Related papers: Tempered distributions and Schwartz functions on d…

We define Schwartz functions, tempered functions and tempered distributions on (possibly singular) real algebraic varieties. We prove that all classical properties of these spaces, defined previously on affine spaces and on Nash manifolds,…

In this paper we extend the notions of Schwartz functions, tempered functions and generalized Schwartz functions to Nash (i.e. smooth semi-algebraic) manifolds. We reprove for this case classically known properties of Schwartz functions on…

We introduce a new category called Quasi-Nash, unifying Nash manifolds and algebraic varieties. We define Schwartz functions, tempered functions and tempered distributions in this category. We show that properties that hold on affine…

We give explicit transforms for Hilbert spaces associated with positive definite functions on $\mathbb{R}$, and positive definite tempered distributions, incl., generalizations to non-abelian locally compact groups. Applications to the…

We present the construction of a theory of distributions (generalized functions) with a ``thick submanifold'', that is, a new theory of thick distributions on $\mathbb{R}^n$ whose domain contains a smooth submanifold on which the test…

The final goal of the present work is to extend the Fourier transform on the Heisenberg group $\H^d,$ to tempered distributions. As in the Euclidean setting, the strategy is to first show that the Fourier transform is an isomorphism on the…

Distribution theory is a cornerstone of the theory of partial differential equations. We report on the progress of formalizing the theory of tempered distributions in the interactive proof assistant Lean, which is the first formalization in…

These notes briefly consider convolutions of tempered distributions with functions in the Schwartz class.

Schwartz functions, or measures, are defined on any smooth semi-algebraic ("Nash") manifold, and are known to form a cosheaf for the semi-algebraic restricted topology. We extend this definition to smooth semi-algebraic stacks, which are…

In this review article we present regularity properties of generalized functions which are useful in the analysis of non-linear problems. It is shown that Schwartz distributions embedded into our new spaces of generalized functions, with…

We present some remarks about the embedding of spaces of Schwartz distributions into spaces of Colombeau generalized functions. We show that the various constructions of such embeddings existing in the literature lead in fact to the same…

In this paper we define Schwartz families in tempered distribution spaces and prove many their properties. Schwartz families are the analogous of infinite dimensional matrices of separable Hilbert spaces, but for the Schwartz test function…

Given a quadratic CR manifold $\mathcal{M}$ embedded in a complex space, we study Paley-Wiener-Schwartz theorems for spaces of Schwartz functions and tempered distributions on $\mathcal{M}$.

We first show that the projection image of a discrete definable set is again discrete for an arbitrary definably complete locally o-minimal structure. This fact together with the results in a previous paper implies tame dimension theory and…

The twisted product of functions on $R^{2N}$ is extended to a $*$-algebra of tempered distributions which contains the rapidly decreasing smooth functions, the distributions of compact support, and all polynomials, and moreover is invariant…

Here we simplify the proof of the de Rham theorem for Schwartz functions on affine Nash manifolds and generalize the result to the case of non affine Nash manifolds.

These notes briefly consider some aspects of the Schwartz class of rapidly decreasing smooth functions, tempered distributions, and harmonic functions of polynomial growth.

We prove a theorem which provides a method for constructing points on varieties defined by certain smooth functions. We require that the functions are definable in a definably complete expansion of a real closed field and are locally…

In this paper we define the Schwartz linear operators among spaces of tempered distributions. These operators are the analogous of linear continuous operators among separable Hilbert spaces, but in the case of spaces endowed with Schwartz…

This paper continues the analysis of Schr\"odinger type equations with distributional coefficients initiated by the authors in [3]. Here we consider coefficients that are tempered distributions with respect to the space variable and are…