19,503 research outputs found
A theoretical insight into morphological operations in surface measurement by introducing the slope transform
As one of the tools for surface analysis, morphological operations, although not as popular as linear convolution operations (e.g. the Gaussian filter), are really useful in mechanical surface reconstruction, surface filtration, functional simulation etc. By introducing the slope transform originally developed for signal processing into the field of surface metrology, an analytic capability is gained for morphological operations, paralleling that of the Fourier transform in the context of linear convolution. Using the slope transform, the tangential dilation is converted into the addition in the slope domain, just as by the Fourier transform, the convolution switches into the multiplication in the frequency domain. Under the theory of the slope transform, the slope and curvature changes of the structuring element to the operated surface can be obtained, offering a deeper understanding of morphological operations in surface measurement. The derivation of the analytical solutions to the tangential dilation of a sine wave and a disk by a disk are presented respectively. An example of the discretized tangential dilation of a sine wave by the disks with two different radii is illustrated to show the consistency and distinction between the tangential dilation and the classical dilation
Boolean convolution of probability measures on the unit circle
We introduce the boolean convolution for probability measures on the unit
circle. Roughly speaking, it describes the distribution of the product of two
boolean independent unitary random variables. We find an analogue of the
characteristic function and determine all infinitely divisible probability
measures on the unit circle for the boolean convolution.Comment: 13 pages, to appear in volume 15 of Seminaires et Congre
Injectivity of sections of convex harmonic mappings and convolution theorems
In the article the authors consider the class of
sense-preserving harmonic functions defined in the unit disk
and normalized so that and , where
and are analytic in the unit disk. In the first part of the article we
present two classes and of
functions from and show that if
and , then the harmonic convolution is a univalent
and close-to-convex harmonic function in the unit disk provided certain
conditions for parameters and are satisfied. In the second
part we study the harmonic sections (partial sums) where , and denote the -th partial sums of
and , respectively. We prove, among others, that if
is a univalent harmonic convex mapping,
then is univalent and close-to-convex in the disk for
, and is also convex in the disk for
and . Moreover, we show that the section of is not convex in the disk but is shown to be convex
in a smaller disk.Comment: 16 pages, 3 figures; To appear in Czechoslovak Mathematical Journa
More on the Density of Analytic Polynomials in Abstract Hardy Spaces
Let be the sequence of the Fej\'er kernels on the unit circle
. The first author recently proved that if is a separable
Banach function space on such that the Hardy-Littlewood maximal
operator is bounded on its associate space , then
for every as . This implies that the set of analytic
polynomials is dense in the abstract Hardy space built
upon a separable Banach function space such that is bounded on . In
this note we show that there exists a separable weighted space such
that the sequence does not always converge to in the norm of
. On the other hand, we prove that the set is dense in
under the assumption that is merely separable.Comment: To appear in the Proceedings of IWOTA 201
- …