Orthogonal polynomials on the disc

• Autores: Shayne Waldron
• Localización: Journal of approximation theory, ISSN 0021-9045, Vol. 150, Nº 2, 2008, págs. 117-131
• Idioma: inglés
• DOI: 10.1016/j.jat.2007.05.001
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• Resumen

  • We consider the space of orthogonal polynomials of degree n on the unit disc for a general radially symmetric weight function. We show that there exists a single orthogonal polynomial whose rotations through the angles , j=0,1,�,n forms an orthonormal basis for , and compute all such polynomials explicitly. This generalises the orthonormal basis of Logan and Shepp for the Legendre polynomials on the disc.

    Furthermore, such a polynomial reflects the rotational symmetry of the weight in a deeper way: its rotations under other subgroups of the group of rotations forms a tight frame for , with a continuous version also holding. Along the way, we show that other frame decompositions with natural symmetries exist, and consider a number of structural properties of including the form of the monomial orthogonal polynomials, and whether or not contains ridge functions.