arXiv subjects

Peter Massopust

Publications and source records attributed to Peter Massopust.

At least 19 recordsLinked to original sources

arXiv2026

Fourier representations of fractional B Splines via generalized Stirling type polynomials

In this paper, we investigate fractional B splines and their connections with Fourier analysis, and establish connections with generalized Stirling-type numbers and distribution theory. Employing a generating function approach inspired by recent results of Simsek [24], we derive a novel Fourier type expansion for fractional B splines that involves generalized Stirling type numbers. Our main contribution is the derivation of a Fourier-type expansion of fractional B splines in terms of generalized Stirling-type numbers. This representation allows us to express fractional B splines as infinite linear combinations of derivatives of the Dirac delta in the distributional sense. Furthermore, we establish an explicit shifted distributional representation and obtain shifted distributional representations that characterize the action of fractional B-splines on test functions. In addition, we introduce a new class of fractional spline polynomials and derive their generating function in terms of the Mittag Leffler function. These results provide a unified framework that connects spline theory, fractional calculus, and combinatorial structures.

math.GM↗

arXiv2023

Complex Box Splines

The novel concept of box spline of complex degree is introduced and several of its properties derived and discussed. These box splines of complex degree generalize and extend the classical box splines. Relations to a class of fractional derivatives defined on certain function spaces are also exhibited.

math.FA↗

arXiv2023

An Integral RB Operator

We introduce the novel concept of integral Read-Bajraktarevi\'c (iRB) operator and discuss some of its properties. We show that this iRB operator generalizes the known Read-Bajraktarevi\'c (RB) operator and we derive conditions for the fixed point of the iRB operator to belong to certain function spaces.

math.FA↗

arXiv2020

Hypercomplex Iterated Function Systems

We introduce the novel concept of hypercomplex iterated function system (IFS) on the complete metric space $(\mathbb{A}_{n+1}^k,d)$ and define its hypercomplex attractor. Systems of hypercomplex function systems arising from hypercomplex IFSs and their backward trajectories are also introduced and it is shown that the attractors of such backward trajectories possess different local (fractal) shapes.

math.MG↗

arXiv2019

Attractors of Trees of Maps and of Sequences of Maps between Spaces with Application to Subdivision

In a previous paper we considered a sequence of maps on a complete metric space $(X,d)$ and derived an extension of the Banach fixed point theorem. We showed that backward trajectories of maps $X\to X$ converge under mild conditions and that they can generate new types of attractors such as scale dependent fractals. Here we present two generalisations of this result and some potential applications. First, we study the structure of an infinite tree of maps $X\to X$ and discuss convergence to a unique "attractor" of the tree. We also consider "staircase" sequences of maps, that is, we consider a countable sequence of metric spaces $\{(X_i,d_i)\}$ and an associated countable sequence of maps $\{T_i\}$, $T_i:X_{i}\to X_{i-1}$. We examine conditions for the convergence of backward trajectories of the $\{T_i\}$ to a unique attractor. An example of such trees of maps are trees of function systems leading to the construction of fractals which are both scale dependent and location dependent. The staircase structure facilitates linking all types of linear subdivision schemes to attractors of function systems.

math.FA↗

arXiv2019

Non-stationary Fractal Interpolation

We introduce the novel concept of a non-stationary iterated function system by considering a countable sequence of distinct set-valued maps $\{\mathcal{F}_k\}_{k\in \mathbb{N}}$ where each $\mathcal{F}_k$ maps $\mathcal{H}(X)\to \mathcal{H}(X)$ and arises from an iterated function system. Employing the recently developed theory of non-stationary versions of fixed points [11] and the concept of forward and backward trajectories, we present new classes of fractal functions exhibiting different local and global behavior, and extend fractal interpolation to this new, more flexible setting.

math.DS↗

arXiv2019

On the Mathematical Validity of the Higuchi Method

In this paper, we discuss the Higuchi algorithm which serves as a widely used estimator for the box-counting dimension of the graph of a bounded function $f : [0,1] \to \R$. We formulate the method in a mathematically precise way and show that it yields the correct dimension for a class of non-fractal functions. Furthermore, it will be shown that the algorithm follows a geometrical approach and therefore gives a reasonable estimate of the fractal dimension of a fractal function. We conclude the paper by discussing the robustness of the method and show that it can be highly unstable under perturbations.

math.MG↗

arXiv2019

On Some Generalizations of B-Splines

In this article, we consider some generalizations of polynomial and exponential B-splines. Firstly, the extension from integral to complex orders is reviewed and presented. The second generalization involves the construction of uncountable families of self-referential or fractal functions from polynomial and exponential B-splines of integral and complex orders. As the support of the latter B-splines is the set $[0,\infty)$, the known fractal interpolation techniques are extended in order to include this setting.

math.MG↗

arXiv2019

Splines and Fractional Differential Operators

Several classes of classical cardinal B-splines can be obtained as solutions of operator equations of the form $Ly = 0$ where $L$ is a linear differential operator of integral order. (Cf., for instance, \cite{akhiezer,Golomb,Krein,micchelli,schoenberg}.) In this article, we consider classes of generalized B-splines consisting of cardinal polynomial B-splines of complex and hypercomplex orders and cardinal exponential B-splines of complex order, and derive the fractional linear differential operators that are naturally associated with them. For this purpose, we also present the spaces of distributions onto which these fractional differential operators act.

math.FA↗

arXiv2016

Self-referential Functions

We introduce the concept of fractels for functions and discuss their analytic and algebraic properties. We also consider the representation of polynomials and analytic functions using fractels, and the consequences of these representations in numerical analysis.

math.CA↗

arXiv2016

Quaternionic B-Splines

We introduce B-splines on the line of quaternionic order $B_q$ ($q$ in the algebra of quaternions) for the purposes of multi-channel signal and image analysis. The functions $B_q$ are defined first by their Fourier transforms, then as the solutions of distributional differential equation of quaternionic order. The equivalence of these definitions requires properties of quaternionic Gamma functions and binomial expansions, both of which we investigate. The relationship between $B_q$ and a backwards difference operator is shown, leading to a recurrence formula. We show that the collection of integer shifts of $B_q$ is a Riesz basis for its span, hence generating a multiresolution analysis. Finally, we demonstrate the pointwise and $L^p$ convergence of the quaternionic B-splines to quarternionic Gaussian functions.

math.FA↗

arXiv2016

Fractional and Complex Pseudo-Splines and the Construction of Parseval Frames

Pseudo-splines of integer order $(m,\ell)$ were introduced by Daubechies, Han, Ron, and Shen as a family which allows interpolation between the classical B-splines and the Daubechies' scaling functions. The purpose of this paper is to generalize the pseudo-splines to fractional and complex orders $(z, \ell)$ with $α:=\re z > 1$. This allows increased flexibility in regard to smoothness: instead of working with a discrete family of functions from $C^m$, $m\in \N_0$, one uses a \emph{continuous} family of functions belonging to the Hölder spaces $C^{α-1}$. The presence of the imaginary part of $z$ allows for direct utilization in complex transform techniques for signal and image analyses. We also show that in analogue to the integer case, the generalized pseudo-splines lead to constructions of Parseval wavelet frames via the unitary extension principle.

math.FA↗

arXiv2015

Multigrid convergence for the MDCA-curvature estimator

We consider the problem of estimating the curvature profile along the boundaries of digital objects in segmented black-and-white images. We start with the curvature estimator proposed by Roussillon et al., which is based on the calculation of \emph{maximal digital circular arcs} (MDCA). We extend this estimator to the $\lambda$-MDCA curvature estimator that considers several MDCAs for each boundary pixel and is therefore smoother than the classical MDCA curvature estimator. We prove an explicit order of convergence result for convex subsets in $\mathbb{R}^2$ with positive, continuous curvature profile. In addition, we evaluate the curvature estimator on various objects with known curvature profile. We show that the observed order of convergence is close to the theoretical limit of $\mathcal{O}(h^{\frac13})$. Furthermore, we establish that the $\lambda$-MDCA curvature estimator outperforms the MDCA curvature estimator, especially in the neighborhood of corners.

math.NA↗

arXiv2015

On the Invalidity of Fourier Series Expansions of Fractional Order

The purpose of this short paper is to show the invalidity of a Fourier series expansion of fractional order as derived by G. Jumarie in a series of papers. In his work the exponential functions $e^{inωx}$ are replaced by the Mittag-Leffler functions $E_α\left (i (nωx)^α\right) ,$ over the interval $[0, M_α/ ω]$ where $0< ω<\infty $ and $M_α$ is the period of the function $E_α\left( ix^α\right),$ i.e., $E_α\left( ix^α\right)=E_α\left( i(x+M_α)^α\right).$

math.CA↗

arXiv2014

Signal Analysis based on Complex Wavelet Signs

We propose a signal analysis tool based on the sign (or the phase) of complex wavelet coefficients, which we call a signature. The signature is defined as the fine-scale limit of the signs of a signal's complex wavelet coefficients. We show that the signature equals zero at sufficiently regular points of a signal whereas at salient features, such as jumps or cusps, it is non-zero. At such feature points, the orientation of the signature in the complex plane can be interpreted as an indicator of local symmetry and antisymmetry. We establish that the signature rotates in the complex plane under fractional Hilbert transforms. We show that certain random signals, such as white Gaussian noise and Brownian motions, have a vanishing signature. We derive an appropriate discretization and show the applicability to signal analysis.

math.NA↗

arXiv2014

Local fractal functions in Besov and Triebel-Lizorkin spaces

Within the new concept of a local iterated function system (local IFS), we consider a class of attractors of such IFSs, namely those that are graphs of functions. These new functions are called local fractal functions and they extend and generalize those that are currently found in the fractal literature. For a class of local fractal functions, we derive explicit conditions for them to be elements of Besov and Triebel--Lizorkin spaces. These two scales of functions spaces play an important role in interpolation theory and for certain ranges of their defining parameters describe many classical function spaces (in the sense of equivalent norms). The conditions we derive provide immediate information about inclusion of local fractal functions in, for instance, Lebesgue, Sobolev, Slobodeckij, Hölder, Bessel potential, and local Hardy spaces.

math.FA↗

arXiv2014

Numerics and Fractals

Local iterated function systems are an important generalisation of the standard (global) iterated function systems (IFSs). For a particular class of mappings, their fixed points are the graphs of local fractal functions and these functions themselves are known to be the fixed points of an associated Read-Bajactarević operator. This paper establishes existence and properties of local fractal functions and discusses how they are computed. In particular, it is shown that piecewise polynomials are a special case of local fractal functions. Finally, we develop a method to compute the components of a local IFS from data or (partial differential) equations.

math.MG↗

arXiv2014

Directional time frequency analysis via continuous frame

Grafakos and Sansing \cite{GS} have shown how to obtain directionally sensitive time-frequency decompositions in $L^2(\mr^n)$ based on Gabor systems in $\ltr;$ the key tool is the "ridge idea," which lifts a function of one variable to a function of several variables. We generalize their result by showing that similar results hold starting with general frames for $L^2(\mr),$ both in the setting of discrete frames and continuous frames. This allows to apply the theory for several other classes of frames, e.g., wavelet frames and shift-invariant systems. We will consider applications to the Meyer wavelet and complex B-splines. In the special case of wavelet systems we show how to discretize the representations using $ε$-nets.

math.FA↗