The estimation of the local phase and local amplitude of 1-D signals can be realized by the construction of the analytic signal. This includes the evaluation of the signal’s Hilbert transform, which performs a phase shift. In the past, different definitions of the analytic signal of multidimensional signals have been proposed, all of which are based on different combinations of partial and tota...

We consider rank one perturbations Aα = A+α( · , φ)φ of a self-adjoint operator A with cyclic vector φ ∈ H−1(A) on a Hilbert space H. The spectral representation of the perturbed operator Aα is given by a singular integral operator of special form. Such operators exhibit what we call ’rigidity’ and are connected with two weight estimates for the Hilbert transform. Also, some results about two w...

The estimation of the local phase and local amplitude of 1-D signals can be realized by the construction of the analytic signal. This includes the evaluation of the signal’s Hilbert transform, which performs a phase shift. In the past, different definitions of the analytic signal of multidimensional signals have been proposed, all of which are based on different combinations of partial and tota...

This paper presents new results in the theory of number theoretic Hilbert (NHT) transforms. New polymorphic solutions have been found for the 14-point and 16-point transforms. Several transform pairs are computed and solutions found for which the sequence and the transform have the same shape. The multiplicity of solutions for the same moduli increases their applicability to cryptography.

We present a survey of mixed norm inequalities for several directional operators, namely, directional Hardy-Littlewood maximal functions and Hilbert transforms (both appearing in the method of rotations of Calderón and Zygmund), X-ray transforms, and directional fractional operators related to Riesz type potentials with variable kernel. In dimensions higher than two several interesting question...

We review traditions and trends in optics and imaging recently arising by applying programmable optical devices or by sophisticated approaches for data evaluation and image reconstruction. Furthermore, a short overview is given about modeling of well-known classical optical elements, and vice versa, about optical realizations of classical mathematical transforms, as in particular Fourier, Hilbe...

This paper presents 10-point and 12-point versions of the recently introduced number theoretic Hilbert (NHT) transforms. Such transforms have applications in signal processing and scrambling. Polymorphic solutions with respect to different moduli for each of the two cases have been found. The multiplicity of solutions for the same moduli increases their applicability to cryptography.