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- and $^{\infty}$ -hypoellipticity of partial differential operators with constant Colombeau coefficients

Claudia Garetto (2010)

Banach Center Publications

We provide a deep investigation of the notions of - and $^{\infty}$ -hypoellipticity for partial differential operators with constant Colombeau coefficients. This involves generalized polynomials and fundamental solutions in the dual of a Colombeau algebra. Sufficient conditions and necessary conditions for - and $^{\infty}$ -hypoellipticity are given

A characterization of sets of functions and distributions on $ℝ^{n}$ described by constraints on the gradient.

Corbo Esposito, Antonio, De Arcangelis, Ricardo (1996)

Journal of Convex Analysis

A characterization of spectral operators of finite type

Charles E. Cleaver (1973)

Compositio Mathematica

A characterization of the generalized Meijer transform.

Deeba, E.Y., Koh, E.L. (1992)

International Journal of Mathematics and Mathematical Sciences

A characterization of the space $D_{F}^{'}$

S. Pilipović (1982)

Matematički Vesnik

A commutative neutrix convolution of distributions and the exchange formula

Brian Fisher, Emin Özçag, Li Chen Kuan (1992)

Archivum Mathematicum

A comparison on the commutative neutrix convolution of distributions and the exchange formula

Adem Kiliçman (2001)

Czechoslovak Mathematical Journal

Let $\tilde{f}$ , $\tilde{g}$ be ultradistributions in $𝒵^{'}$ and let ${\tilde{f}}_{n} = \tilde{f} * δ_{n}$ and ${\tilde{g}}_{n} = \tilde{g} * σ_{n}$ where ${δ_{n}}$ is a sequence in $𝒵$ which converges to the Dirac-delta function $δ$ . Then the neutrix product $\tilde{f} ⋄ \tilde{g}$ is defined on the space of ultradistributions $𝒵^{'}$ as the neutrix limit of the sequence ${\frac{1}{2} ({\tilde{f}}_{n} \tilde{g} + \tilde{f} {\tilde{g}}_{n})}$ provided the limit $\tilde{h}$ exist in the sense that $\underset{n \to \infty}{N - l i m} \frac{1}{2} 〈 {\tilde{f}}_{n} \tilde{g} + \tilde{f} {\tilde{g}}_{n}, ψ 〉 = 〈 \tilde{h}, ψ 〉$ for all $ψ$ in $𝒵$ . We also prove that the neutrix convolution product $f ♦ * g$ exist in $𝒟^{'}$ , if and only if the neutrix product $\tilde{f} ⋄ \tilde{g}$ exist in $𝒵^{'}$ and the exchange formula $F (f ♦ * g) = \tilde{f} ⋄ \tilde{g}$ is then satisfied.

A Complex Characterization of the Schwartz Space D (...).

M.A. Dostal (1971)

Mathematische Annalen

A contribution to the equivalence results for the product of distributions

Jiří Jelínek (1994)

Commentationes Mathematicae Universitatis Carolinae

Products $[S] \cdot [T]$ and $[S] \cdot T$ , defined by model delta-nets, are equivalent.

A Convolution Connected with the Kontorovich-Lebedev Transform.

Hans-Jürgen, Heß, Albrecht Glaeske (1986)

Mathematische Zeitschrift

A convolution operation for a distributional Hankel transformation

J. Betancor, B. González (1995)

Studia Mathematica

We investigate the Hankel transformation and the Hankel convolution on new spaces of generalized functions.

A convolution product of $(2 j)$ th derivative of Dirac's delta in $r$ and multiplicative distributional product between $r^{- k}$ and $\nabla (▵^{j} δ)$ .

Aguirre Téllez, Manuel A. (2003)

International Journal of Mathematics and Mathematical Sciences

A distribution Hardy transformation.

Pathak, R.S., Pandey, J.N. (1979)

International Journal of Mathematics and Mathematical Sciences

A Distribution - Theoretic Proof of Kirillov's Character Formula for Nilpotent Lie Groups.

Linda Preiss Rothschild (1974)

Mathematische Zeitschrift

A distributional representation of strip analytic functions.

Mitrovic, Dragisa (1982)

International Journal of Mathematics and Mathematical Sciences

A distributional summation formula of Euler-MacLaurin type in two variables.

Ulate, Carlos Ml., Estrada, Ricardo (1998)

Divulgaciones Matemáticas

A family of Kučera spaces

Kelly McKennon (1979)

Czechoslovak Mathematical Journal

A functional equation of Aczél and Chung in generalized functions.

Chung, Jae-Young (2008)

Advances in Difference Equations [electronic only]

A general approximation theorem of Whitney type.

Michael Langenbruch (2003)

RACSAM

We show that Whitney?s approximation theorem holds in a general setting including spaces of (ultra)differentiable functions and ultradistributions. This is used to obtain real analytic modifications for differentiable functions including optimal estimates. Finally, a surjectivity criterion for continuous linear operators between Fréchet sheaves is deduced, which can be applied to the boundary value problem for holomorphic functions and to convolution operators in spaces of ultradifferentiable functions...

A General Character Formula for Irreducible Projections on L2 of a Nilmanifold.

Frederick P. Greenleaf, Lawrence Corwin, Richard Penney (1977)

Mathematische Annalen

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