[Paper Review] Orthogonal polynomials on the disk in the absence of finite moments

Peter C. Gibson|arXiv (Cornell University)|Feb 5, 2015

Electromagnetic Scattering and Analysis6 references3 citations

TL;DR

This paper introduces scattering polynomials—orthogonal on the unit disk with respect to the measure $ dxdy/(1 - x^2 - y^2) $, which has infinite even moments—proving they form an orthogonal basis for $ L^2( ext{disk}, dx hinspace dy/(1 - x^2 - y^2)) $. The construction uses a Rodrigues-type formula and connects to Jacobi polynomials, enabling completeness despite the absence of finite moments.

ABSTRACT

We introduce a new family of orthogonal polynomials on the disk that has emerged in the context of wave propagation in layered media. Unlike known examples, the polynomials are orthogonal with respect to a measure all of whose even moments are infinite.

Motivation & Objective

• To establish a new family of orthogonal polynomials on the unit disk where the underlying measure has infinite even moments, specifically $ (1 - x^2 - y^2)^{-1} dx hinspace dy $.
• To resolve the challenge of defining orthogonal polynomials when standard $ L^2 $ inner products diverge due to non-integrable moments.
• To prove that scattering polynomials, defined via a Rodrigues-type formula, form an orthogonal basis in the weighted $ L^2 $ space with measure $ dx hinspace dy/(1 - x^2 - y^2) $.
• To demonstrate completeness and orthogonality of these polynomials despite the absence of finite moments, using spectral theory and orthogonal polynomial connections.

Proposed method

• Defining scattering polynomials via a Rodrigues-type formula: $ ho^{(p,q)}(z) = rac{(-1)^p}{q(p+q-1)!}(1 - zar{z}) rac{ au^{p+q}}{ au z^p au ar{z}^q}(1 - zar{z})^{p+q-1} $ for $ p, q eq 0 $.
• Showing that scattering polynomials are eigenfunctions of the operator $ - ilde{ abla} = -(1 - x^2 - y^2) abla/4 $, with eigenvalues $ pq $, ensuring orthogonality across distinct eigenvalues.
• Expressing the radial part of the polynomials in terms of Jacobi polynomials $ P^{(1, |p-q|)}_{ u}(2r^2 - 1) $, where $ u = ext{min}iglrace{p,qigrace} - 1 $, linking them to classical orthogonal polynomials.
• Using the transformation $ h(r, heta) = ilde{g}(r, heta) ilde{ ho}(r) $ with $ ilde{ ho}(r) = ilde{1} - r^2 $ to relate the weighted $ L^2 $ space to standard $ L^2 $, enabling use of known orthogonal bases.
• Proving completeness by showing that if a function is orthogonal to all scattering polynomials, it must vanish almost everywhere, using the density of $ iglrace{Q^{(1,|n|)}_{ u}(u)igrace} $ in $ L^2([-1,1], du) $.
• Establishing orthogonality between polynomials with different $ pq $-products via distinct eigenvalues and between same-$ pq $-product polynomials via angular frequency separation $ e^{i(q-p) heta} $.

Experimental results

Research questions

• RQ1 Can orthogonal polynomials be constructed on the unit disk when the weight measure $ (1 - x^2 - y^2)^{-1} dx hinspace dy $ has infinite moments, rendering standard $ L^2 $ inner products ill-defined?
• RQ2 Do the scattering polynomials defined via a Rodrigues-type formula form a complete orthogonal basis in the space $ L^2( ext{disk}, dx hinspace dy/(1 - x^2 - y^2)) $, despite the non-integrability of monomials?
• RQ3 How are scattering polynomials related to classical orthogonal polynomials, particularly Jacobi polynomials, given the absence of a corresponding $ eta $-parameter in the standard family?
• RQ4 Can the spectral properties of the modified Laplacian $ - ilde{ abla} $ be used to prove orthogonality and completeness of the scattering polynomial system?
• RQ5 What is the role of angular frequency $ e^{i(q-p) heta} $ in ensuring orthogonality between polynomials with different $ (p,q) $ indices?

Key findings

• Scattering polynomials $ ho^{(p,q)} $ defined by the Rodrigues formula are orthogonal in $ L^2( ext{disk}, dx hinspace dy/(1 - x^2 - y^2)) $, with orthogonality arising from distinct eigenvalues of $ - ilde{ abla} $ or distinct angular frequencies.
• The radial components of $ ho^{(p,q)} $ are expressible as $ f^{(p,q)}(r) = c imes (1 - r^2) r^{|p-q|} P^{(1,|p-q|)}_{ u}(2r^2 - 1) $, linking them to Jacobi polynomials with $ eta = |p - q| $, $ u = ext{min}iglrace{p,qigrace} - 1 $.
• The system $ iglrace{ ho^{(p,q)} igm| ext{min}iglrace{p,qigrace} eq 0 igrrace} $ forms an orthogonal basis for $ L^2( ext{disk}, r hinspace dr hinspace d heta / (1 - r^2)) $, proven via spectral theory and completeness of $ Q^{(1,m)}_{ u}(u) $ in $ L^2([-1,1], du) $.
• The space $ L^2( ext{disk}, dx hinspace dy/(1 - x^2 - y^2)) $ is isometric to $ ilde{1} - r^2 imes L^2( ext{disk}, r hinspace dr hinspace d heta) $, enabling transfer of completeness from standard $ L^2 $ to the weighted space.
• If a function $ h $ is orthogonal to all $ ho^{(p,q)} $, then the associated $ g $ in $ L^2( ext{disk}, r hinspace dr hinspace d heta) $ must vanish a.e., proving that the orthogonal complement of the span of $ ho^{(p,q)} $ is trivial.
• The scattering polynomials provide a basis for $ L^2( ext{disk}, dx hinspace dy) $ that is consistent with Dirichlet boundary conditions, unlike Zernike polynomials which are non-zero on the boundary.