Orthogonal structure on a quadratic curve
Sheehan Olver, Yuan Xu
Abstract
Abstract Orthogonal polynomials on quadratic curves in the plane are studied. These include orthogonal polynomials on ellipses, parabolas, hyperbolas and two lines. For an integral with respect to an appropriate weight function defined on any quadratic curve, an explicit basis of orthogonal polynomials is constructed in terms of two families of orthogonal polynomials in one variable. Convergence of the Fourier orthogonal expansions is also studied in each case. We discuss applications to the Fourier extension problem, interpolation of functions with singularities or near singularities and the solution of Schrödinger’s equation with nondifferentiable or nearly nondifferentiable potentials.
Topics & Concepts
MathematicsOrthogonal polynomialsHyperbolaClassical orthogonal polynomialsMathematical analysisGravitational singularityQuadratic equationInterpolation (computer graphics)Fourier transformComplex planeEllipseOrthogonal functionsMehler–Heine formulaDiscrete orthogonal polynomialsPure mathematicsGegenbauer polynomialsGeometryComputer graphics (images)AnimationComputer scienceMathematical functions and polynomialsIterative Methods for Nonlinear EquationsFractional Differential Equations Solutions