On the structure of Mellin distributions
The Taylor transformation of analytic functionals with non-bounded carrier
Let L be a closed convex subset of some proper cone in ℂ. The image of the space of analytic functionals Q'(L) with non-bounded carrier in L under the Taylor transformation as well as the representation of analytic functionals from Q'(L) as the boundary values of holomorphic functions outside L are given. Multipliers and operators in Q'(L) are described.
Generalized analytic functions of Bogdan Ziemian
Definitions, properties, examples and applications of generalized analytic functions introduced by B. Ziemian are presented.
Laplace integrals in partial differential equations in papers of Bogdan Ziemian
Fundamental solutions to linear partial differential equations with constant coefficients are represented in the form of Laplace type integrals.
Laplace ultradistributions on a half line and a strong quasi-analyticity principle
Several representations of the space of Laplace ultradistributions supported by a half line are given. A strong version of the quasi-analyticity principle of Phragmén-Lindelöf type is derived.
On 1-regular ordinary differential operators
Solutions to singular linear ordinary differential equations with analytic coefficients are found in the form of Laplace type integrals.
A Phragmén-Lindelöf type quasi-analyticity principle
Quasi-analyticity theorems of Phragmén-Lindelöf type for holomorphic functions of exponential type on a half plane are stated and proved. Spaces of Laplace distributions (ultradistributions) on ℝ are studied and their boundary value representation is given. A generalization of the Painlevé theorem is proved.
Generalized Taylor expansions of singular functions
Nonanalyticity of solutions to
It is proved that the solution to the initial value problem , u(0,x) = 1/(1+x²), does not belong to the Gevrey class in time for 0 ≤ s < 1. The proof is based on an estimation of a double sum of products of binomial coefficients.
Borel summable solutions of the Burgers equation
We give necessary and sufficient conditions for the formal power series solutions to the initial value problem for the Burgers equation to be convergent or Borel summable.
Uniform analytic-Gevrey regularity of solutions to a semilinear heat equation
We study the Gevrey regularity down to t = 0 of solutions to the initial value problem for a semilinear heat equation . The approach is based on suitable iterative fixed point methods in based Banach spaces with anisotropic Gevrey norms with respect to the time and the space variables. We also construct explicit solutions uniformly analytic in t ≥ 0 and x ∈ ℝⁿ for some conservative nonlinear terms with symmetries.
A characterization of elliptic operators
We give a characterization of constant coefficients elliptic operators in terms of estimates of their iterations on smooth functions.
Bogdan Ziemian (1953-1997)
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