Positivity 7: 323–334, 2003.
© 2003 Kluwer Academic Publishers. Printed in the Netherlands.
Bochner Schwartz Type Theorem for Conditionally
Positive Deﬁnite Fourier Hyperfunctions
, SOON-YEONG CHUNG and DOHAN KIM
Department of Mathematics, Kunsan National University, Kunsan 573–701, Korea. E-mail:
Department of Mathematics, Sogang University, Seoul 121–742, Korea.
Department of Mathematics, Seoul National University, Seoul
151–747, Korea. E-mail: firstname.lastname@example.org
Received 20 September 2001; accepted 18 April 2002
Abstract. We prove the Bochner–Schwartz type theorem for conditionally positive deﬁnite Four-
ier hyperfunctions which generalizes the result of Gelfand-Vilenkin in their treatise Generalized
functions, vol. IV for distributions.
Key words: Conditionally positive deﬁnite, distributions, hyperfunctions
It is well known in positive deﬁnite (generalized) functions that:
(1) (Bochner) Every positive deﬁnite continuous function is the Fourier transform
of a ﬁnite positive measure,
(2) (Bochner-Schwartz) Every positive deﬁnite (tempered) distribution is the Fou-
rier transform of a positive tempered measure.
Recall that a continuous function f(x)on R
is positive deﬁnite if
∈ C, and that a distri-
bution (tempered distribution, resp.) u is positive deﬁnite if u, ϕ ∗ ϕ
ϕ ∈ C
(ϕ ∈ S, resp.), where ϕ
(x) =¯ϕ(−x). See  for more details on distri-
butions. Also, recall that a positive measure µ is tempered if
dµ < ∞
for some p ≥ 0.
The above Bochner-Schwartz theorem for distributions was generalized again
in [1,5] for the spaces of Fourier hyperfunctions and hyperfunctions, which states
that every positive deﬁnite (Fourier) hyperfunction is the Fourier transform of a
positive measure of infra-exponential type. Here, a positive measure µ is of infra-
exponential type if
dµ <∞ for every ≥ 0.
This work was supported by Korea Research Foundation Grant(KRF-99-015-DP0017).