From: schleeha@aol.com (Schleeha) Newsgroups: sci.math [1] Proof of Completeness of Sturm-Liouville Solutions? Date: 22 Dec 1996 21:55:50 GMT Organization: AOL http://www.aol.com Lines: 16 Message-ID: <19961222215400.QAA26777@ladder01.news.aol.com> NNTP-Posting-Host: ladder01.news.aol.com X-Admin: news@aol.com I have heard it asserted that the solutions to any Sturm-Liouville problem forms a complete set of orthogonal functions. However, I have not been successful at finding a proof that the solutions form a complete set, meaning that any (well-behaved) arbitrary function on the interval of orthogonality can be expressed as a linear combination of the solutions. All references to this assertion have included statements like, "The proof is too complicated for inclusion in this text...the interested reader can refer to XYZ." I have followed all leads, and it turns out that the references "XYZ" do not contain the proof either. Any help in this area would be much appreciated. End of article 173515 (of 173787) -- what next? [npq] sci.math #173666 (18 + 1051 more) (1)+-[1] From: kochman@harder.ccr-p.ida.org (Fred Kochman) \-[1] [1] Re: Proof of Completeness of Sturm-Liouville Solutions? Proofs can be found in many textbooks. Two that come to mind are the ones by Birkhoff and Rota, and the one by Phillip Hartman. End of article 173666 (of 173787) -- what next? [npq] sci.math #173708 (17 + 1051 more) (1)+-(1) From: "N. Shamsundar" \-[1] [1] Re: Proof of Completeness of Sturm-Liouville Solutions? For a proof, based on Parseval's equality, see Redheffer's "Differential Equations", Jones & Bartlett, 1991, p.447. He excuses himself from the proof of the converse theorem for the reason that it requires the Lebesgue integral.