Peter Massopust
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View article: The theory of deep convolutional neural networks and a data approximation problem based on the fractional Fourier transform
The theory of deep convolutional neural networks and a data approximation problem based on the fractional Fourier transform Open
View article: Semi-Rings, Semi-Vector Spaces, and Fractal Interpolation
Semi-Rings, Semi-Vector Spaces, and Fractal Interpolation Open
In this paper, we introduce fractal interpolation on complete semi-vector spaces. This approach is motivated by the requirements of the preservation of positivity or monotonicity of functions for some models in approximation and interpolat…
View article: Fractal cubic multiquadric quasi-interpolation
Fractal cubic multiquadric quasi-interpolation Open
View article: Solutions of differential equations using fractal multiquadric RBF networks
Solutions of differential equations using fractal multiquadric RBF networks Open
In this article, we introduce a novel class of self-referential fractal multiquadric (MQ) functions that exhibit symmetry about the origin. To ensure the differentiability of the original classical multiquadric function, we carefully limit…
View article: A Short Note on Fractal Interpolation in the Space of Convex Lipschitz Functions
A Short Note on Fractal Interpolation in the Space of Convex Lipschitz Functions Open
In this short note, we consider fractal interpolation in the Banach space Vθ(I) of convex Lipschitz functions defined on a compact interval I⊂R. To this end, we define an appropriate iterated function system and exhibit the associated Read…
View article: Complex Box Splines
Complex Box Splines Open
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 de…
View article: An integral RB operator
An integral RB operator Open
We introduce the novel concept of integral Read–Bajraktarević (iRB) operator and discuss some of its properties. We show that this iRB operator generalizes the known Read–Bajraktarević (RB) operator and we derive conditions for the fixed p…
View article: Deep convolutional neural networks and data approximation using the fractional Fourier transform
Deep convolutional neural networks and data approximation using the fractional Fourier transform Open
In the first part of this paper, we define a deep convolutional neural network connected with the fractional Fourier transform (FrFT) using the $θ$-translation operator, the translation operator associated with the FrFT. Subsequently, we s…
View article: Approximation with fractal radial basis functions
Approximation with fractal radial basis functions Open
The article reports on the construction of a general class of fractal radial basis functions (RBFs) in the literature. The fractal RBFs is defined through fractal perturbation of a RBF through suitable choice of iterated function system (I…
View article: Shape preserving fractal multiquadric quasi-interpolation
Shape preserving fractal multiquadric quasi-interpolation Open
In this article, we construct a novel self-referential fractal multiquadric function which is symmetric about the origin. The scaling factors are suitably restricted to preserve the differentiability and the convexity of the underlying cla…
View article: Deep Convolutional Neural Networks and Data Approximation Using the Fractional Fourier Transform
Deep Convolutional Neural Networks and Data Approximation Using the Fractional Fourier Transform Open
View article: Systems of left translates and oblique duals on the Heisenberg group
Systems of left translates and oblique duals on the Heisenberg group Open
In this paper, we characterize the system of left translates $\{L_{(2k,l,m)}g:k,l,m\in\mathbb{Z}\}$, $g\in L^2(\mathbb{H})$, to be a frame sequence or a Riesz sequence in terms of the twisted translates of the corresponding function $g^\la…
View article: Fractal hypersurfaces, affine Weyl groups, and wavelet sets
Fractal hypersurfaces, affine Weyl groups, and wavelet sets Open
In this expository paper, we present some fundamental connections between iterated function systems, in particular those whose attractors are the graphs of multivariate real-valued fractal functions, and foldable figures, affine Weyl group…
View article: Complex Box Splines
Complex Box Splines Open
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 de…
View article: An Integral RB Operator
An Integral RB Operator Open
We introduce the novel concept of integral Read-Bajraktarević (iRB) operator and discuss some of its properties. We show that this iRB operator generalizes the known Read-Bajraktarević (RB) operator and we derive conditions for the fixed p…
View article: Clifford-Valued Fractal Interpolation
Clifford-Valued Fractal Interpolation Open
View article: Multivariate Zipper Fractal Functions
Multivariate Zipper Fractal Functions Open
View article: Approximation by Quantum Meyer-König-Zeller Fractal Functions
Approximation by Quantum Meyer-König-Zeller Fractal Functions Open
In this paper, a novel class of quantum fractal functions is introduced based on the Meyer-König-Zeller operator Mq,n. These quantum Meyer-König-Zeller (MKZ) fractal functions employ Mq,nf as the base function in the iterated function syst…
View article: Multivariate Zipper Fractal Functions
Multivariate Zipper Fractal Functions Open
A novel approach to zipper fractal interpolation theory for functions of several variables is proposed. We develop multivariate zipper fractal functions in a constructive manner. We then perturb a multivariate function to construct its zip…
View article: Approximation by Quantum Meyer König and Zeller-Fractal Functions
Approximation by Quantum Meyer König and Zeller-Fractal Functions Open
In this paper, a novel class of quantum fractal functions is introduced based on the Meyer-König-Zeller operator $M_{q,n}$. These quantum Meyer-König-Zeller (MKZ) fractal functions employ $M_{q,n} f$ as the base function in the iterated fu…
View article: Dimension preserving approximation
Dimension preserving approximation Open
View article: Fractal Interpolation over Curves
Fractal Interpolation over Curves Open
This paper introduces the novel concept of fractal interpolation over curves in Banach spaces. The contents are based on the usual methodologies involving the fractal interpolation problem over intervals but the current approach considerab…
View article: Fractal interpolation over nonlinear partitions
Fractal interpolation over nonlinear partitions Open
View article: Fractal Interpolation: From Global to Local, to Nonstationary and Quaternionic
Fractal Interpolation: From Global to Local, to Nonstationary and Quaternionic Open
This chapter presents an introduction to fractal interpolation beginning with a global set-up and then extending to a local, a non-stationary, and finally the novel quaternionic setting. Emphasis is placed on the overall perspective with r…
View article: Fractal Interpolation Over Nonlinear Partitions
Fractal Interpolation Over Nonlinear Partitions Open
View article: Fractal Interpolation: From Global to Local, to Nonstationary and Quaternionic
Fractal Interpolation: From Global to Local, to Nonstationary and Quaternionic Open
We present an introduction to fractal interpolation beginning with a global set-up and then extending to a local, a non-stationary, and finally the novel quaternionic setting. Emphasis is placed on the overall perspective with references g…
View article: Interpolation and Sampling with Exponential Splines of Real Order
Interpolation and Sampling with Exponential Splines of Real Order Open
View article: B-splines on the Heisenberg group
B-splines on the Heisenberg group Open
In this paper, we introduce a class of $B$-splines on the Heisenberg group $\mathbb{H}$ and study their fundamental properties. Unlike the classical case, we prove that there does not exist any sequence $\{α_n\}_{n\in\mathbb{N}}$ such that…
View article: Twisted B-splines in the complex plane
Twisted B-splines in the complex plane Open
In this paper, we introduce the new class of twisted $B$-splines and study some properties of these B-splines. We also investigate the system of twisted translates and the wavelets corresponding to these twisted $B$-splines.
View article: Splines and fractional differential operators
Splines and fractional differential operators Open
Several classes of classical cardinal B-splines can be obtained as solutions of operator equations of the form [Formula: see text] where [Formula: see text] is a linear differential operator of integral order. In this paper, we consider cl…