The Hilbert Transform 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...

The Schwartz space and the Fourier transform

-boundedness of the Hilbert transform

The Hilbert transform is essentially the only singular operator in dimension 1. This undoubtedly makes it one of the the most important linear operators in harmonic analysis. The Hilbert transform has had a profound bearing on several theoretical and physical problems across a wide range of disciplines; this includes problems in Fourier convergence, complex analysis, potential theory, modulatio...

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...

The Number Theoretic Hilbert Transform

Abstract This paper presents a general expression for a number-theoretic Hilbert transform (NHT). The transformations preserve the circulant nature of the discrete Hilbert transform (DHT) matrix together with alternating values in each row being zero and non-zero. Specific examples for 4point, 6-point, and 8-point NHT are provided. The NHT transformation can be used as a primitive to create cry...