Related papers: Curvilinearity and Orthogonality
Weighted cone-volume functionals are introduced for the convex polytopes in $\mathbb{R}^n$. For these functionals, geometric inequalities are proved and the equality conditions are characterized. A variety of corollaries are derived,…
We introduce two explicit examples of polynomials orthogonal on the unit circle. Moments and the reflection coefficients are expressed in terms of Jacobi elliptic functions. We find explicit expression for these polynomials in terms of a…
We study a Lax pair in a $2$-parameter Lie algebra in various representations. The overlap coefficients of the eigenfunctions of $L$ and the standard basis are given in terms of orthogonal polynomials and orthogonal functions. Moreover,…
In our recent publication we obtained a series expansion of the arctangent function involving complex numbers. In this work we show that this formula can also be expressed as a real rational function.
The existence or non-existence of positive orthogonal functions for subspaces of almost periodic functions has important applications in studying the oscillatory behavior of vibrations. Cazenave, Haraux and Komornik have obtained a number…
We show how to use the method of orthogonal polynomials for integrating, in the planar approximation, the partition function of one-matrix models with a potential with even or odd vertices, or any combination of them.
In this paper, we introduce new classes of functions that extend the known classes of functions of complex variable, such as entire functions, meromorphic functions, rational functions and polynomial functions and take values in the set of…
We provide an elementary derivation of an orthogonal coordinate system for boundary layers around evolving smooth surfaces and curves based on the signed-distance function. We go beyond previous works on the signed-distance function and…
Finite dimensional subspaces spanned by exponential functions in the space of square integrable functions on a finite interval of the real line are considered. Their limiting positions are studied and described in terms of expo-polynomials.
This paper provides a finite pair of biorthogonal matrix polynomials and their finite biorthogonality, several recurrence relations, matrix differential equation, generating function and integral representation.
The differential systems satisfied by orthogonal polynomials with arbitrary semiclassical measures supported on contours in the complex plane are derived, as well as the compatible systems of deformation equations obtained from varying such…
Generalizations of some known results on the best, best linear and best one-sided approxima- tions by trigonometric polynomials of the classes of 2\pi - periodic functions presented in the form of convolutions to the case of set-valued…
We present a formulation of quantum mechanics based on orthogonal polynomials. The wavefunction is expanded over a complete set of square integrable basis in configuration space where the expansion coefficients are orthogonal polynomials in…
We prove that linearizing certain families of polynomial optimization problems leads to new functorial operations in real convex sets. We show that under some conditions these operations can be computed or approximated in ways amenable to…
This study presents miscellaneous properties of pseudo-factorials, which are numbers whose recurrence relation is a twisted form of that of usual factorials. These numbers are associated with special elliptic functions, most notably, a…
In this article, we further explore convex functions by revealing new bounds, resulting from stronger convexity behavior. In particular, we define the so called radical convex functions and study their properties. We will see that such…
Algorithms and underlying mathematics are presented for numerical computation with periodic functions via approximations to machine precision by trigonometric polynomials, including the solution of linear and nonlinear periodic ordinary…
We seek random versions of some classical theorems on complex approximation by polynomials and rational functions, as well as investigate properties of random compact sets in connection to complex approximation.
We extend the piecewise orthogonal collocation method to computing periodic solutions of coupled renewal and delay differential equations. Through a rigorous error analysis, we prove convergence of the relevant finite-element method and…
The analysis of complex nonlinear systems is often carried out using simpler piecewise linear representations of them. A principled and practical technique is proposed to linearize and evaluate arbitrary continuous nonlinear functions using…