Search

Establishing the accurate relationship between fractional calculus and fractals is an important research content of fractional calculus theory. In the present paper, we investigate the relationship between fractional calculus and fractal functions, based only on fractal dimension considerations. Fractal dimension of the Riemann–Liouville fractional integral of continuous functions seems no more than fractal dimension of functions themselves. Meanwhile fractal dimension of the Riemann–Liouville fractional differential of continuous functions seems no less than fractal dimension of functions themselves when they exist. After further discussion, fractal dimension of the Riemann–Liouville fractional integral is at least linearly decreasing and fractal dimension of the Riemann–Liouville fractional differential is at most linearly increasing for the Hölder continuous functions. Investigation about other fractional calculus, such as the Weyl-Marchaud fractional derivative and the Weyl fractional integral has also been given elementary. This work is helpful to reveal the mechanism of fractional calculus on continuous functions. At the same time, it provides some theoretical basis for the rationality of the definition of fractional calculus. This is also helpful to reveal and explain the internal relationship between fractional calculus and fractals from the perspective of geometry. [ABSTRACT FROM AUTHOR]

This paper investigates fractal dimension of product of continuous functions with Box dimension on [ 0 , 1 ]. For two continuous functions with different Box dimensions, the Box dimension of their product has been proved to be the larger one. Furthermore, the Box dimension of product of two continuous functions with the same Box dimension may not exist. Definitions of regular fractal and local fractal functions have been given. Product of a regular fractal function and a local fractal function with the same Box dimension must still be the original Box dimension. [ABSTRACT FROM AUTHOR]

In this paper, we mainly investigate continuous functions with unbounded variation on closed intervals. Given the increasing number of proposals and definitions of different kinds of functions with fractal structure in the scope of fractal functions, it is important to introduce a systematic classification and discuss elementary definitions of fractal functions. Different examples of fractal functions with certain fractal dimensions have been given. Then, we have introduced different kind of fractal function such as singular fractal function, irregular fractal function, regular fractal function and complete regular fractal function. We also introduce fractal functions of integer dimension with one and two, respectively. Other continuous and discontinuous functions with fractal structure have been given. Applications of fractal functions in interpolation, approximation and relationship between fractional calculus have been investigated. Finally, we studied fractal dimension spaces with different fractal dimensions. [ABSTRACT FROM AUTHOR]

The primary goal of this article is devoted to the study of fractal bases and fractal frames for L 2 (I × J) , the collection of all square integrable functions on the rectangle I × J . The fractal function when recognized as an internal binary operation paved way for the construction of right and left partial fractal convolution operators on L 2 (I) , for any real compact interval I. The aforementioned operators defined on one dimensional space have been applied to obtain operators on the space L 2 (I × J) by the identification of L 2 (I × J) with the tensor product space L 2 (I) ⊗ L 2 (J) . In this paper, we establish properties of this bounded linear operator which eventually helps to prove that L 2 (I × J) admits Bessel sequences, Riesz bases and frames consisting of products of fractal (self-referential) functions in a nice way. [ABSTRACT FROM AUTHOR]

Two classes of Taylor-type formulas for arbitrary continuous functions on intervals are obtained using Bernstein polynomials. These formulas are applicable to both smooth functions and functions that have neither finite nor infinite derivatives at any point. The Taylor-type formulas are considered in close connection with Dini derivatives, which exist for any continuous function. An example is given in which these formulas are applied to the problem of controlling a distributed oscillatory system whose dynamics obeys the d'Alembert representation. [ABSTRACT FROM AUTHOR]

Fractal approximation is a well studied concept, but the convergence of all the existing fractal approximants towards the original function follows usually if the magnitude of the corresponding scaling factors approaches zero. In this article, for a given function f ∈ C (I) , by exploiting fractal approximation theory and considering the classical q-Bernstein polynomials as base functions, we construct a sequence { f n (q , α) (x) } n = 1 ∞ of (q , α) -fractal functions that converges uniformly to f even if the norm/magnitude of the scaling functions/scaling factors does not tend to zero. The convergence of the sequence { f n (q , α) (x) } n = 1 ∞ of (q , α) -fractal functions towards f follows from the convergence of the sequence of q-Bernstein polynomials of f towards f. If we consider a sequence { f m (x) } m = 1 ∞ of positive functions on a compact real interval that converges uniformly to a function f, we develop a double sequence { { f m , n (q , α) (x) } n = 1 ∞ } m = 1 ∞ of (q , α) -fractal functions that converges uniformly to f. [ABSTRACT FROM AUTHOR]

The Koch curve was first described by the Swedish mathematician Helge von Koch in 1904 as an example of a continuous but nowhere differentiable curve. Such functions are now characterised as fractal since their graphs are in general fractal sets. Furthermore, it can be obtained as the graph of an appropriately chosen iterated function system. On the other hand, a fractal interpolation function can be seen as a special case of an iterated function system thus maintaining all of its characteristics. Fractal interpolation functions are continuous functions that can be used to model continuous signals. An in-depth discussion on the theory of affine fractal interpolation functions generating the Koch Curve by using fractal analysis as well as its recent development including some of the research made by the authors is provided. We ensure that the graph of fractal interpolation functions on the Koch Curve are attractors of an iterated function system constructed by non-constant harmonic functions. [ABSTRACT FROM AUTHOR]

In this article, taking the quantum Bernstein functions as base functions, we have proposed the class of quantum Bernstein fractal functions. When f∈𝒞(I), the base function is taken as the classical q‐Bernstein polynomials, we propose the class of quantum fractal functions through a multivalued quantum fractal operator. When f∈ℒp(I),1≤p≤∞, the base function is assumed as q‐Kantorovich‐Bernstein polynomial to construct the sequence of (q,α)‐Kantorovich‐Bernstein fractal functions that converges uniformly to f. Some approximation properties of these new class of quantum fractal interpolants have been studied. [ABSTRACT FROM AUTHOR]

Subdivision schemes generate self-similar curves and surfaces for which it has a familiar connection between fractal curves and surfaces generated by iterated function systems (IFS). Overveld [Comput.-Aided Des. 22(9) (1990) 591–597] proved that the subdivision matrices can be perturbated in such a way that it is possible to get fractal-like curves that are perturbated Bézier cubic curves. In this work, we extend the Overveld scheme to n th degree curves, and deduce the condition for curvature continuity and convex hull property. We find the conditions for positive preserving fractal-like Bézier curves in the proposed subdivision matrices. The resulting 2D/3D curves from these binary subdivision matrices resemble with fractal images. Finally, the dependence of the shape of these fractal-like curves on the elements of subdivision matrices is demonstrated with suitably chosen examples. [ABSTRACT FROM AUTHOR]

On the one hand, the notion of mixed norm spaces has attracted considerable attention in fields such as harmonic analysis and PDE. On the other hand, a particular modification of the Bernstein operator, the so-called Bernstein–Kantorovich operator, has been of special interest for the approximation of the classical L p -functions. This note has a double purpose. First, we record some elementary approximation properties of the Bernstein–Kantorovich operators on the mixed norm Lebesgue spaces. In the second part, we construct self-referential (fractal) counterparts to the functions belonging to the mixed norm Lebesgue spaces and introduce fractal operators on these spaces. With the help of the Bernstein–Kantorovich operators, we obtain a fractal approximation process on the mixed norm Lebesgue spaces. Furthermore, using the multivariate Haar system, we provide a Schauder basis consisting of self-referential functions for the mixed norm Lebesgue spaces, which we call the Bernstein–Kantorovich fractal Haar system. [ABSTRACT FROM AUTHOR]

In this paper, we study solutions of a variation of a classical integral equation (based on the Picard operator) in which Lebesgue measure is replaced by a self-similar measure μ. Our main interest is in the fractal nature of the solutions and we use Iterated Function System (IFS) tools to investigate the behavior and self-similarity of these solutions. Both the integral and differential forms of the equation are discussed since each brings useful insights. Several convergence results are provided along with illustrative examples that show the applications of the theory when the underlying fractal object is the celebrated Cantor set. Additionally we show that the solution to our integral equation inherits self-similarity from the defining measure μ. [ABSTRACT FROM AUTHOR]

Through appropriate choices of elements in the underlying iterated function system, the methodology of fractal interpolation enables us to associate a family of continuous self-referential functions with a prescribed real-valued continuous function on a real compact interval. This procedure elicits what is referred to as an α-fractal operator on , the space of all real-valued continuous functions defined on a compact interval I. With an eye towards connecting fractal functions with other branches of mathematics, in this paper we continue to investigate the fractal operator in more general spaces such as the space of all bounded functions and the Lebesgue space , and in some standard spaces of smooth functions such as the space of k-times continuously differentiable functions, Hölder spaces and Sobolev spaces . Using properties of the α-fractal operator, the existence of Schauder bases consisting of self-referential functions for these function spaces is established. [ABSTRACT FROM PUBLISHER]

Fractal interpolation function defined with the aid of iterated function system can be employed to show that any continuous real-valued function defined on a compact interval is a special case of a class of fractal functions (self-referential functions). Elements of the iterated function system can be selected appropriately so that the corresponding fractal function enjoys certain properties. In the first part of the paper, we associate a class of self-referential L p -functions with a prescribed L p -function. Further, we apply our construction of fractal functions in L p -spaces in some approximation problems, for instance, to derive fractal versions of the full Müntz theorems in L p -spaces. The second part of the paper is devoted to identify parameters so that the fractal functions affiliated to a given continuous function satisfy certain conditions, which in turn facilitate them to find applications in some one-sided uniform approximation problems. [ABSTRACT FROM AUTHOR]

Orthogonal wavelets on the Cantor dyadic group are identified with multiwavelets on the real line consisting of piecewise fractal functions. A tree algorithm for analysis using these wavelets is described. Multiwavelet systems with algorithms of similar structure include certain orthogonal compactly supported multiwavelets in the linear double-knot spline space S1,2.

Recently, using the Walsh-Dirichlet type kernels, the first author has defined periodic dyadic wavelets on the positive semiaxis which are similar to the Chui-Mhaskar trigonometric wavelets. In this paper we generalize this construction and give examples of applications of periodic dyadic wavelets for coding the Riemann, Weierstrass, Schwarz, van der Waerden, Hankel, and Takagi fractal functions. [ABSTRACT FROM AUTHOR]

Abstract: Let be the Thue–Morse sequence with denoting the sum of the digits in the binary expansion of j. A well-known result of Newman [On the number of binary digits in a multiple of three, Proc. Amer. Math. Soc. 21 (1969) 719–721] says that for all . In the first part of the paper we show that and for , where equality is characterized by means of an automaton. This sharpens results given by Dumont [Discrépance des progressions arithmétiques dans la suite de Morse, C. R. Acad. Sci. Paris Sér. I Math. 297 (1983) 145–148]. In the second part we study more general settings. For let and , where denotes the sum of digits in the g-ary digit expansion of j. We observe trivial Newman-like phenomena whenever . Furthermore, we show that the case inherits many Newman-like phenomena for every even and large classes of arithmetic progressions of indices. This, in particular, extends results by Drmota and Skałba [Rarified sums of the Thue–Morse sequence, Trans. Amer. Math. Soc. 352 (2000) 609–642] to the general g-case. [Copyright &y& Elsevier]

Abstract: Geronimo, Hardin et al. have previously constructed orthogonal and biorthogonal scaling vectors by extending a spline scaling vector with functions supported on . Many of these constructions occurred before the concept of balanced scaling vectors was introduced. This paper will show that adding functions on is insufficient for extending spline scaling vectors to scaling vectors that are both orthogonal and balanced. We are able, however, to use this technique to extend spline scaling vectors to balanced, biorthogonal scaling vectors, and we provide two large classes of this type of scaling vector, with approximation order two and three, respectively, with two specific constructions with desirable properties in each case. The constructions will use macroelements supported on , some of which will be fractal functions. [Copyright &y& Elsevier]

The singular behavior of functions is generally characterized by their Hölder exponent. However, we show that this exponent poorly characterizes oscillating singularities. We thus introduce a second exponent that accounts for the oscillations of a singular behavior and we give a characterization of this exponent using the wavelet transform. We then elaborate on a “grand-canonical” multifractal formalism that describes statistically the fluctuations of both the Hölder and the oscillation exponents. We prove that this formalism allows us to recover the generalized singularity spectrum of a large class of fractal functions involving oscillating singularities. [ABSTRACT FROM AUTHOR]

We extend the multifractal formalism for the local dimension spectrum of a Gibbs measure μ supported on the attractor Λ of a conformal iterated functions system on the real line. Namely, for α ∈ R , we establish the multifractal formalism for the Hausdorff dimension of the set of x ∈ Λ for which the μ -measure of a ball of radius r n centred at x obeys a power law r n α , for a sequence r n → 0. This allows us to investigate the Hölder regularity of various fractal functions, such as distribution functions and conjugacy maps associated with conformal iterated function systems. [ABSTRACT FROM AUTHOR]

The objective of this paper is to study the box-counting dimension of graphs of fractal interpolation functions and harmonic functions on the Sierpiński gasket. Firstly, we give construction of a fractal interpolation function on the Sierpiński gasket and then with the help of fractal interpolation functions we show the existence of fractal functions in the space dom (E) consisting of all finite energy functionals on the Sierpiński gasket. Later, we provide bounds for the box-counting dimension of graphs of some functions belonging to the family of continuous functions which arise as fractal interpolation functions. Moreover, we also obtain bounds for the box-counting dimension of graphs of harmonic functions and piecewise harmonic functions. Also, we obtain upper and lower bounds for the box-counting dimension of graphs of functions that belong to dom (E). [ABSTRACT FROM AUTHOR]