Browsing by Author "Kushpel, Alexander"

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Interpolation of Exponential-Type Functions on a Uniform Grid by Shifts of a Basis Function
(American Institute of Mathematical Sciences, 2021) Sun, Xinping; Kushpel, Alexander; Levesley, Jeremy; Jarad, Fahd
Interpolation of Exponential-Type Functions on a Uniform Grid by Shifts of a Basis Function
(Amer inst Mathematical Sciences-aims, 2021) Jarad, Fahd; Kushpel, Alexander; Levesley, Jeremy; Sun, Xinping
In this paper, we present a new approach to solving the problem of interpolating a continuous function at (n + 1) equally-spaced points in the interval [0, 1], using shifts of a kernel on the (1/n)-spaced infinite grid. The archetypal example here is approximation using shifts of a Gaussian kernel. We present new results concerning interpolation of functions of exponential type, in particular, polynomials on the integer grid as a step en route to solve the general interpolation problem. For the Gaussian kernel we introduce a new class of polynomials, closely related to the probabilistic Hermite polynomials and show that evaluations of the polynomials at the integer points provide the coefficients of the interpolants. Finally we give a closed formula for the Gaussian interpolant of a continuous function on a uniform grid in the unit interval (assuming knowledge of the discrete moments of the Gaussian).
Citation - WoS: 3
Citation - Scopus: 3
The Lebesgue Constants on Projective Spaces
(2021) Kushpel, Alexander
We give the solution of a classical problem of Approximation Theory on sharp asymptotic of the Lebesgueconstants or norms of the Fourier-Laplace projections on the real projective spaces Pd(R). In particular, these resultsextend sharp asymptotic found by Fejer [2] in the case of S1in 1910 and by Gronwall [4] in 1914 in the case of S2. Thecase of spheres, Sd, complex and quaternionic projective spaces, Pd(C), Pd(H) and the Cayley elliptic plane P16(Cay)was considered by Kushpel [8].
Citation - WoS: 3
Citation - Scopus: 3
Lower Bounds of Cowidths and Widths of Multiplier Operators
(Academic Press inc Elsevier Science, 2022) Kushpel, Alexander
The main objective of this article is to present new results on optimal reconstruction of function classes on probability spaces (Omega, A, nu) in the standard L-q spaces. We consider the problem of optimal reconstruction in the sense of the respective cowidths of standard function classes Lambda U-p generated by multiplier or pseudo differential operators Lambda : L-p -> L-q, 1 <= p, q <= infinity. Our approach is based on the estimates of volumes of John-Lowner ellipsoids and expectations of norms induced by orthonormal systems on (Omega, A, nu). It is shown that the results obtained are order sharp in many cases. In particular, we obtain sharp orders of entropy of Sobolev classes W-infinity(gamma), gamma > 0 in L-1 and n-widths of Lambda U-p in L-q, 1 < q <= p < infinity in the case of two-point homogeneous spaces and torus. (C) 2021 Elsevier Inc. All rights reserved.
Methods of Financial Mathematics
(LAMBERT Academic Publishing, 2020) Kushpel, Alexander
Citation - Scopus: 1
On the Problem of Schoenberg On Rn
(Univ Prishtines, 2024) Kushpel, Alexander; Tas, Kenan
In 1946 Schoenberg introduced splines on R, which play now one of the central roles in Numerical Analysis, and posed the problem on spline interpolation. The main aim of this article is to establish explicit representations of fundamental splines on Rn and give a positive solution of the problem of Schoenberg on Rn
Citation - Scopus: 1
On The Problem Of Schoenberg On Rn
(University of Prishtina, 2024) Kushpel, Alexander; Taş, Kenan
Optimal Approximation and Pricing of High- Dimensional Options
(Lambert Academic Publishing, 2019) Kushpel, Alexander
The book introduces an original general approach to the problem of multidimensional pricing which is applicable for a wide range of practically important examples. It gives a comprehensive and self-contained treatment of the problem of multidimensional pricing which provides the reader with all technical details. The book reflects a new stage of research in Quantitative Finance. It gives a particular fascination when apparently disjoint areas turn out to have a meaningful connection to each other. It demonstrates on concrete examples deep connections between Quantitative Finance and Numerical Analysis, Topology, Functional Analysis and Complexity Theory. The book can be considered as an inspirational source for practitioners, graduate and postgraduate students, MSc and PhD projects, to those working in Quantitative Finance and Economics.
Citation - WoS: 1
Citation - Scopus: 1
Optimal Recovery and Volume Estimates
(Academic Press inc Elsevier Science, 2023) Kushpel, Alexander
We study volumes of sections of convex origin-symmetric bodies in Rn induced by orthonormal systems on probability spaces. The approach is based on volume estimates of John-Lowner ellipsoids and expectations of norms induced by the respective systems. The estimates obtained allow us to establish lower bounds for the radii of sections which gives lower bounds for Gelfand widths (or linear cowidths). As an application we offer a new method of evaluation of Gelfand and Kolmogorov widths of multiplier operators. In particular, we establish sharp orders of widths of standard Sobolev classes Wp & gamma;, & gamma; > 0 in Lq on two-point homogeneous spaces in the difficult case, i.e. if 1 < q < p < oo.& COPY; 2023 Elsevier Inc. All rights reserved.
Citation - WoS: 6
Citation - Scopus: 6
The Radii of Sections of Origin-Symmetric Convex Bodies and Their Applications
(Academic Press inc Elsevier Science, 2021) Tas, Kenan; Kushpel, Alexander
Let V and W be any convex and origin-symmetric bodies in R-n . Assume that for some A is an element of L (R-n -> R-n), det A not equal 0, V is contained in the ellipsoid A(-1)B((2))(n), where B-(2)(n) is the unit Euclidean ball. We give a lower bound for the W-radius of sections of A(-1) V in terms of the spectral radius of AA and the expectations of parallel to . parallel to(V) and parallel to . parallel to(W)0 with respect to Haar measure on Sn-1 subset of R-n. It is shown that the respective expectations are bounded as n -> infinity in many important cases. As an application we offer a new method of evaluation of n-widths of multiplier operators. As an example we establish sharp orders of n-widths of multiplier operators Lambda : L-p (M-d) -> L-q (M-d), 1 < q <= 2 <= p < infinity on compact homogeneous Riemannian manifolds M-d. Also, we apply these results to prove the existence of flat polynomials on M-d. (c) 2020 Elsevier Inc. All rights reserved.
The Lebesgue Constants on Projective Spaces
(Tubitak Scientific & Technological Research Council Turkey, 2021) Kushpel, Alexander
We give the solution of a classical problem of Approximation Theory on sharp asymptotic of the Lebesgue constants or norms of the Fourier-Laplace projections on the real projective spaces P-d (R). In particular, these results extend sharp asymptotic found by Fejer [2] in the case of S-1 in 1910 and by Gronwall [4] in 1914 in the case of S-2. The case of spheres, S-d, complex and quaternionic projective spaces, P-d(C), P-d(H) and the Cayley elliptic plane P-16 (Cay) was considered by Kushpel [8].
Citation - WoS: 5
Citation - Scopus: 5
Widths and Entropy of Sets of Smooth Functions on Compact Homogeneous Manifolds
(Tubitak Scientific & Technological Research Council Turkey, 2021) Levesley, Jeremy; Tas, Kenan; Kushpel, Alexander
We develop a general method to calculate entropy and n-widths of sets of smooth functions on an arbitrary\rcompact homogeneous Riemannian manifold Md\r. Our method is essentially based on a detailed study of geometric\rcharacteristics of norms induced by subspaces of harmonics on Md\r. This approach has been developed in the cycle\rof works [1, 2, 10–19]. The method’s possibilities are not confined to the statements proved but can be applied in\rstudying more general problems. As an application, we establish sharp orders of entropy and n-widths of Sobolev’s\rclasses Wγ\rp\r(\rMd\r)\rand their generalisations in Lq\r(\rMd\r)\rfor any 1 < p, q < ∞. In the case p, q = 1, ∞ sharp in the power\rscale estimates are presented.