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Currently displaying 1 – 13 of 13

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On the structure of Mellin distributions

Grzegorz Łysik — 1990

Annales Polonici Mathematici

The Taylor transformation of analytic functionals with non-bounded carrier

Grzegorz Łysik — 1994

Studia Mathematica

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

Grzegorz Łysik — 2000

Annales Polonici Mathematici

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

Grzegorz Łysik — 2000

Annales Polonici Mathematici

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

Grzegorz Łysik — 1996

Annales Polonici Mathematici

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

Grzegorz Łysik — 2000

Annales Polonici Mathematici

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

Grzegorz Łysik — 1997

Studia Mathematica

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

Grzegorz Łysik — 1991

Studia Mathematica

Nonanalyticity of solutions to $\partial_{t} u = \partial ²_{x} u + u ²$

Grzegorz Łysik — 2003

Colloquium Mathematicae

It is proved that the solution to the initial value problem $\partial_{t} u = \partial ²_{x} u + u ²$ , u(0,x) = 1/(1+x²), does not belong to the Gevrey class $G^{s}$ 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

Grzegorz Łysik — 2009

Annales Polonici Mathematici

We give necessary and sufficient conditions for the formal power series solutions to the initial value problem for the Burgers equation $\partial_{t} u - \partial_{x} ² u = \partial_{x} (u ²)$ to be convergent or Borel summable.

Uniform analytic-Gevrey regularity of solutions to a semilinear heat equation

Todor Gramchev; Grzegorz Łysik — 2008

Banach Center Publications

We study the Gevrey regularity down to t = 0 of solutions to the initial value problem for a semilinear heat equation $\partial_{t} u - Δ u = u^{M}$ . The approach is based on suitable iterative fixed point methods in $L^{p}$ 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.