Pointwise kernels of Schwartz distributions

Pointwise Kernels of Schwartz Distributions

We show that Schwartz distributions have kernels in the class of the pointwise nonstandard functions. The main purpose of this note is to show that every Schwartz distribution T G 3' has a kernel /: *Rd —> *C in the class of the pointwise nonstandard functions in the sense that (1) (T,p)= [ f(xy<p(x)dx for all <p G 3, where *<p is the nonstandard extension of (p. Recall that, in general, the Sc...

Asymptotic Functions as Kernels of the Schwartz Distributions

Using a version of the sequential method we introduce a class of generalized functions called here "asymptotic functions''. This class contains kernels of all Schwartz distri­ butions and is equipped with a correctly defined multiplication operation. So, in a sense, one solves the problem of "multiplication of Schwartz distributions" although the solution refers to the class of the asymptotic f...

Sharp Pointwise Estimates on Heat Kernels

Conformal predictive distributions with kernels

This paper reviews the checkered history of predictive distributions in statistics and discusses two developments, one from recent literature and the other new. The first development is bringing predictive distributions into machine learning, whose early development was so deeply influenced by two remarkable groups at the Institute of Automation and Remote Control. The second development is com...

Functional Equations in Schwartz Distributions and Their Stabilities

Employing two methods we consider a class of n-dimensional functional equations in the space of Schwartz distributions. As the first approach, employing regularizing functions we reduce the equations in distributions to classical ones of smooth functions and find the solutions. Secondly, using differentiation in distributions, converting the functional equations to differential equations and fi...