Generalized Hilbert Transforms in Tempered Distributions

Generalized multidimensional Hilbert transforms in Clifford analysis

During the last fifty years, Clifford analysis has gradually developed to a comprehensive theory offering a direct, elegant, and powerful generalization to higher dimension of the theory of holomorphic functions in the complex plane. In its most simple but still useful setting, flat m-dimensional Euclidean space, Clifford analysis focusses on the so-called monogenic functions, that is, null sol...

Generalized Fourier Transforms of Distributions

In [1], R. A. Kunze has presented a notion of generalized Fourier transform of functions on locally compact abelian groups. The point of this paper is to extend this idea in the direction of distribution theory and to present some initial results on this generalized Fourier transform of distributions. In a later paper we hope to investigate in detail the domain, range, and kernel of this transf...

Multiresolution Expansion and Approximation Order of Generalized Tempered Distributions

Let KMM 󸀠󸀠(R) be the generalized tempered distributions of eeMMMMMM-growth with restricted order rr r N0, where the function MM(MM) grows faster than any linear functions as |MM| x x. We show the convergence of multiresolution expansions ofKMM(R) in the test function spaceKMM(R) ofKMM(R). In addition, we show that the kernel of an integral operatorKK K KMM(R) x KMM(R) provides approximation ord...

2D Image Analysis by Generalized Hilbert Transforms in Conformal Space

This work presents a novel rotational invariant quadrature filter approach called the conformal monogenic signal for analyzing i(ntrinsic)1D and i2D local features of any curved 2D signal such as lines, edges, corners and junctions without the use of steering. The conformal monogenic signal contains the monogenic signal as a special case for i1D signals and combines monogenic scale space, phase...

Generalized Frames in Hilbert Spaces