Related papers: Recovering orthogonality from Quasi-type Kernel Po…
Orthogonal polynomials in two variables on cubic curves are considered, including the case of elliptic curves. For an integral with respect to an appropriate weight function defined on a cubic curve, an explicit basis of orthogonal…
I propose an orthogonalization procedure preserving the grading of the initial graded set of linearly independent vectors. In particular, this procedure is applicable for orthonormalization of any countable set of polynomials in several…
This article studies bivariate multiple orthogonal polynomials of the mixed type on the step-line. The analysis is based on the LU factorization of a moment matrix specifically adapted to this framework. The orthogonality and…
Computing many eigenpairs of the Schr{\"o}dinger operator presents a computational bottleneck in large-scale quantum simulations due to the global communication overhead of explicit orthogonalization. To address this issue, we propose a…
Kernels are key in machine learning for modeling interactions. Unfortunately, brute-force computation of the related kernel sums scales quadratically with the number of samples. Recent Fourier-slicing methods lead to an improved linear…
Representations of polynomial covariance type commutation relations by linear integral operators on $L_p$ over measures spaces are investigated. Necessary and sufficient conditions for integral operators to satisfy polynomial covariance…
In this paper, we present a new method via the transfer matrix approach to obtain asymptotic formulae of orthogonal polynomials with asymptotically identical coefficients of bounded variation. We make use of the hyperbolicity of the…
The metohod of ortogonal rotations introduced in the previous papers of the author is used for construction of the explicit form the generators of the simple roots for quantum (and ussual) semisimple algebras. All calculations are presented…
We report a new method to calculate the quasinormal modes of rotating black holes, using a spectral decomposition to solve the partial differential equations that result from introducing linear metric perturbations to a rotating background.…
It is an open question whether the fractional parts of nonlinear polynomials at integers have the same fine-scale statistics as a Poisson point process. Most results towards an affirmative answer have so far been restricted to almost sure…
We study orthogonal polynomials for a weight function defined over a domain of revolution, where the domain is formed from rotating a two-dimensional region and goes beyond the quadratic domains. Explicit constructions of orthogonal bases…
Given $\{P_n \}$ a sequence of monic orthogonal polynomials, we analyze their linear combinations $\{Q_n \}$with constant coefficients and fixed length $k+1$. Necessary and sufficient conditions are given for the orthogonality of the monic…
The use of kernel functions is a common technique to extract important features from data sets. A quantum computer can be used to estimate kernel entries as transition amplitudes of unitary circuits. Quantum kernels exist that, subject to…
In the paper "Infinite product representations for kernels and iterations of functions", the authors associate certain Fatou subsets with reproducing kernel Hilbert spaces. They also present a method for constructing an orthonormal basis…
We study orthogonal polynomial ensembles whose weights are deformations of exponential weights, in the limit of a large number of particles. The deformation symbols we consider affect local fluctuations of the ensemble around a bulk point…
The polynomial kernels are widely used in machine learning and they are one of the default choices to develop kernel-based classification and regression models. However, they are rarely used and considered in numerical analysis due to their…
Discrete spectral transformations of skew orthogonal polynomials are presented. From these spectral transformations, it is shown that the corresponding discrete integrable systems are derived both in 1+1 dimension and in 2+1 dimension.…
We investigate a series of learning kernel problems with polynomial combinations of base kernels, which will help us solve regression and classification problems. We also perform some numerical experiments of polynomial kernels with…
We list the so-called Askey-scheme of hypergeometric orthogonal polynomials. In chapter 1 we give the definition, the orthogonality relation, the three term recurrence relation and generating functions of all classes of orthogonal…
We study quantum sparse recovery in non-orthogonal, overcomplete dictionaries: given coherent quantum access to a state and a dictionary of vectors, the goal is to reconstruct the state up to $\ell_2$ error using as few vectors as possible.…