Fractional Operators, Dirichlet Averages, and Splines

Fractional differential and integral operators, Dirichlet averages, and splines of complex order are three seemingly distinct mathematical subject areas addressing different questions and employing different methodologies. It is the purpose of this paper to show that there are deep and interesting relationships between these three areas. First a brief introduction to fractional differential and integral operators defined on Lizorkin spaces is presented and some of their main properties exhibited. This particular approach has the advantage that several definitions of fractional derivatives and integrals coincide. We then introduce Dirichlet averages and extend their definition to an infinite-dimensional setting that is needed to exhibit the relationships to splines of complex order. Finally, we focus on splines of complex order and, in particular, on cardinal B-splines of complex order. The fundamental connections to fractional derivatives and integrals as well as Dirichlet averages are presented

Sputter Crater Contour Mapping with Multilayered Films

Multilayered films composed of alternating 200 Å Al and 267 Å Al203 layers are made by physical vapor deposition. Twenty-two pairs of these films are deposited on a polished Si wafer. Ion beam sputtering is used to form craters in the multilayered film. When a crater is viewed or photographed in situ by scanning electron microscopy, the Al2O3 layers appear bright and the Al layers appear dark. In the scanning electron microscope (SEM) the Al2O3 layers have a high secondary electron yield compared to Al. In secondary ion mass spectrometry (SIMS), using Cs+ as the ion beam, imaging with O- produces an image with Al2O3 layers appearing white and with Al layers appearing dark. Scanning Auger microscopy (SAM) imaging of oxygen produces the same result. In all cases, the alternating bright and dark layers along the wall of the sputter crater form a contour map. The width of each bright band represents a change of depth corresponding to the thickness of the Al2O3 layer and similarly for the dark Al bands. Therefore, the operator of a SEM, SAM or SIMS unit can determine the depth as well as the shape of a sputter crater in situ by using a multilayered film. The main requirement is that the films be smooth on a scale that is small compared to the thickness of each layer and that alternate films have high contrast in the imaging process

Stability of Coalescence Hidden variable Fractal Interpolation Surfaces

In the present paper, the stability of Coalescence Hidden variable Fractal Interpolation Surfaces(CHFIS) is established. The estimates on error in approximation of the data generating function by CHFIS are found when there is a perturbation in independent, dependent and hidden variables. It is proved that any small perturbation in any of the variables of generalized interpolation data results in only small perturbation of CHFIS. Our results are likely to be useful in investigations of texture of surfaces arising from the simulation of surfaces of rocks, sea surfaces, clouds and similar natural objects wherein the generating function depends on more than one variable

Multiwavelets: Some approximation-theoretic properties, sampling on the interval, and translation invariance.

In this survey paper, some of the basic properties of multiwavelets are reviewed. Particular emphasis is given to approximation-theoretic issues and sampling on compact intervals. In addition, a translation invariant multiwavelet transform is discussed and the regularity and approximation order of the associated correlation matrices, which satisfy a particular matrix-valued refinement equation, are presented

Interpolation and approximation with splines and fractals.

This textbook is intended to supplement the classical theory of uni- and multivariate splines and their approximation and interpolation properties with those of fractals, fractal functions, and fractal surfaces. This synthesis will complement currently required courses dealing with these topics and expose the prospective reader to some new and deep relationships. In addition to providing a classical introduction to the main issues involving approximation and interpolation with uni- and multivariate splines, cardinal and exponential splines, and their connection to wavelets and multiscale analysis, which comprises the first half of the book, the second half will describe fractals, fractal functions and fractal surfaces, and their properties. This also includes the new burgeoning theory of superfractals and superfractal functions. The theory of splines is well-established but the relationship to fractal functions is novel. Throughout the book, connections between these two apparently different areas will be exposed and presented. In this way, more options are given to the prospective reader who will encounter complex approximation and interpolation problems in real-world modeling. Numerous examples, figures, and exercises accompany the material

On local fractal functions in Banach spaces.

We give a short introduction to local fractal functions defined on several classes of Banach spaces. The emphasis is on the Banach space of bounded functions B, the Lebesgue spaces Lp, 1 ≤ p ≤ ∞, and the Sobolev spaces Wn,p, n ε N0and 1 ≤ p ≤ ∞