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An overview of the basic results on Macdonald(-Koornwinder) polynomials and double affine Hecke algebras is given. We develop the theory in such a way that it naturally encompasses all known cases. Among the basic properties of the…
In this chapter are given necessary and sufficient conditions for the regularity of solutions of the functional equation appearing in the theory of classical orthogonal polynomials. In addition, we also present the functional Rodrigues…
The present paper is devoted to the description of finite-dimensional semisimple Leibniz algebras over complex numbers, their derivations and automorphisms.
We provide a complete description of normal affine varieties with effective algebraic torus action in terms of what we call proper polyhedral divisors on semiprojective varieties. Our theory extends classical cone constructions of…
A real-valued function on R^n is k-regulous, where k is a nonnegative integer, if it is of class C^k and can be represented as a quotient of two polynomial functions on R^n. Several interesting results involving such functions have been…
In a recent paper, we gave a topological description of Colombeau type algebras introducing algebras of sequences with exponential weights. Embeddings of Schwartz' spaces into the Colombeau algebra G are well known, but for…
We introduce and study the notion of plurisubharmonic functions in calibrated geometry. These functions generalize the classical plurisubharmonic functions from complex geometry and enjoy their important properties. Moreover, they exist in…
We provide a unified, elementary, topological approach to the classical results stating the continuity of the complex roots of a polynomial with respect to its coefficients, and the continuity of the coefficients with respect to the roots.…
In this paper we consider generalized monomial functions $f, g\colon \mathbb{F}\to \mathbb{C}$ (of possibly different degree) that also fulfill \[ f(P(x))= Q(g(x)) \qquad \left(x\in \mathbb{F}\right), \] where $P\in \mathbb{F}[x]$ and $Q\in…
We consider a semi-algebraic function defined on a closed semi-algebraic set X. We give formulas relating the topology of X to the indices of the critical points of the function and to the topological behavior of the function at infinity.…
We provide a classification of Fueter-regular quaternionic functions $f$ in terms of the degree of complex linearity of their real differentials $df$. Quaternionic imaginary units define orthogonal almost-complex structures on the tangent…
We introduce the class of semiweak Cullen-regular quaternionic functions by interpreting Cullen-regular functions as solutions to an inhomogeneous PDE in terms of the Fueter operator.
In this paper, we introduce and study the locally strongly convex equiaffine isoparametric hypersurfaces and equiaffine isoparametric functions on the affine space $A^{n+1}$. Motivated by the case on the Euclidean space $E^{n+1}$, we first…
This article offers a comprehensive treatment of polynomial functional regression, culminating in the establishment of a novel finite sample bound. This bound encompasses various aspects, including general smoothness conditions, capacity…
In this paper we propose a novel family of weighted orthonormal rational functions on a semi-infinite interval. We write a sequence of integer-coefficient polynomials in several forms and derive their corresponding differential equations.…
Homogeneous rotation symmetric Boolean functions have been extensively studied in recent years because of their applications in cryptography. Little is known about the basic question of when two such functions are affine equivalent. The…
In this paper, approximation by means of algebraic polynomials of classes of functions defined by a generalised modulus of smoothness of operators of differentiation of these functions is considered. We give structural characteristics of…
We give an elementary characterization of rational functions among meromorphic functions in the complex plane.
This work sets the statistical affine shape theory in the context of real normed division algebras. The general densities apply for every field: real, complex, quaternion, octonion, and for any noncentral and non-isotropic elliptical…
We investigate density of various subalgebras of regular generalized functions in the special Colombeau algebra of generalized functions.