The workshop aims to offer a multi-faceted overview of contemporary research in orthogonal polynomials and special functions. A number of talks focus on the analysis of orthogonal polynomials themselves, such as asymptotic analysis of large degree behaviour. A second class of topics cover generalizations of orthogonal polynomials, such as polynomials of several variables, matrix-valued orthogonal polynomials or polynomials over curves and surfaces. Finally, polynomials as a tool in applications feature in talks on numerical simulations and other computational settings.
The study of orthogonal polynomials links to several branches of mathematics. In applications, they play a pivotal role in approximation methods and numerical simulations. As part of the
latter the workshop links to the other workshops “Approximation Theory and Computational Harmonic Analysis” and “Foundations of Numerical PDEs”, and to a lesser extent also with “Foundations of Data Science and Machine Learning” and “Information-based Complexity”.
Organizers
Speakers
Semi-plenary speakers
National University of Cordoba
University of Montreal
Invited speakers
Ankara University
University of Sevilla
City University of Hong Kong
Lancaster University
TU Delft
KU Leuven
University of South-Africa
Chuo University
University of Kent
University of Edinburg
Imperial College London
Chiara Paulsen
University of Wisconsin-Madison
University Of Oxford
University of Almeria
JKU Linz
University of Vienna
Universidade de São Paulo
Carlos III University
KU Leuven
University of Granada
Thursday, 9 July
![[HS 11 – 2nd Floor]](https://focm2026.univie.ac.at/wp-content/uploads/2026/06/HS11.png){kind=link}
14:00-15:00 semi-plenary talk
Luc Vinet (University of Montreal)
Leonard pairs that will be briefly reviewd have a close connection with the finite families of orthogonal polynomials of the Askey scheme. This concept will be extended to the notion of a Leonard trio with the aim of generating an algebraic framework for biorthogonal rational functions. Elements of classification based on Leonard pairs and their associated Heun operators will be presented. This will lead to the characterization of the Wilson and other rational functions.
15:00-15:30
Wolter Groenevelt (Delft University Of Technology)
In this talk I show how rational functions of ${}_4\phi_3$-type appear in the representation theory of the quantum algebra $\mathcal U_q(\mathfrak{sl}_2)$. Using this algebraic interpretation we show that the rational functions satisfy discrete biorthogonality relations and are solutions of an explicit generalized eigenvalue problem. The coalgebra structure naturally leads to a construction of multivariate extensions. This then gives biorthogonal rational functions in $N$ variables that solve $N$ independent generalized eigenvalue problems. This is joint work with Carel Wagenaar.
15:30-16:00
Guilherme Silva (Universidade De São Paulo)
We survey how deformations of orthogonal polynomials, inspired by free fermions in finite temperature, led to the construction of solutions to the Boussinesq equation, and connections to statistics of certain random matrix models.
16:00-16:30 Coffee Break
16:30-17:00
Veronika Pillwein (JKU Linz)
Over the past decades computer algebraists have delevoped methods that can be applied for discovering and proving statements on special functions. In this talk, we show how to use some of these algorithms while presenting a result on a sequence of polynomials generated by a Kapteyn series of the second kind. This is joint work with Diego Dominici.
17:00-17:30
Michael Schlosser (University of Vienna)
We introduce the bilateral $q$-ultraspherical functions, a bilateral-series extension of the continuous $q$-ultraspherical polynomials. They are defined by specific bilateral basic hypergeometric $_2\psi_2$ series, are analytic in the variable $x=\cos\theta$, and depend on two parameters $\beta$ and $\gamma$ and on a base $q$. We derive a product formula for their bilateral generating function, a three-term recurrence relation, their transformation under the Askey–Wilson divided difference operator, and explicit large-order asymptotic expansions. We also prove full orthogonality relations with respect to explicit orthogonality functionals involving analytic mass aggregates, together with shifted orthogonality relations. In the limit $\gamma\to1$, the construction and identities reduce to the classical results for the continuous $q$-ultraspherical polynomials.
17:30-18:00
Makoto Katori (Chuo University)
One of the fundamental problems in machine learning is that the internal structures of hidden layers formed through learning processes remain a black box. Following Sonoda and Murata [Appl. Comput. Harmon. Anal. 43, 233—268 (2017)], we treat the learning equation of depth-2 neural networks as an integral transform from the distribution functions of hidden parameters to target functions.
Sonoda and Murata introduced another integral transform called ridgelet transform which can invert this learning equation. We fix the activation function to the hyperbolic tangent. Then we show that an infinite series of ridgelet transforms can be generated using integral kernels expressed through Hermite polynomials. This result implies that the patterns of hidden parameters are structured, but these structures are hidden by their redundancy.
In the present talk, I will propose a new application of the theory of orthogonal polynomials and special functions to reveal such hidden structures in machine learning.
This talk is based on joint work with Raian Suzuki (Chuo Univ.) and Saori Morimoto (NEC).
18:00-18:30
Max van Horssen (KU Leuven)
We study a one-parameter family of tiling models of non-regular hexagons with doubly periodic weights, in which two side lengths are equal and their ratio to the third varies. These models exhibit three phases in the large-size limit: solid, liquid, and gas. Depending on the side length ratio, there are either one or two gas phases, and we observe that these gas phases can merge in two distinct ways.
We present an overview of a systematic method for obtaining large-size asymptotics of the correlation kernel and employ it to establish a scaling limit of this kernel in a special case. The method includes non-linear steepest descent analysis of a Riemann-Hilbert problem and saddle point analysis on the (double) amoeba of a genus-one Harnack curve.
This talk is based on joint work with Arno Kuijlaars.
Friday, 10 July
14:00-14:30
Adri Olde-Daalhuis (The University of Edinburgh)
In solving $q$-difference equations, and in the definition of $q$-special functions, we encounter formal power series in which the $n$th coefficient is of size $q^{-\binom{n}{2}}$ with $q\in(0,1)$ fixed. To make sense of these formal series, a $q$-Borel-Laplace resummation is required. There are three candidates for the $q$-Laplace transform, resulting in three different resummations. Surprisingly, the differences between these resummations have hardly been discussed in the literature. Our main result provides explicit formulas for these $q$-exponentially small differences. We also give simple Mellin–Barnes integral representations for all the basic hypergeometric ${}_r\phi_s$ functions and derive a third (discrete) orthogonality condition for the Stieltjes–Wigert polynomials.
14:30-15:00
Kerstin Jordaan (University Of South Africa)
The zeros of orthogonal polynomials are well known to satisfy rich interlacing properties, but interlacing phenomena also occur in more general families of real-rooted polynomials. In this talk I will describe a mechanism by which a single, naturally occurring point can complete an interlacing pattern between zeros of polynomials from different sequences. The method is based on mixed recurrence relations in which one coefficient has a single sign change. In the orthogonal setting, this leads to results for families such as Laguerre, Jacobi, Meixner, Meixner–Pollaczek, Pseudo-Jacobi and Continuous Hahn polynomials, where the zero of a linear coefficient acts as an added interlacing point. More generally, the method does not rely on orthogonality, and this opens the way to applications to non-orthogonal families, including Narayana polynomials. I will focus on the common structural feature behind these examples: interlacing is controlled by the recurrence relation and the location of the added point. The talk will conclude with a brief discussion of why this perspective may be useful in a broader study of real-rooted polynomial sequences.
15:00-15:30
Benjamin Eichinger (Lancaster University)
Universality describes the phenomena that the local behavior of zeros of orthogonal polynomial depends only on general characteristics of the system, rather than on its precise properties. From a modern perspective, such local behavior is analyzed through local scaling limits of the Christoffel–Darboux kernel. Recent works have established necessary and sufficient conditions for the most studied universality classes, including bulk universality and hard-edge universality. In all these settings, the rescaled Christoffel–Darboux kernel converges to a single limit kernel.
In this talk, we consider scaling limits in which there is not a single limit kernel, but rather an entire cycle of limit kernels. We show that this type of behavior arises naturally when studying scaling limits for orthogonal polynomials associated with the equilibrium measure of the real Julia set of an expanding polynomial or with the Cantor measure on the classical middle-third Cantor set.
This talk is based on joint work with Alexander Kheifets, Milivoje Lukić and Peter Yuditskii.
15:30-16:00
Chiara Paulsen (University Of Wisconsin-Madison)
A classical theorem of Szegö states that for any probability measure $\mu=w\frac{d\theta}{2\pi}+\mu_s$ on the unit circle the polynomials are dense in $L^2(\mathbb{T},\mu)$ if and only if $\log(w)
otin L^1(\mathbb{T})$. A related question asks whether the monomials with exponents in some subset $\Lambda\subseteq \mathbb{N}_0$ already span $L^2(\mathbb{T},\mu)$ if $\log(w)
otin L^1(\mathbb{T})$. A result by Olevskii and Ulanovskii gives an answer if $\mu$ belongs to a class of absolutely continuous measures. We investigate the same question for Markoff measures.
16:00-16:30 Coffee Break
16:30-17:00
Ana Loureiro (University Of Kent)
The discussion will be centred on the study of special solutions of a selection of discrete Painlevé. They are linked to orthogonal polynomials, via their recurrence coefficients. The problem arises in a variety of contexts, other than orthogonal polynomials, most notably within the context of random matrix theory. Hidden and gorgeous patterns will emerge.
17:00-17:30
Mirta María Castro Smirnova (Universidad de Sevilla)
We consider a few examples of differential operators obtained from the Hermite differential operator $\mathcal{H}_x$, namely $\mathcal{H}_x=\partial_x^2-2x\partial_x
$, by applications of the Darboux process, and consider the issue of bispectrality in these situations with some emphasis on the role played by trivial monodromy. We are interested in the full two-dimensional space of eigenfunctions of the differential operator and not just in polynomial eigenfunctions. Here we display examples with trivial monodromy associated with rank-two bundles of common eigenfunctions giving a bispectral situation, showing that the situation in the seminal work of Duistermaat and Grünbaum, üDifferential equations in the spectral parameter, Comm. Math. Phys. 103 (1986), where simple monodromy went only with rank-one bispectral families, does not hold.
This is a joint work with A. Grünbaum.
17:30-18:30 semi-plenary talk
Pablo Roman (Universidad Nacional De Cordoba)
Matrix-valued orthogonal polynomials that are eigenfunctions of second-order matrix differential and difference operators are a rich generalization of the classical Askey scheme. This talk is an overview of the theory of such polynomials, with an emphasis on the matrix Bochner problem: the classification of all matrix-valued inner products whose associated orthogonal polynomials satisfy a second-order differential equation.
We will discuss several families that originate from different constructions, including those arising from the representation theory of compact Lie groups, as well as more recent constructions independent of group theory.
Throughout, representation theory acts as a guiding thread. It points the way toward new families, new symmetries, and natural extensions.
Saturday, 11 July
14:00-14:30
Sheehan Olver (Imperial College London)
We construct vector-valued multivariate orthogonal polynomials that capture the structure of the de Rham complex (i.e., the gradient, curl and divergence) with boundary conditions in disks and cylinders (both periodic and finite) in a way that respects rotational symmetry. This construction naturally leads to bases in cylinders with simple recurrences relating their gradient, curl and divergence, which leads to an optimal complexity solver for Maxwell and Hodge-Laplacian equations in a cylinder.
14:30-15:00
Niel Van Buggenhout (Universidad Carlos III De Madrid)
Orthogonal polynomials on the real line satisfy a three term recurrence relation and their zeros correspond to the nodes of Gaussian quadrature rules. For classical orthogonal polynomials an analytical expression for the coefficients of the three term recurrence relation is known and can be used to obtain the zeros of the orthogonal polynomials. A reliable way to obtain these zeros is by grouping the coefficients in a tridiagonal matrix and computing the eigenvalues of this matrix numerically. For Sobolev orthogonal polynomials the situation is more complicated, grouping their recurrence coefficients together results in a Hessenberg matrix, and an analytical expression for these coefficients is not always available, or numerical evaluation of this expression results in a large error due to rounding error accumulation. The eigenvalues of the Hessenberg matrix are the zeros of the Sobolev orthogonal polynomials, but rounding errors can lead to large errors on the numerically computed eigenvalues. We discuss a general purpose numerical algorithm that works for a large and important class of Sobolev orthogonal polynomials, namely, those corresponding to sequentially dominated diagonal Sobolev inner products. This algorithm starts from a quadrature rule, which is used to formulate a structured matrix problem. By solving this problem, we obtain the Hessenberg matrix. Then we remark on the advantages of developing specialized numerical algorithms for specific families of Sobolev orthogonal polynomials. A few examples of specialized numerical algorithms are given.
15:00-15:30
Cristina Rodríguez Perales (Universidad De Almería)
In this talk, we consider the orthogonal polynomials with respect to a Sobolev inner product involving a Freud weight which due to its historical relevance, we will refer to as Nevai weight. Our first objective is to study the local asymptotics known as Mehler-Heine asymptotics for this family of polynomials. Moreover, this result together with the well-known Hurwitz’s Theorem allows us to establish the asymptotic behavior of the zeros of these polynomials. Motivated by this consequence, we also address the problem of computing the zeros of these polynomials. With that purpose, we construct a generalized eigenvalue problem from the five-term recurrence relation satisfied by these polynomials, so that the computation of the zeros is equivalent to computing the generalized eigenvalues. We also present numerical experiments. Finally, if we have time we will deduce the fourth-order differential equation satisfied by these polynomials. Joint work with Juan F. Mañas-Mañas and Juan J. Moreno-Balcázar.
15:30-16:00
Ioannis Papadopoulos (University Of Oxford)
We present a sparse high-order hierarchical hp-finite element method, tailored for disk and annulus domains, with radial discontinuities in the right-hand side and data. The basis leverages Zernike annular polynomials and the overall 2D solution process, including the expansion of the right-hand side and the evaluation of the solution on a grid, achieves quasi-optimal complexity. We validate the effectiveness of this approach on PDEs, such as the time-dependent Schrödinger equation and demonstrate that the basis can be combined with the Alternating Direction Implicit (ADI) method of Fortunato and Townsend, extending the method’s applicability to 3D cylindrical domains. Reference: doi.org/10.1007/s10915-025-02964-4.
16:00-16:30 Coffee Break
16:30-17:00
Dan Dai (City University of Hong Kong)
In this talk, we study relations between the Fredholm determinant of the generalized Bessel kernel with both ordinary differential equations (ODEs) and partial differential equations (PDEs). The ODEs belong to a special class of Fuchsian equations: a Garnier system in two variables, while the PDEs belong to certain specific Ablowitz–Kaup–Newell–Segur (AKNS) flows. This is a joint work with Shuai-Xia Xu and Xiaolu Yue.
17:00-17:30
Astrid Herremans (KU Leuven)
The efficient approximation of functions with singularities is central to many problems in computational mathematics, including the numerical solution of elliptic PDEs with geometric singularities. Methods such as hp-finite element methods, rational approximation, variable transform methods and even neural networks achieve fast convergence by concentrating approximation power near singularities. We prove an impossibility result: no approximation method can simultaneously be well-conditioned, converge rapidly and avoid using information clustered near the singularities. This is joint work with Ben Adcock.
17:30-18:00
Juan Antonio Villegas (Universidad de Granada)
We propose a new numerical method that combines interpolation and regression to approximate and integrate functions on certain 2D shapes, such as polygons, annuli, and ellipses. By extending existing mock-Chebyshev techniques developed for the disk, and by building on recent advances in sampling patterns for the mentioned planar domains, which in turn are based on generalizations of Zernike polynomials to non-standard geometries, we construct a novel framework that allows for accurate function approximation and integration on these domains. Our method leverages the strengths of both interpolation and regression, providing a powerful tool for numerical analysis in complex geometries. We demonstrate the effectiveness of our approach through various examples and applications, testing its accuracy as well as its computational efficiency.
18:00-18:30
Rabia Aktas Kamaran (Ankara University)
Some systems of univariate orthogonal polynomials are related to one another through the Fourier transform. The most extensively studied example involves the Hermite functions, which are eigenfunctions of the Fourier transform [1]. Previous studies have shown that the Jacobi and continuous Hahn polynomials can also be connected in a similar way [2]. Recently, in the univariate setting, the Fourier transforms of generalized ultraspherical polynomials, generalized Hermite polynomials, finite classical orthogonal polynomials, and symmetric sequences of finite orthogonal polynomials have been investigated. Using these Fourier transforms together with Parseval’s identity, several families of orthogonal functions have been obtained [3-5]. In this talk, we study the multivariate case by applying the Fourier transform and Parseval’s identity to construct several examples of orthogonal systems and to analyze their orthogonality relations. We first focus on orthogonal polynomials on the unit ball, and then investigate the Fourier transforms of multivariate orthogonal polynomials on the cone and on the paraboloid [6-9]. Bibliography [1] T.H. Koornwinder, Special orthogonal polynomial systems mapped onto each other by the Fourier-Jacobi transform. In: Orthogonal polynomials and applications (Bar-le-Duc, 1984). (Lecture Notes in Math.; Vol. 1171). Springer, Berlin, p. 174-183, 1985. [2] H.T. Koelink, On Jacobi and continuous Hahn polynomials, Proc. Amer. Math. Soc., 124(3), 887-898, 1996. [3] W. Koepf and M. Masjed Jamei, Two classes of special functions using Fourier transforms of some finite classes of classical orthogonal polynomials, Proc. Amer. Math. Soc., 135(11), 3599-3606, 2007. [4] M. Masjed Jamei and W. Koepf, Two classes of special functions using Fourier transforms of generalized ultraspherical and generalized Hermite polynomials, Proc. Amer. Math. Soc., 140(6), 2053-2063, 2012. [5] M. Masjed Jamei and W. Koepf, Two finite classes of orthogonal functions, Applicable Analysis, 92(11), 2392-2403, 2013. [6] E. Guldogan, R. Aktas and I. Area, Some classes of special functions using Fourier transforms of some twovariable orthogonal polynomials, Integral Transforms and Spec. Funct., 31(6), 437-470, 2020. [7] E. Guldogan Lekesiz, R. Aktas and I. Area, Fourier transform of the orthogonal polynomials on the unit ball and continuous Hahn polynomials, Axioms 2022, 11, 558. [8] R. Aktas Karaman and I. Area, Fourier transforms of orthogonal polynomials on the cone, Numer. Algor., https://doi.org/10.1007/s11075-025-02112-x, 2025. [9] H. O. Cetin and R. Aktas Karaman, Fourier transforms of orthogonal structures on the paraboloid, Mediterr. J. Math., 23, 119, https://doi.org/10.1007/s00009-026-03086-3, 2026.
Posters
![[PZ11 – 2nd Floor]](https://focm2026.univie.ac.at/wp-content/uploads/2026/06/poster_zones_2-stock.png){kind=link}
Benedikt Buchecker
(KU Leuven)
It is a long-standing problem to compute asymptotics of Chebyshev polynomials for smooth Jordan arcs in the complex plane, i.e., homoeomorphic images of the unit interval. For a given compact set $K$, these are monic polynomials minimizing the sup-norm on $K$. Following a 1969 work of Widom on extremal polynomials, the Widom factors are defined by rescaling the norm of the $n$-th Chebyshev polynomial by the logarithmic capacity of $K$ taken to the $n$-th power.
While the limits of Widom factors as $n\to\infty$ are well-understood if the compact set is a closed $C^{2+}$-Jordan curve, even for analytic Jordan arcs it is not known whether the sequence converges. Widom conjectured that the limit is $2$ based on his results for intervals. However, this was later proven to be wrong when the limit was computed for circular arcs. Based on these results, a suitable candidate for the limit was found recently, given by the reproducing kernel of an associated Hardy space. We will confirm that this candidate indeed provides an asymptotic upper bound for the Widom factors on smooth Jordan arcs. This reports on ongoing joint work with Benjamin Eichinger, Olof Rubin, and Aron Wennman.
Svetlana Grigoryan
(National Polytechnic University of Armenia)
Walsh functions constitute a fundamental orthogonal system for the analysis of Boolean functions on finite dyadic groups. Through the Walsh transform, they provide a spectral representation that is especially effective for studying discrete mappings relevant to cryptography. In particular, Walsh spectra encode essential information about linear structure, correlation properties, and nonlinearity, making them a natural tool in the investigation of cryptographic transformations.
This poster concerns the spectral theory of Boolean permutations arising in substitution–permutation network (SPN) cryptography. Such permutations, realized as S-boxes, are central nonlinear components in block cipher designs, and their security depends on a combination of algebraic, combinatorial, and spectral properties.
The research focuses on the Walsh-spectral behavior of S-boxes and on the extent to which spectral characteristics can be used to assess cryptographic strength. Particular attention is given to nonlinearity and resistance to linear approximation. The poster highlights how Walsh analysis and discrete harmonic methods provide a coherent mathematical framework for evaluating Boolean permutations and clarifying the role of spectral distribution in SPN cryptography.
Lennart Hübner
(KU Leuven)
We study the strong asymptotic behavior of planar orthonormal polynomials associated with smooth, strictly positive (non-varying) weights on smooth Jordan domains in the complex plane. Complementing prior results by Carleman, Suetin, and Hedenmalm–Wennman, we establish a full asymptotic expansion of these polynomials that remains valid up to the boundary for all smooth (non-analytic) weights. This work bridges the gap between the Hölder-continuous and real-analytic settings, and further extends it to smooth instead of analytic Jordan domains. Our method combines conformal mapping techniques with a refined Fourier approximation scheme adapted to smooth weights, allowing for uniform asymptotic control both pointwise and in (L^2)-norm. As an application, we obtain precise asymptotics for the associated reproducing kernels, including quantitative error bounds in microscopic scaling limits near the boundary. This project is joint work with Aron Wennman.
Dmitrii Karp
(Holon Institute Of Technology)
A product of two hypergeometric series is generally not hypergeometric. However, there is a handful of examples of parameter restrictions reducing such product to a single hypergeometric series. The oldest result of this type is Euler’s transformation for the Gauss hypergeometric function, another important example is the celebrated Clausen’s identity for the square of a special case of the Gauss function. By equating power series coefficients each product identity corresponds to a summation theorem for terminating series. Over the last two decades Euler’s transformations and many summation theorems have been extended by introducing additional top-bottom parameter pairs differing by positive integers. This amounts to multiplication of the power series coefficients by values of a fixed polynomial at non-negative integers. The poster is devoted to extensions of Clausen’s, Orr’s and other hypergeometric product identities by introducing such polynomial perturbations in one or both factors. In many cases we are able to show that the new product is a polynomial or rational perturbation of the original unperturbed product and to find explicit forms of the perturbing polynomial or rational function. The poster is based on a joint work with Vinay Shulka (Shanghai Jiao Tong University).
Juan José Moreno-Balcázar
(University of Almería)
We consider the sequence of quasi-orthogonal polynomials of order one associated with the Hahn difference operator. Our main goal is to derive the second–order difference equation satisfied by these polynomials.
Teresa Pérez
(University of Granada)
The main objective of this work is to show how Sobolev orthogonal polynomials emerge as a useful tool within the frWe introduce a stable and efficient polynomial reconstruction operator for functions defined on the unit sphere $\mathbb{S}^2$ from discrete samples. The proposed method is based on a mixed interpolation–regression strategy that combines exact interpolation on a carefully selected subset of nodes with a simultaneous least-squares fit over a larger sampling grid. This approach is motivated by the fact that, in practical applications, the available data are typically collected at a number of sampling points that does not match the dimension of a spherical polynomial space, making global interpolation either impossible or severely ill-conditioned.
Starting from a structured set of sampling nodes $X_n \subset \mathbb{S}^2$, an appropriate polynomial degree $m$ is selected in order to extract a Fekete-like interpolation subset $X_m \subset X_n$, chosen so as to guarantee numerical stability. Exact interpolation is enforced only on $X_m$, while the remaining data in $X_n\setminus X_m$ are used by increasing the polynomial degree and determining the coefficients through a constrained least-squares formulation.
Fixing a (orthogonal) basis of the chosen spherical polynomial space, the coefficients of the approximating polynomial are obtained as the solution of a Karush–Kuhn–Tucker system that simultaneously encodes the interpolation constraints and the regression conditions. The resulting operator yields spherical polynomials of intermediate degree, balancing local exactness with global error reduction.
The proposed framework extends interpolation-regression techniques previously developed for intervals and planar domains to the spherical setting and applies to any sampling configuration yielding a full-rank spherical Vandermonde matrix. Numerical experiments confirm that the method delivers stable and high-order accurate reconstructions, making it suitable for applications in geophysics, climate modeling, and scientific computing.
This is a join work with F. Dell’Accio (U. Calabria, Italy), F. Nudo (U. Calabria, Italy), and M. A. Piñar (U. Granada, Spain).amework of spectral methods for boundary-value problems. The solution of a boundary-value problem for a stationary Schrödinger equation on the unit ball can be studied from a variational perspective. In this variational formulation, a Sobolev inner product naturally arises. As test functions, we consider the linear space of the polynomials satisfying the boundary conditions on the sphere, and a basis of mutually orthogonal polynomials with respect to the Sobolev inner product is provided. The basis of the proposed method is given in terms of spherical harmonics and univariate Sobolev orthogonal polynomials. The connection formula between these Sobolev orthogonal polynomials and the classical orthogonal polynomials on the ball is established. Consequently, the Sobolev Fourier coefficients of a function satisfying the boundary value problem are recursively derived. Finally, one numerical experiment is presented.
Miguel Pinar
(University of Granada)
The main objective of this work is to show how Sobolev orthogonal polynomials emerge as a useful tool within the framework of spectral methods for boundary-value problems. The solution of a boundary-value problem for a stationary Schrödinger equation on the unit ball can be studied from a variational perspective. In this variational formulation, a Sobolev inner product naturally arises. As test functions, we consider the linear space of the polynomials satisfying the boundary conditions on the sphere, and a basis of mutually orthogonal polynomials with respect to the Sobolev inner product is provided. The basis of the proposed method is given in terms of spherical harmonics and univariate Sobolev orthogonal polynomials. The connection formula between these Sobolev orthogonal polynomials and the classical orthogonal polynomials on the ball is established. Consequently, the Sobolev Fourier coefficients of a function satisfying the boundary value problem are recursively derived. Finally, one numerical experiment is presented.