proceedings of the american mathematical society Volume 123, Number 2, February 1995 A UNIFIED APPROACH TO SOME PREDICTION PROBLEMS STEPHEND. ABBOTT (Communicated by Palle E. T. Jorgensen) Abstract. In this paper we solve a general extremal problem for a nonnega- tive operator in Hubert space. We show that it contains the classical infimum problems of Szegö and Kolmogorov for bounded weight functions on the circle and also prove some new prediction theorems. 1. Introduction Let C be the complex numbers, D = {z G C : |z| < 1}, and dD = {z £ C : |z| = 1}. Define the function % on dD by xie'e) = e'e > and let a be the normalized Lebesgue measure on dD. Given a nonnegative function co(e'e) G Lx[dD], we state two classical theorems from prediction theory: (i) Szegö's infimum: inf < / \l - xp\2ojdo : p = ^2cjXj \ = exp I / logo)da j ; (ii) Kolmogorov's infimum: 1 - p\2œda :p = y^ cix1, / pda = 0} = \ —da .£*» ho [ [Jor>w -i \J\<« If the function appearing on the right-hand side of either equation above is not integrable, the infimum is understood to be zero. We now pose an abstract problem that essentially contains both of these. Let AAf be a Hilbert space with inner product (•,•), and let ¿%(A¿A) be the set of bounded operators on A?. Let F be the orthogonal projection onto a closed subspace fê C 5A, and let IF be a nonnegative operator in ^(^A). Now fix k £ %? and consider iaf{(W(k - f), (k - f)) : f £¿f, Pf = 0}. Received by the editors December 18, 1992 and, in revised form, May 3, 1993. 1991 Mathematics Subject Classification. Primary47B65, 47N30,60G25;Secondary 47B35. Key words and phrases. Szegö's infimum, Kolmogorov's infimum, prediction theory, Hardy space, Toeplitz operator, Hilbert space. The author would like to acknowledge and thank his advisor, Professor Marvin Rosenblum, for many helpful suggestions and conversations. © 1994 American Mathematical Society 0002-9939/94 $1.00+ $.25 per page 425 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use