Orthogonal Polynomials on a Planar Quartic Curve

The orthogonal structure in two variables on the quartic curve \(y^2=a x^4+bx^2+c\) $y^2=a{x}^4+b{x}^2+c$ is considered. For an even weight function on the curve, we show that orthogonal polynomials can be expressed in terms of two families of orthogonal polynomials in one variable. We establish relationships between the partial Fourier sum on the curve and partial Fourier sums in one variable. We also investigate the quadrature rule and polynomial interpolation on the curve.