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A Course In Functional Analysis

Preface 1. The approximation problem and existence of best approximations 2. The uniqueness of best approximations 3. Approximation operators and some approximating functions 4. Polynomial interpolation 5. Divided differences 6. The uniform convergence of polynomial approximations 7. The theory of minimax approximation 8. The exchange algorithm 9. The convergence of the exchange algorithm 10. Rational approximation by the exchange algorithm 11. Least squares approximation 12. Properties of orthogonal polynomials 13. Approximation of periodic functions 14. The theory of best L1 approximation 15. An example of L1 approximation and the discrete case 16. The order of convergence of polynomial approximations 17. The uniform boundedness theorem 18. Interpolation by piecewise polynomials 19. B-splines 20. Convergence properties of spline approximations 21. Knot positions and the calculation of spline approximations 22. The Peano kernel theorem 23. Natural and perfect splines 24. Optimal interpolation Appendices Index.

Approximation theory and methods, by M. J. D. Powell. Pp 339. £25 (hardcover), £8·50 (paperback). 1981. ISBN 0-521-22472-1/29514-9 (Cambridge University Press)

高等学校計算数学学報 = Numerical mathematics

Fundamentals of natural computing: an overview

Full length article: Fractal interpolation functions with variable parameters and their analytical properties

We construct a generalized Cr-Fractal Interpolation Function (Cr-FIF) f by prescribing any combination of r values of the derivatives f(k), k=1,2,\dots,r, at boundary points of the interval I = [x0,xN]. Our approach to construction settles several questions of Barnsley and Harrington [J. Approx Theory, 57 (1989), pp. 14-34] when construction is not restricted to prescribing the values of $f^{(k)}$ at only the initial endpoint of the interval $I.$ In general, even in the case when r equations involving $f^{(k)}(x_0)$ and $f^{(k)}(x_N)$, $k=1,2,\dots,r,$ are prescribed, our method of construction of the $C^r$-FIF works equally well. In view of wide ranging applications of the classical cubic splines in several mathematical and engineering problems, the explicit construction of cubic spline FIF $f_{\Delta}(x)$ through moments is developed. It is shown that the sequence $\{f_{\Delta_k}(x)\}$ converges to the defining data function $\Phi(x)$ on two classes of sequences of meshes at least as rapidly as the square of the mesh norm $\| \Delta_k \|$ approaches to zero, provided that $\Phi^{(r)}(x)$ is continuous on I for r=2,3, or 4.

Generalized Cubic Spline Fractal Interpolation Functions

The theory of splines is a well studied topic, but the kinship of splines with fractals is novel We introduce a simple explicit construction for a Open image in new window-cubic Hermite Fractal Interpolation Function (FIF) Under some suitable hypotheses on the original function, we establish a priori estimates (with respect to the Lp-norm, 1≤p≤∞) for the interpolation error of the Open image in new window-cubic Hermite FIF and its first derivative Treating the first derivatives at the knots as free parameters, we derive suitable values for these parameters so that the resulting cubic FIF enjoys Open image in new window global smoothness Consequently, our method offers an alternative to the standard moment construction of Open image in new window-cubic spline FIFs Furthermore, we identify appropriate values for the scaling factors in each subinterval and the derivatives at the knots so that the graph of the resulting Open image in new window-cubic FIF lies within a prescribed rectangle These parameters include, in particular, conditions for the positivity of the cubic FIF Thus, in the current article, we initiate the study of the shape preserving aspects of fractal interpolation polynomials We also provide numerical examples to corroborate our results

A constructive approach to cubic Hermite Fractal Interpolation Function and its constrained aspects

We construct hidden variable bivariate fractal interpolation surfaces (FIS). The vector valued iterated function system (IFS) is constructed in ℝ4 and its projection in ℝ3 is taken. The extra degree of freedom coming from ℝ4 provides hidden variable, which is an important factor for flexibility and diversity in the interpolated surface. In the present paper, we construct an IFS that generates both self-similar and non-self-similar FIS simultaneously and show that the hidden variable fractal surface may be self-similar under certain conditions.

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Hidden Variable Bivariate Fractal Interpolation Surfaces

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.

A. K. B. Chand

Papers

Generalized Cubic Spline Fractal Interpolation Functions

A constructive approach to cubic Hermite Fractal Interpolation Function and its constrained aspects

Hidden Variable Bivariate Fractal Interpolation Surfaces

Fundamental Sets of Fractal Functions

Shape preservation of scientific data through rational fractal splines