It is shown that, under some general algebraic conditions on fixed real numbers a,b,α,β, every solution f:ℝ → ℝ of the system of functional inequalities f(x+a) ≤ f(x)+α, f(x+b) ≤ f(x)+β that is continuous at some point must be a linear function (up to an additive constant). Analogous results for three other similar simultaneous systems are presented. An application to a characterization of -norm is given.
Let P be a hypoelliptic polynomial. We consider classes of ultradifferentiable functions with respect to the iterates of the partial differential operator P(D) and prove that such classes satisfy a Paley-Wiener type theorem. These classes and the corresponding test spaces are nuclear.
In this paper, we study the finite Hankel transformation on spaces of generalized functions by developing a new procedure. We consider two Hankel type integral transformations and connected by the Parseval equation A space of functions and a space of complex sequences are introduced. is an isomorphism from onto when . We propose to define the generalized finite Hankel transform of by
Motivated by applications to transition semigroups, we introduce the notion of a norming dual pair and study a Pettis-type integral on such pairs. In particular, we establish a sufficient condition for integrability. We also introduce and study a class of semigroups on such dual pairs which are an abstract version of transition semigroups. Using our results, we give conditions ensuring that a semigroup consisting of kernel operators has a Laplace transform which also consists of kernel operators....
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.
We give an example of a fourth degree polynomial which does not satisfy Rolle’s Theorem in the unit ball of .
Introduction. For bounded domains in satisfying the cone condition there are many embedding and module structure theorem for Sobolev spaces which are of great importance in solving partial differential equations. Unfortunately, most of them are wrong on arbitrary unbounded domains or on open manifolds. On the other hand, just these theorems play a decisive role in foundations of nonlinear analysis on open manifolds and in solving partial differential equations. This was pointed out by the author...
For a bounded operator T on a separable infinite-dimensional Banach space X, we give a "random" criterion not involving ergodic theory which implies that T is frequently hypercyclic: there exists a vector x such that for every non-empty open subset U of X, the set of integers n such that Tⁿx belongs to U, has positive lower density. This gives a connection between two different methods for obtaining the frequent hypercyclicity of operators.