is the most fundamental example. Much like in the classical case of the Lebesgue di erentiation theorem, pointwise convergence almost everywhere of the inverse Fourier transform to f ∈ Lp (R) can be reduced to Lp bounds for the maximal operator C. Weak type L2 bounds were rst obtained by Lennart Carleson in 1966 [8], thus providing a surprising a rmative solution to the question of pointwise co...

We define the concept of instability index of an isolated eigenvalue of a non-self-adjoint operator, and prove some of its general properties. We also describe a stable procedure for computing this index for Schrödinger operators in one dimension, and apply it to the complex resonances of a typical operator with a dilation analytic potential. AMS subject classification: 34L05, 35P05, 47A75, 49R...

Two procedures to compute the output distribution φS of certain stack filters S (so called erosion-dilation cascades) are given. One rests on the disjunctive normal form of S and also yields the rank selection probabilities. The other is based on inclusion-exclusion and e.g. yields φS for some important LULU -operators S. Properties of φS can be used to characterize smoothing properties.

A central feature in the Hilbert space formulation of classical mechanics is the quantisation of classical Liouville densities, leading to what may be termed term Groenewold operators. We investigate the spectra of the Groenewold operators that correspond to Gaussian and to certain uniform Liouville densities. We show that when the classical coordinatemomentum uncertainty product falls below He...

This paper presents the operation of tangential dilation, which describes the touching of differentiable surfaces. It generalizes the classical dilation, but is invertible. It is shown that line segments are eigenfunctions of this dilation, and are parallel transported, and that curvature is additive. We then present the slope transform which provides tangential morphology with the analytical p...

A proper time observable for a quantum clock is introduced and it found that the read by one conditioned on another reading different obeys classical dilation in accordance with special relativistic kinematical dilation. Here, we extend this proposal to weak gravitational field order investigate whether equivalence principle holds matter. We find general states similar relativity. However, unli...

This paper presents a new application of mathematical morphology (MM) operators for low speed slew bearing condition monitoring. The MM operators were used as a signal processing step and feature extraction method for bearing vibration signals. Four basic MM operators; erosion, dilation, closing and opening, were studied. This paper also investigates another potential MM operator, namely gradie...

Refinable functions and distributions with integer dilations have been studied extensively since the pioneer work of Daubechies on wavelets. However, very little is known about refinable functions and distributions with non-integer dilations, particularly concerning its regularity. In this paper we study the decay of the Fourier transform of refinable functions and distributions. We prove that ...

Paul Halmos’ work in dilation theory began with a question and its answer: Which operators on a Hilbert space H can be extended to normal operators on a larger Hilbert space K ⊇ H? The answer is interesting and subtle. The idea of representing operator-theoretic structures in terms of conceptually simpler structures acting on larger Hilbert spaces has become a central one in the development of ...

In this work, we obtain a relation for average quantum time dilation between two clocks A and B in the near-horizon region of black hole supported by Rindler metric conformal tortoise coordinate. It is indicated that identified with classical flat background limits.