Related papers: Fourier orthogonal series on a paraboloid

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…

In this paper we propose a way to construct classical type Sobolev orthogonal polynomials. We consider two families of hypergeometric polynomials: ${}_2 F_2(-n,1;q,r;x)$ and ${}_3 F_2(-n,n-1+a+b,1;a,c;x)$ ($a,b,c,q,r>0$, $n=0,1,...$), which…

A rigorous and powerful theoretical framework is proposed to obtain systems of orthogonal functions (or shape modes) to represent optical surfaces. The method is general so it can be applied to different initial shapes and different…

The sequence $\{\,_2\phi_1(q^{-k},q^{b+1};\,q^{-\overline{b}-k+1};\, q, q^{-\overline{b}+1/2} z)\}_{k \geq 0}$ of basic hypergeometric polynomials is known to be orthogonal on the unit circle with respect to the weight function…

The use of orthonormal polynomial bases has been found to be efficient in preventing ill-conditioning of the system matrix in the primal formulation of Virtual Element Methods (VEM) for high values of polynomial degree and in presence of…

We consider for structure groups ${\rm SU}(n)\,\subset\, {\rm SL}(n,\mathbb C)$ a densely defined toric structure on the moduli of framed parabolic sheaves on a three-punctured sphere, which degenerates to an actual toric structure. In…

Using the orthogonal connectedness, we introduce the notion of orthogonal decomposability of convex polytopes and study it in the case of Platonic and Archimedean solids. While doing so, we also encounter polytopes which are not…

We study connections between orthogonal polynomials, reproducing kernel functions, and polynomials $p$ minimizing Dirichlet-type norms $\|pf-1\|_{\alpha}$ for a given function $f$. For $\alpha\in [0,1]$ (which includes the Hardy and…

In this work, the concept of quasi-type Kernel polynomials with respect to a moment functional is introduced. Difference equation satisfied by these polynomials along with the criterion for orthogonality conditions are discussed. The…

We present computational methods for constructing orthogonal/orthonormal polynomials over arbitrary polygonal domains in $\mathbb{R}^2$ using bivariate spline functions. Leveraging a mature MATLAB implementation which generates spline…

A set of orthogonal polynomials on the unit disk $B(0,1)$ known as Zernike polynomials are commonly used in the analysis and evaluation of optical systems. Here Zernike polynomials are used to construct wavelets for polynomial subspaces of…

Persymmetric Jacobi matrices are invariant under reflection with respect to the anti-diagonal. The associated orthogonal polynomials have distinctive properties that are discussed. They are found in particular to be also orthogonal on the…

We introduce a two parameter ($\alpha, \beta>-1$) family of interacting particle systems with determinantal correlation kernels expressible in terms of Jacobi polynomials $\{ P^{(\alpha, \beta)}_k \}_{k \geq 0}$. The family includes…

By characterizing all orthogonal polynomials sequences $(P_n)_{n\geq 0}$ for which $$ (ax+b)(\triangle +2\,\mathrm{I})P_n(x(s-1/2))=(a_n x+b_n)P_n(x)+c_n P_{n-1}(x),\quad n=0,1,2,\dots, $$ where $\,\mathrm{I}$ is the identity operator, $x$…

For orthogonal polynomials defined by compact Jacobi matrix with exponential decay of the coefficients, precise properties of orthogonality measure is determined. This allows showing uniform boundedness of partial sums of orthogonal…

The purpose of this note is to extend in a simple and unified way the known results on interlacing of zeros of paraorthogonal polynomials on the unit circle. These polynomials can be regarded as the characteristic polynomials of any matrix…

Orthogonal polynomials on the product domain $[a_1,b_1] \times [a_2,b_2]$ with respect to the inner product $$ \langle f,g \rangle_S = \int_{a_1}^{b_1} \int_{a_2}^{b_2} \nabla f(x,y)\cdot \nabla g(x,y)\, w_1(x)w_2(y) \,dx\, dy + \lambda…

We study skew-orthogonal polynomials with respect to the weight function $\exp[-2V(x)]$, with $V(x)=\sum_{K=1}^{2d}(u_{K}/{K})x^{K}$, $u_{2d} > 0$, $d > 0$. A finite subsequence of such skew-orthogonal polynomials arising in the study of…

Given a function $b$, holomorphic on the disc and bounded by 1, one can construct an associated reproducing kernel Hilbert space called the de Branges--Rovnyak space $H(b)$. We explore representations of such spaces via descriptions of the…

We consider the systems of rational functions $\{\Phi_n(z)\}, ~n \in \mathbb{Z}$, defined by fixed set points ${\bf a}:=\{a_k\}_{k=0}^{\infty}, ~ (\mathop{\rm Im} a_k>0)$, ${\bf b}:=\{b_k\}_{k=1}^{\infty}, ~ (\mathop{\rm Im} b_k<0)$ and is…