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Balanced Colombeau products of the distributions x ± - p and x - p

Blagovest Damyanov (2005)

Czechoslovak Mathematical Journal

Results on singular products of the distributions x ± - p and x - p for natural p are derived, when the products are balanced so that their sum exists in the distribution space. These results follow the pattern of a known distributional product published by Jan Mikusiński in 1966. The results are obtained in the Colombeau algebra of generalized functions, which is the most relevant algebraic construction for tackling nonlinear problems of Schwartz distributions.

Bases in spaces of analytic germs

Michael Langenbruch (2012)

Annales Polonici Mathematici

We prove precise decomposition results and logarithmically convex estimates in certain weighted spaces of holomorphic germs near ℝ. These imply that the spaces have a basis and are tamely isomorphic to the dual of a power series space of finite type which can be calculated in many situations. Our results apply to the Gelfand-Shilov spaces S ¹ α and S ₁ α for α > 0 and to the spaces of Fourier hyperfunctions and of modified Fourier hyperfunctions.

Between the Paley-Wiener theorem and the Bochner tube theorem

Zofia Szmydt, Bogdan Ziemian (1995)

Annales Polonici Mathematici

We present the classical Paley-Wiener-Schwartz theorem [1] on the Laplace transform of a compactly supported distribution in a new framework which arises naturally in the study of the Mellin transformation. In particular, sufficient conditions for a function to be the Mellin (Laplace) transform of a compactly supported distribution are given in the form resembling the Bochner tube theorem [2].

BMO-scale of distribution on ℝ n

René Erlín Castillo, Julio C. Ramos Fernández (2008)

Czechoslovak Mathematical Journal

Let S ' be the class of tempered distributions. For f ∈ S ' we denote by J - α f the Bessel potential of f of order α . We prove that if J - α f ∈ B M O , then for any λ ∈ ( 0 , 1 ) , J - α ( f ) λ ∈ B M O , where ( f ) λ = λ - n f ( φ ( λ - 1 · ) ) , φ ∈ S . Also, we give necessary and sufficient conditions in order that the Bessel potential of a tempered distribution of order α > 0 belongs to the V M O space.

Boehmians of type S and their Fourier transforms

R. Bhuvaneswari, V. Karunakaran (2010)

Annales UMCS, Mathematica

Function spaces of type S are introduced and investigated in the literature. They are also applied to study the Cauchy problem. In this paper we shall extend the concept of these spaces to the context of Boehmian spaces and study the Fourier transform theory on these spaces. These spaces enable us to combine the theory of Fourier transform on these function spaces as well as their dual spaces.

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