### On the structure of Mellin distributions

Skip to main content (access key 's'),
Skip to navigation (access key 'n'),
Accessibility information (access key '0')

Back to Simple Search
# Advanced Search

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.

Definitions, properties, examples and applications of generalized analytic functions introduced by B. Ziemian are presented.

Fundamental solutions to linear partial differential equations with constant coefficients are represented in the form of Laplace type integrals.

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.

Solutions to singular linear ordinary differential equations with analytic coefficients are found in the form of Laplace type integrals.

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.

It is proved that the solution to the initial value problem ${\partial}_{t}u=\partial {\xb2}_{x}u+u\xb2$, 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.

We give necessary and sufficient conditions for the formal power series solutions to the initial value problem for the Burgers equation ${\partial}_{t}u-{\partial}_{x}\xb2u={\partial}_{x}\left(u\xb2\right)$ to be convergent or Borel summable.

We study the Gevrey regularity down to t = 0 of solutions to the initial value problem for a semilinear heat equation ${\partial}_{t}u-\Delta u={u}^{M}$. 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.

We give a characterization of constant coefficients elliptic operators in terms of estimates of their iterations on smooth functions.

**Page 1**