Gabor Szeg: Collected Papers

1. 1. Definition of L-forms. In the years 1907-1911 O. Toeplitz [21, 22, 23, 24]* studied a class of quadratic forms whose matrix is of the follO\\"ing type: (Ll) C-2 C_I Co C-n-I Cn-I The elements Cn are given complex constants. Toeplitz designated these forms as L-forms and investigated in detail their relation to the analytic function defined in a neighborhood of the unit circle by the Laurent series 2; C z", n = n - 00, . . . , 00; this series is assumed to be convergent in a certain circular ring rl < I z I < r2, rl < 1 < r2. It is obvious that these matrices are connected with the infinite cyclic group, just as the finite cyclic matrix CO CI C2 C Co CI n r (1. 2) Cn-I C Co n L. c, c, Co is associated with the finite cyclic group. The main result of Toeplitz is that the spectrum of the L-form is identical with the complex values the Laurent series assumes on the unit circle I z I = 1. 1. 2. Hermitian forms. The case C = en is of particular importance; the n matrix (1. 1) is in this case a Hermitian one and the associated Laurent series i8 represents a real function f(8) on the unit circle z = e , -'II" ~ 8 < '11".