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A Course In Functional Analysis

高等学校計算数学学報 = Numerical mathematics

nonlinear functional analysis and its applications is available in our book collection an online access to it is set as public so you can get it instantly. Our books collection hosts in multiple countries, allowing you to get the most less latency time to download any of our books like this one. Merely said, the nonlinear functional analysis and its applications is universally compatible with any devices to read.

Nonlinear Functional Analysis And Its Applications

Most engineering and scientific phenomena, such as the surface of a landscape or the continuously changing temperature at a location are inherently infinite in space or time or both. We cannot measure all the data. Generally it is possible to record surface elevation values or the temperature only at some specific locations and times.

Interpolation and Approximation

Fundamentals of natural computing: an overview

https://rgmia.org/papers/v7n3/Hermite.pdf

Generalization of Hermite functions by fractal interpolation

Fractal methodology provides a general frame for the understanding of real-world phenomena. In particular, the classical methods of real-data interpolation can be generalized by means of fractal techniques. In this paper, we describe a procedure for the construction of smooth fractal functions, with the help of Hermite osculatory polynomials. As a consequence of the process, we generalize any smooth interpolant by means of a family of fractal functions. In particular, the elements of the class can be defined so that the smoothness of the original is preserved. Under some hypotheses, bounds of the interpolation error for function and derivatives are obtained. A set of interpolating mappings associated to a cubic spline is defined and the density of fractal cubic splines in
 is proven.

/pdf/smooth-fractal-interpolation-epusjpdjzm.pdf

Smooth fractal interpolation

Fractal interpolants constructed through iterated function systems prove more general than classical interpolants. In this paper, we assign a family of fractal functions to several classes of real mappings like, for instance, maps defined on sets that are not intervals, maps integrable but not continuous and may be defined on unbounded domains. In particular, based on fractal interpolation functions, we construct fractal Muntz polynomials that successfully generalize classical Muntz polynomials. The parameters of the fractal Muntz system enable the control and modification of the properties of original functions. Furthermore, we deduce fractal versions of classical Muntz theorems. In this way, the fractal methodology generalizes the fundamental sets of the classical approximation theory and we construct complete systems of fractal functions in spaces of continuous and p-integrable mappings on bounded domains.

Fundamental Sets of Fractal Functions

Fractal interpolation is a modern technique in approximation theory to fit and analyze scientific data. We develop a new class of $$\mathcal C ^1$$ - rational cubic fractal interpolation functions, where the associated iterated function system uses rational functions of the form $$\frac{p_i(x)}{q_i(x)},$$ where $$p_i(x)$$ and $$q_i(x)$$ are cubic polynomials involving two shape parameters. The rational cubic iterated function system scheme provides an additional freedom over the classical rational cubic interpolants due to the presence of the scaling factors and shape parameters. The classical rational cubic functions are obtained as a special case of the developed fractal interpolants. An upper bound of the uniform error of the rational cubic fractal interpolation function with an original function in $$\mathcal C ^2$$ is deduced for the convergence results. The rational fractal scheme is computationally economical, very much local, moderately local or global depending on the scaling factors and shape parameters. Appropriate restrictions on the scaling factors and shape parameters give sufficient conditions for a shape preserving rational cubic fractal interpolation function so that it is monotonic, positive, and convex if the data set is monotonic, positive, and convex, respectively. A visual illustration of the shape preserving fractal curves is provided to support our theoretical results.

M. A. Navascués

Papers

Generalization of Hermite functions by fractal interpolation

Smooth fractal interpolation

Fundamental Sets of Fractal Functions

Shape preservation of scientific data through rational fractal splines

Fractal perturbation preserving fundamental shapes: Bounds on the scale factors