Fractal functions using weak contraction theory in some function spaces and generalized 𝛼-fractal functions

Constructions of the (global) fractal interpolation functions on standard function spaces got a lot of attention in the last centuries. Motivated by the newly introduced local fractal functions corresponding to a local iterated functions system which is the generalization of the traditional iterated functions system we construct the local non-affine alpha - fractal functions in this article. A few examples of the graphs of these functions are provided. A fractal operator which takes the classical function to its local fractal counterpart is defined and some of its properties are also studied.

The fractal interpolation functions with appropriate iterated function systems provide a method to perturb and approximate a continuous function on a compact interval I. This method produces a class of functions $$f^{\varvec{\alpha }}\in {\mathcal {C}}(I)$$ , where $$\varvec{\alpha }$$ is a vector with functional components. The presence of scaling function in these fractal functions helps to get a wide variety of mappings for approximation problems. The current article explores the shape-preserving properties of the $$\varvec{\alpha }$$ -fractal functions with variable scalings, where the optimal ranges of the scaling functions are derived for fundamental shapes of the germ f. We provide several examples to illustrate the shape preserving results and apply our fractal methodologies in approximation problems. Also, it is shown that the order of convergence of the $$\varvec{\alpha }$$ -fractal polynomial to the original shaped function matches with that of polynomial approximation. Further, based on the shape preserving properties of the $$\varvec{\alpha }$$ -fractal functions, we provide the fractal analogue of the Chebyshev alternation theorem. To the end, we deduce the fractal version of the classical full Muntz theorem in $${\mathcal {C}}[0,1]$$ .

In this paper, we explore the concept of dimension preserving approximation of continuous multivariate functions defined on the domain [Formula: see text] (q-times) where [Formula: see text] is a natural number). We establish a few well-known multivariate constrained approximation results in terms of dimension preserving approximants. In particular, we indicate the construction of multivariate dimension preserving approximants using the concept of [Formula: see text]-fractal interpolation functions. We also prove the existence of one-sided approximation of multivariate function using fractal functions. Moreover, we provide an upper bound for the fractal dimension of the graph of the [Formula: see text]-fractal function. Further, we study the approximation aspects of [Formula: see text]-fractal functions and establish the existence of the Schauder basis consisting of multivariate fractal functions for the space of all real valued continuous functions defined on [Formula: see text] and prove the existence of multivariate fractal polynomials for the approximation.

A subspace projection method for the implementation of interface conditions in a single-drop flow problem

Fractal Functions and Wavelet Expansions Based on Several Scaling Functions

We prove that the existence of a solution operator for a convolution operator from the space of ultradifferntiable functions to the corresponding space of ultradistributions is equivalent to the existence of a continuous solution operator in the space of functions. Our results are in the spirit of a classical characterization of the surjectivity of convolution operators due to Hormander. The behaviour of a fixed convolution operator in different classes of ultradifferentiable functions of Beurling type concerning the existence of a continuous linear right inverse is also considered.

Fractal interpolation functions are fixed points of contraction maps on suitable function spaces. In this paper, we introduce the Kantorovich-Bernstein α-fractal operator in the Lebesgue space 𝓛p(I), 1 ≤ p ≤ ∞. The main aim of this article is to study the convergence of the sequence of Kantorovich-Bernstein fractal functions towards the original functions in 𝓛p(I) spaces and Lipschitz spaces without affecting the non-linearity of the fractal functions. In the first part of this paper, we introduce a new family of self-referential fractal 𝓛p(I) functions from a given function in the same space. The existence of a Schauder basis consisting of self-referential functions in 𝓛p spaces is proven. Further, we derive the fractal analogues of some 𝓛p(I) approximation results, for example, the fractal version of the classical Müntz-Jackson theorem. The one-sided approximation by the Bernstein α-fractal function is developed.

Chapter 6 - Dimension of Fractal Functions

Zipper fractal interpolation function (ZFIF) is a generalization of fractal interpolation function through an
 improved version of iterated function system by using a binary parameter called a signature. The signature
 allows the horizontal scalings to be negative. ZFIFs have a complex geometric structure, and they can
 be non-differentiable on a dense subset of an interval I. In this paper, we construct k-times continuously
 differentiable ZFIFs with variable scaling functions on I. Some properties like the positivity, monotonicity,
 and convexity of a zipper fractal function and the one-sided approximation for a continuous function by a
 zipper fractal function are studied. The existence of Schauder basis of zipper fractal functions for the space
 of k-times continuously differentiable functions and the space of p-integrable functions for p ∈ [1,∞) are
 studied. We introduce the zipper versions of full Müntz theorem for continuous function and p-integrable
 functions on I for p ∈ [1,∞).

A general procedure to define non-smooth versions of classical approximants by means of fractal interpolation functions is proposed. A complete and explicit description in the frequency domain of the functions constructed is obtained through their exact Fourier transforms. In particular, the generalization of the polynomial interpolation is developed. The Lagrange basis of the space of polynomials of degree lower or equal than N is generalized to a basis of fractal polynomials. As a consequence of the process, the density of the polynomial fractal interpolation functions with non-null scale vector in the space of continuous functions in a compact interval is deduced. Furthermore, a method for the interpolation of real data is proposed, by the construction of a fractal function coming from any classical approximant. The convergence of the process when the partition is refined is proved, supposing the convergence of the smooth interpolant.

In this paper, for a given \(f\in \mathcal {C}(I),\) we construct a sequence of Bernstein \(\alpha \)-fractal functions with variable scaling factors. Bernstein \(\alpha \)-fractal functions proposed in this paper are fixed points of the Read-Bajraktarevic operators defined on a suitable function space. The convergence of the Bernstein \(\alpha \)-fractal function with variable scaling factors towards the original function f follows from the convergence of Bernstein polynomials of f toward f. Owing this reason, Bernstein \(\alpha \)-fractal function with variable scaling factors proposed in this paper converge to the original function for any choice of the variable scalings whereas the existing fractal approximants converge to the original function only if the magnitude of the scaling factors goes to zero. Using the Bernstein \(\alpha \)-fractal functions, we study the Bernstein \(\alpha \)-fractal trigonometric polynomials with variable scaling factors and their constrained approximation properties. For a given sequence \(\{f_n(x)\}^{\infty }_{n=1}\) of continuous periodic functions defined on \([-l,l]\) that converges uniformly to a function f, we investigate the existence of a double sequence of Bernstein \(\alpha \)-fractal trigonometric polynomials that converges uniformly to f.

The construction of fractal versions of classical functions as polynomials, trigonometric maps, etc. by means of a particular Iterated Function System of the plane is tackled. The closeness between the classical function and its fractal analogue provides good properties of approximation and interpolation to the latter. This type of methodology opens the use of non-smooth and fractal functions in approximation. The procedure involves the definition of an operator mapping standard functions into their dual fractals. The transformation is linear and bounded and some bounds of its norm are established. Through this operator we define families of fractal functions that generalize the classical Schauder systems of Banach spaces and the orthonormal bases of Hilbert spaces. With an appropriate election of the coefficients of Iterated Function System we define sets of fractal maps that span the most important spaces of functions as \({\mathcal {C}}[a,b]\) or \({\mathcal {L}}^p[a,b]\).

We prove the existence of a family of compact subdomains \Omega of the flat cylinder \mathbb{R}^{N}\times \mathbb{R}/2\pi\mathbb{Z} for which the Neumann eigenvalue problem for the Laplacian on \Omega admits eigenfunctions with constant Dirichlet values on \partial \Omega . These domains \Omega have the property that their boundaries \partial \Omega have nonconstant principal curvatures. In the context of ambient Riemannian manifolds, our construction provides the first examples of such domains whose boundaries are neither homogeneous nor isoparametric hypersurfaces. The functional analytic approach we develop in this paper overcomes an inherent loss of regularity of the problem in standard function spaces. With the help of this approach, we also construct a related family of subdomains of the 2-sphere S^{2} . By this we disprove a conjecture by Souam [Ann. Global Anal. Geom. 27, 341–354 (2005)].

Coupled 3D-1D problems arise in many practical applications, in an attempt to reduce the computational burden in simulations where cylindrical inclusions with a small section are embedded in a much larger domain. Nonetheless the resolution of such problems can be non trivial, both from a mathematical and a geometrical standpoint. Indeed 3D-1D coupling requires to operate in non standard function spaces, and, also, simulation geometries can be complex for the presence of multiple intersecting domains. Recently, a PDE-constrained optimization based formulation has been proposed for such problems, proving a well posed mathematical formulation and allowing for the use of non conforming meshes for the discrete problem. Here an unconstrained optimization formulation of the problem is derived and an efficient gradient based solver is proposed for such formulation, along with a suitable preconditioner to speed up the iterative solver. Some numerical tests on quite complex configurations are discussed to show the viability of the method .

We consider an optimal control problem for the heat conductivity equation with an integral boundary condition. In addition to the instability of the problem in standard function spaces, it is necessary to take into account that the operator of the problem is not self-adjoint To obtain stable (regularized) solutions to the problem posed, we propose to solve a close stable problem with a small parameter in the overdetermination conditions. For the constructed approximate solution, an exact estimate of its deviation from the accurate solution is derived.