291 online
© 2010 Taylor & Francis
DOI: 10.1080/10652460903016141
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