Orthogonal Polynomials on Planar Cubic Curves

Foundations of Computational Mathematics https://doi.org/10.1007/s10208-021-09540-w Orthogonal Polynomials on Planar Cubic Curves Marco Fasondini1 · Sheehan Olver1 · Yuan Xu2 Received: 22 November 2020 / Revised: 14 June 2021 / Accepted: 30 July 2021 © The Author(s) 2021 Abstract Orthogonal polynomials in two variables on cubic curves are considered. For an integral with respect to an appropriate weight function defined on a cubic curve, an explicit basis of orthogonal polynomials is constructed in terms of two families of orthogonal polynomials in one variable. We show that these orthogonal polynomials can be used to approximate functions with cubic and square root singularities, and demonstrate their usage for solving differential equations with singular solutions. Keywords Orthogonal polynomials · Orthogonal series · Cubic curve · Approximation · Singular functions Mathematics Subject Classification 33C50 · 42C05 · 42C10 · 65M70 1 Introduction We study orthogonal polynomials of two variables with respect to an inner product defined on a planar cubic curve. This is a continuation of recent work by the last two authors that studies orthogonal polynomials on simple one-dimensional geometries Communicated by Alan Edelman. The first and second authors were supported by the Leverhulme Trust Research Project Grant RPG-2019-144 “Constructive approximation theory on and inside algebraic curves and surfaces”. B Sheehan Olver Marco Fasondini Yuan Xu 1 Department of Mathematics, Imperial College, London SW7 2AZ, UK 2 Department of Mathematics, University of Oregon, Eugene, OR 97403, USA 123 Foundations of Computational Mathematics embedded in two-dimensional space, including wedges [15] and quadratic curves [17], as well as higher-dimensional cases constructed via surfaces of revolution [16,22,23]. It is assumed the cubic curve γ is of the standard form y 2 = φ(x), where φ is a cubic polynomial of one variable. We consider polynomials that are orthogonal with respect to the inner product   f , gγ = Ωγ f (x, y)g(x, y)w(x)dσ (x, y), where Ωγ is the set on which the cubic curve γ is defined and w is an appropriate weight function, and the inner product is well defined on the space R[x, y]/y 2 −φ(x). These orthogonal polynomials are algebraic polynomials of two variables restricted to γ and their structure is determined by the characteristics of the curve. In particular, the dimension of the space of the orthogonal polynomials of degree n is 3 for all n ≥ 3, so that it does not increase with n, as with orthogonal polynomials of two variables on either an algebraic surface or on a domain with non-empty interior [4]. Our main result shows that the orthogonal structure on the cubic curve can be understood, by making use of symmetry, through a mixture of two univariate orthogonal systems. To wit, we are able to construct orthogonal polynomials on the curve explicitly in terms of univariate orthogonal polynomials. Moreover, this structural connection propagates to quadrature rules and polynomial interpolation based on the roots of the orthogonal polynomials. Hence, we have developed a toolbox for the computational and analytical study of functions on cubic curves. We provide two applications to showcase the usage of our results. The first is the approximation of functions with cubic singularities. We compare the results to Hermite–Padé approximation, a common technique for approximating functions with singularities, demonstrating that our approximation converges faster and is more robust to degeneracies. The second application is differential equations, which demonstrates the effectiveness of a spectral collocation method, based on our toolbox, for solving differential equations with singular solutions. The examples include the computation of an elliptic integral that can be expressed in terms of the Legendre’s incomplete integral of the first kind. The paper is organized as follows. The orthogonal structure on the cubic curve is established in the next section, where we clarify the families of cubic curves that we consider, which leads to several distinguished cases, and show how orthogonal polynomials can be constructed explicitly in terms of univariate orthogonal polynomials; the section also contains several families of examples. In the third section we consider quadrature rules on the cubic curve as well as polynomial interpolation based on the nodes of the quadrature rules, both from a theoretical and a computational perspective. The applications are discussed in the fourth section and the final section is on possible future work. 123 Foundations of Computational Mathematics 2 Orthogonal Polynomials on Cubic Curves 2.1 Cubic Curves Throughout this paper we let φ be a cubic polynomial defined by φ(x) = a0 x 3 + a1 x 2 + a2 x + a3 , ai ∈ R, a0 = 0. (1) We consider the cubic curve γ on the plane defined by the standard form y 2 = φ(x), (x, y) ∈ R2 , since all irreducible bivariate cubics can be transformed1 into this form [1,2]. We let γ = {(x, y) : y 2 = φ(x)} be the graph of the curve. Without loss of generality, we assume that a0 > 0, so that φ(x) > 0 for sufficiently large x > 0. Let Ωγ := {x : φ(x) > 0}, which is the set on which the cubic curve is defined. The cubic polynomial φ can have either one real zero or three real zeros, so that Ωγ can be either one interval or the union of two intervals. This leads to three possibilities: (I) Ωγ = (A, ∞): the curve has one component; (II) Ωγ = (A1 , B1 ) ∪ (A2 , ∞) with A1 < B1 < A2 : the curve has two disjoint components; (III) Ωγ = (A, B) ∪ (B, ∞): the curve has two touching components. In the first case, φ has one real zero A. In the second case, φ has three real zeros A1 < B1 < A2 . In the third case, φ has a real zero at A and a double zero at B. For examples, see Figs. 1 and 2 below. One important family of cubic curves included in our definition is that of elliptic curves [10]. An elliptic curve is a plane curve defined by y 2 = x 3 + ax + b, (2) where a and b are real numbers and the curve has no cusps, self-intersections, or isolated points. This holds if and only if the discriminant Δ E = −16(4a 3 + 27b2 ) is not equal to zero. The graph has two components if Δ E > 0 and one component if Δ E < 0. Two elliptic curves are depicted in Fig. 1, the left-hand one has one component, whereas the right-hand one has two components. 1 While the transformation to canonical form will not necessarily map polynomials to polynomials, it will still provide an orthogonal expansion on other cubic curves. 123 Foundations of Computational Mathematics Fig. 1 Elliptic curves. Left: y 2 = x 3 − x + 1. Right: y 2 = x 3 − 3x + 1 We can write the elliptic curve in a different form. Let φ(x) = x 3 + ax + b. By the definition, φ(A) = 0 or b = −A3 − a A, so that φ(x) = x 3 + ax − A3 − a A = (x − A)(x 2 + Ax + A2 + a). We need x 2 + Ax + A2 + a ≥ 0 for x ≥ A, which holds only if its discriminant A2 − 4(A2 + a) < 0 o (...truncated)