The Eigenfunctions of the Hilbert Matrix

Summary, in English

For each noninteger complex number lambda, the Hilbert matrix H-lambda = (1/n+m+lambda)(n,m >= 0) defines a bounded linear operator on the Hardy spaces H-p, 1 < p < a, and on the Korenblum spaces , A(-tau), tau > 0. In this work, we determine the point spectrum with multiplicities of the Hilbert matrix acting on these spaces. This extends to complex lambda results by Hill and Rosenblum for real lambda. We also provide a closed formula for the eigenfunctions. They are in fact closely related to the associated Legendre functions of the first kind. The results will be achieved through the analysis of certain differential operators in the commutator of the Hilbert matrix.

• Författare: Alexandru Aleman; Alfonso Montes-Rodriguez; Andreea Sarafoleanu
• Avdelning/ar: Matematik (naturvetenskapliga fakulteten)
• Publiceringsår: 2012
• Språk: Engelska
• Sidor: 353-374
• Publikation/Tidskrift/Serie: Constructive Approximation
• Volym: 36
• Issue: 3
• Länkar: http://dx.doi.org/10.1007/s00365-012-9157-z
• Dokumenttyp: Artikel i tidskrift
• Förlag: Springer
• Ämne: Mathematics
• Nyckelord: Hilbert matrix; Integral operator; Eingenvalues; Eigenfunctions; Differential operators; Hypergeometric function; Associated Legendre; functions of the first kind
• Status: Published
• ISBN/ISSN/Övrigt: ISSN: 0176-4276