Related papers: Orthogonal polynomial method and odd vertices in m…
We present a spectral method for one-sided linear fractional integral equations on a closed interval that achieves exponentially fast convergence for a variety of equations, including ones with irrational order, multiple fractional orders,…
A general method of obtaining linear differential equations having polynomial solutions is proposed. The method is based on an equivalence of the spectral problem for an element of the universal enveloping algebra of some Lie algebra in the…
Using a lemma of Davis on Gram matrices applied to the classical Orthogonal Polynomials to generate reproducing kernel interpolation over the classical domains for polynomials. These kernels have terms which are exact over the rational…
We consider the problem of approximating the solution of variational problems subject to the constraint that the admissible functions must be convex. This problem is at the interface between convex analysis, convex optimization, variational…
In this paper, we achieve three primary objectives related to the rigorous computational analysis of nonlinear PDEs posed on complex geometries such as disks and cylinders. First, we introduce a validated Matrix Multiplication Transform…
One way of constructing explicit expressions of solutions of integrable systems of Partial Differential Equations (PDEs) goes via the Darboux method. This requires the construction of Darboux matrices. Here we introduce a novel algorithm to…
In many problems in Computational Physics and Chemistry, one finds a special kind of sparse matrices, termed "banded matrices". These matrices, which are defined as having non-zero entries only within a given distance from the main…
Several recently discovered properties of multiple families of special polynomials (some orthogonal and some not) that satisfy certain differential, difference or q-difference equations are reviewed. A general method of construction of…
Orthogonal Fractional Factorial Designs and in particular Orthogonal Arrays are frequently used in many fields of application, including medicine, engineering and agriculture. In this paper we present a methodology and an algorithm to find…
By using Fourier transforms of two symmetric sequences of finite orthogonal polynomials, we introduce two new classes of finite orthogonal functions and obtain their orthogonality relations via Parseval's identity.
In this short lecture, we compute asymptotics of orthogonal polynomials, from a saddle point approximation. This is an example of a calculation which shows the link between integrability, algebraic geometry and random matrices.
One can improve predictability in the unknown domain by combining forecasts of imperfect complex computational models using a Bayesian statistical machine learning framework. In many cases, however, the models used in the mixing process are…
We propose a general method for optimization with semi-infinite constraints that involve a linear combination of functions, focusing on the case of the exponential function. Each function is lower and upper bounded on sub-intervals by…
Let $d\nu$ be a measure in $\mathbb{R}^d$ obtained from adding a set of mass points to another measure $d\mu$. Orthogonal polynomials in several variables associated with $d\nu$ can be explicitly expressed in terms of orthogonal polynomials…
We present two new classes of orthogonal functions, log orthogonal functions (LOFs) and generalized log orthogonal functions (GLOFs), which are constructed by applying a $\log$ mapping to Laguerre polynomials. We develop basic approximation…
We extend the use of piecewise orthogonal collocation to computing periodic solutions of renewal equations, which are particularly important in modeling population dynamics. We prove convergence through a rigorous error analysis. Finally,…
In this paper, models that approximate stochastic processes from the space $Sub_\varphi(\Omega)$ with given reliability and accuracy in $L_p(T)$ are considered for some specific functions $\varphi(t)$. For processes that are decomposited in…
The speed of convergence of the R-linear GMRES is bounded in terms of a polynomial approximation problem on a finite subset of the spectrum. This result resembles the classical GMRES convergence estimate except that the matrix involved is…
We derive formulas for characterizing bounded orthogonally additive polynomials in two ways. Firstly, we prove that certain formulas for orthogonally additive polynomials derived in \cite{Kusa} actually characterize them. Secondly, by…
Linear differential equations of arbitrary order with polynomial coefficients are considered. Specifically, necessary and sufficient conditions for the existence of polynomial solutions of a given degree are obtained for these equations. An…