THE SMOOTHNESS AND DIMENSION OF FRACTAL INTERPOLATION FUNCTIONS

In this paper, we investigate the smoothness of non-equidistant fractal interpolation functions We obtain the Holder exponents of such fractal interpolation functions by using the technique of operator approximation. At last, We discuss the series expressiong of these functions and give a Box-counting dimension estimation of “critical” fractal interpohltion functions by using our smoothness results.     

A Construction of Compactly Supported Biorthogonal Scaling Vectors and Multiwavelets on R

In Kessler (Appl. Comput. Harmonic Anal.9 (2000), 146–165), a construction was given for a class of orthogonal compactly supported scaling vectors on R², called short scaling vectors, and their associated multiwavelets. The span of the translates of the scaling functions along a triangular lattice includes continuous piecewise linear functions on the lattice, although the scaling functions are fractal interpolation functions and possibly nondifferentiable. In this paper, a similar construction will be used to create biorthogonal scaling vectors and their associated multiwavelets. The additional freedom will allow for one of the dual spaces to consist entirely of the continuous piecewise linear functions on a uniform subdivision of the original triangular lattice.     

Variational Principles in Curve Design

The paper considers Hermite interpolation for vector-valued functions. Corresponding to the interpolating functions f we define functionals I which contain function values of f^(r) and integrals of f^(r) where 0 ≤ r ≤ m for some integer m. The main purpose of the paper is to characterize those functions which satisfy the interpolation problem and have a minimal value of I. These characterizations contain several results of the literature including splines in tension and geometric splines.     

Bernstein Fractal Trigonometric Approximation

Fractal interpolation and approximation received a lot of attention in the last thirty years. The main aim of the current article is to study a fractal trigonometric approximants which converge to the given continuous function even if the magnitude of the scaling factors does not approach zero. In this paper, we first introduce a new class of fractal approximants, namely, Bernstein \(\alpha \)-fractal functions using the theory of fractal approximation and Bernstein polynomial. Using the proposed class of fractal approximants and imposing no condition on corresponding scaling factors, we establish that the set of Bernstein \(\alpha \)-fractal trigonometric functions is fundamental in the space of continuous periodic functions. Fractal version of Gauss formula of trigonometric interpolation is obtained by means of Bernstein trigonometric fractal polynomials. We study the Bernstein fractal Fourier series of a continuous periodic function \(f\) defined on \([-l,l]\). The Bernstein fractal Fourier series converges to \(f\) even if the magnitude of the scaling factors does not approach zero. Existence of the \(\mathcal{C}^{r}\)-Bernstein fractal functions is investigated, and Bernstein cubic spline fractal interpolation functions are proposed based on the theory of \(\mathcal{C}^{r}\)-Bernstein fractal functions.

     

A de Montessus type convergence study of a least-squares vector-valued rational interpolation procedure

In a recent paper of the author [A. Sidi, A new approach to vector-valued rational interpolation, J. Approx. Theory 130 (2004) 177–187], three new interpolation procedures for vector-valued functions F(z), where F:C→C^N, were proposed, and some of their algebraic properties were studied. One of these procedures, denoted IMPE, was defined via the solution of a linear least-squares problem. In the present work, we concentrate on IMPE, and study its convergence properties when it is applied to meromorphic functions with simple poles and orthogonal vector residues. We prove de Montessus and Koenig type theorems when the points of interpolation are chosen appropriately.     

A de Montessus type convergence study for a vector-valued rational interpolation procedure

In a recent paper of the author [8], three new interpolation procedures for vector-valued functions F(z), where F: ℂ → ℂ^N, were proposed, and some of their algebraic properties were studied. In the present work, we concentrate on one of these procedures, denoted IMMPE, and study its convergence properties when it is applied to meromorphic functions. We prove de Montessus and Koenig type theorems in the presence of simple poles when the points of interpolation are chosen appropriately. We also provide simple closed-form expressions for the error in case the function F(z) in question is itself a vector-valued rational function whose denominator polynomial has degree greater than that of the interpolant.     

Vector-valued fractal functions: Fractal dimension and fractional calculus

There are many research available on the study of a real-valued fractal interpolation function and fractal dimension of its graph. In this paper, our main focus is to study the dimensional results for a vector-valued fractal interpolation function and its Riemann–Liouville fractional integral. Here, we give some results which ensure that dimensional results for vector-valued functions are quite different from real-valued functions. We determine interesting bounds for the Hausdorff dimension of the graph of a vector-valued fractal interpolation function. We also obtain bounds for the Hausdorff dimension of the associated invariant measure supported on the graph of a vector-valued fractal interpolation function. Next, we discuss more efficient upper bound for the Hausdorff dimension of measure in terms of probability vector and contraction ratios. Furthermore, we determine some dimensional results for the graph of the Riemann–Liouville fractional integral of a vector-valued fractal interpolation function.     

Fractional Calculus on Fractal Interpolation for a Sequence of Data with Countable Iterated Function System

In recent years, the concept of fractal analysis is the best nonlinear tool towards understanding the complexities in nature. Especially, fractal interpolation has flexibility for approximation of nonlinear data obtained from the engineering and scientific experiments. Random fractals and attractors of some iterated function systems are more appropriate examples of the continuous everywhere and nowhere differentiable (highly irregular) functions, hence fractional calculus is a mathematical operator which best suits for analyzing such a function. The present study deals the existence of fractal interpolation function (FIF) for a sequence of data \({\{(x_n,y_n):n\geq 2\}}\) with countable iterated function system, where \({x_n}\) is a monotone and bounded sequence, \({y_n}\) is a bounded sequence. The integer order integral of FIF for sequence of data is revealed if the value of the integral is known at the initial endpoint or final endpoint. Besides, Riemann–Liouville fractional calculus of fractal interpolation function had been investigated with numerical examples for analyzing the results.     

Recursive Kernels

This paper is an extension of earlier papers [8, 9] on the “native” Hilbert spaces of functions on some domain Ω ⊂ R ^d in which conditionally positive definite kernels are reproducing kernels. Here, the focus is on subspaces of native spaces which are induced via subsets of Ω, and we shall derive a recursive subspace structure of these, leading to recursively defined reproducing kernels. As an application, we get a recursive Neville-Aitken-type interpolation process and a recursively defined orthogonal basis for interpolation by translates of kernels.     

Generalized Interpolation in Bergman Spaces and Extremal Functions

In a recent paper A. Schuster and K. Seip [SchS] have characterized interpolating sequences for Bergman spaces in terms of extremal functions (or canonical divisors). As these are natural analogues in Bergman spaces of Blaschke products, this yields a Carleson type condition for interpolation. We intend to generalize this idea to generalized free interpolation in weighted Bergman spaces B^{p, α} as was done by V. Vasyunin [Va1] and N. Nikolski [Ni1] (cf.also [Ha2]) in the case of Hardy spaces. In particular we get a strong necessary condition for free interpolation in B^{p, α} on zero–sets of B^{p, α}–functions that in the special case of finite unions of B^{p, α}–interpolating sequences turns out to be also sufficient.     

An interpolation problem in tube domains

Some recent results concerning theL-problem of moments in two variables are related via the Fourier-Laplace transform to an interpolation problem in the tube domain over a quadrant inR ². The class of analytic functions for which the interpolation problem is posed is identified with the symbols of all bounded analytic Wiener-Hopf operators acting on theH ²-Hardy space of the tube domain. The extremal solutions of the corresponding truncated problem are computed and the related uniqueness phenomenon is also discussed.     

Fundamental Sets of Fractal Functions

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 Müntz polynomials that successfully generalize classical Müntz polynomials. The parameters of the fractal Müntz system enable the control and modification of the properties of original functions. Furthermore, we deduce fractal versions of classical Müntz 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. This work is supported by the project No: SB 2005-0199, Spain.     

Convergence of trajectories in fractal interpolation of stochastic processes

The notion of fractal interpolation functions (FIFs) can be applied to stochastic processes. Such construction is especially useful for the class of α-self-similar processes with stationary increments and for the class of α-fractional Brownian motions. For these classes, convergence of the Minkowski dimension of the graphs in fractal interpolation of the Hausdorff dimension of the graph of original process was studied in [Herburt I, Małysz R. On convergence of box dimensions of fractal interpolation stochastic processes. Demonstratio Math 2000;4:873–88. [11]], [Małysz R. A generalization of fractal interpolation stochastic processes to higher dimension. Fractals 2001;9:415–28. [15]], and [Herburt I. Box dimension of interpolations of self-similar processes with stationary increments. Probab Math Statist 2001;21:171–8. [10]].We prove that trajectories of fractal interpolation stochastic processes converge to the trajectory of the original process. We also show that convergence of the trajectories in fractal interpolation of stochastic processes is equivalent to the convergence of trajectories in linear interpolation.     

Least Frobenius norm updating of quadratic models that satisfy interpolation conditions

Quadratic models of objective functions are highly useful in many optimization algorithms. They are updated regularly to include new information about the objective function, such as the difference between two gradient vectors. We consider the case, however, when each model interpolates some function values, so an update is required when a new function value replaces an old one. We let the number of interpolation conditions, m say, be such that there is freedom in each new quadratic model that is taken up by minimizing the Frobenius norm of the second derivative matrix of the change to the model. This variational problem is expressed as the solution of an (m+n+1)×(m+n+1) system of linear equations, where n is the number of variables of the objective function. Further, the inverse of the matrix of the system provides the coefficients of quadratic Lagrange functions of the current interpolation problem. A method is presented for updating all these coefficients in ({m+n}²) operations, which allows the model to be updated too. An extension to the method is also described that suppresses the constant terms of the Lagrange functions. These techniques have a useful stability property that is investigated in some numerical experiments.     

The interpolation problem for ridge functions

It is shown that the interpolation problem for ridge functions can be solved if and only if the rank of a certain matrix A equals the number of interpolation points. The elements of the matrix A are either 0 or 1 and can be easilyfound from the arguments of the unknown functions. It is shown that Sun's Characteristic, or incidence matrix C is given by C = AA ^T . From this it follows that the rank condition is equivalent to Sun's positive definite C condition.     

Lipschitz Continuity and Interpolation Properties of Some Classes of Increasing Functions

We study local Lipschitz continuity and interpolation properties of some classes of increasing functions defined on the cone Rⁿ ₊₊ of n-vectors with positive coordinates. We also study the so-called self-conjugate increasing positively homogeneous functions.     

Discrete-Time Dichotomous Well-Posed Linear Systems and Generalized Schur–Nevanlinna–Pick Interpolation

We introduce a class of matrix-valued functions W called “L²- regular”. In case W is J-inner, this class coincides with the class of “strongly regular J-inner” matrix functions in the sense of Arov–Dym. We show that the class of L²-regular matrix functions is exactly the class of transfer functions for a discrete-time dichotomous (possibly infinite-dimensional) input-state-output linear system having some additional stability properties. When applied to J-inner matrix functions, we obtain a state-space realization formula for the resolvent matrix associated with a generalized Schur–Nevanlinna–Pick interpolation problem. Communicated by Daniel Alpay Submitted: August 20, 2006; Accepted: September 13, 2006     

How good can polynomial interpolation on the sphere be?

This paper explores the quality of polynomial interpolation approximations over the sphere S ^r−1⊂R ^r in the uniform norm, principally for r=3. Reimer [17] has shown there exist fundamental systems for which the norm ‖Λ _n ‖ of the interpolation operator Λ _n , considered as a map from C(S ^r−1) to C(S ^r−1), is bounded by d _n , where d _n is the dimension of the space of all spherical polynomials of degree at most n. Another bound is d _n ^1/2(λ_avg/λ_min)^1/2, where λ_avg and λ_min are the average and minimum eigenvalues of a matrix G determined by the fundamental system of interpolation points. For r=3 these bounds are (n+1)² and (n+1)(λ_avg/λ_min)^1/2, respectively. In a different direction, recent work by Sloan and Womersley [24] has shown that for r=3 and under a mild regularity assumption, the norm of the hyperinterpolation operator (which needs more point values than interpolation) is bounded by O(n ^1/2), which is optimal among all linear projections. How much can the gap between interpolation and hyperinterpolation be closed?

For interpolation the quality of the polynomial interpolant is critically dependent on the choice of interpolation points. Empirical evidence in this paper suggests that for points obtained by maximizing λ_min, the growth in ‖Λ _n ‖ is approximately n+1 for n<30. This choice of points also has the effect of reducing the condition number of the linear system to be solved for the interpolation weights. Choosing the points to minimize the norm directly produces fundamental systems for which the norm grows approximately as 0.7n+1.8 for n<30. On the other hand, ‘minimum energy points’, obtained by minimizing the potential energy of a set of (n+1)² points on S ², turn out empirically to be very bad as interpolation points.

This paper also presents numerical results on uniform errors for approximating a set of test functions, by both interpolation and hyperinterpolation, as well as by non-polynomial interpolation with certain global basis functions.

     

On Interpolation Spaces with the Gelfand-Phillips Property

In this paper we study interpolation spaces generated by some interpolation functors. We show that under some conditions for Banach couples X and Y the spaces dual to the orbits of elements are Gelfand—Philips spaces. Consequently, the ideal of nuclear operators from X to Y contains a copy of l¹. We give also an interpolation theorem for limited operators.