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".

Inhalt
Volume 3.- (19451) Inequalities for the Capacity of a Condenser (with G. Pólya).- (19452) On the Capacity of a Condenser.- (19481) On an Inequality of P. Turán Concerning Legendre Polynomials.- (19491) The Virtual Mass of Nearly Spherical Solids.- (19492) Virtual Mass and Polarization (with M. Schiffer).- (19501)Über eine Verallgemeinerung des Dirichletschen Integrals.- (19502) On Membranes and Plates.- (19503) Extremum Problems for Non-Negative Sine Polynomials (with W. W. Rogosinski).- (19504) Conformal Mapping of the Interior of an Ellipse onto a Circle.- (19505) On the Relative Extrema of Legendre Polynomials.- (19506) On Certain Special Sets of Orthogonal Polynomials.- (19511) Special Conformal Mappings (with L. Fejér).- (19512) Principal Frequency, Torsional Rigidity and Electrostatic Capacity.- (19521) On Certain Set Functions Defined by Extremum Properties in the Theory of Functions and in Mathematical Physics.- (19522) On Certain Hermitian Forms Associated with the Fourier Series of a Positive Function.- (19523) Conformal Mapping Related to Torsional Rigidity, Principal Frequency, and Electrostatic Capacity.- (19524) Recent Contributions of the Hungarian School to Conformal Mapping.- (19531) A Note on the Reciprocal of a Fourier Series (with A. Edrei).- (19532) On the Vibrations of a Clamped Plate.- (19533) Remark on the Preceding Paper of Charles Loewner.- (19534) On the Eigen-values of Certain Hermitian Forms (with M. Kac and W. L. Murdock).- (19541) Inequalities for Certain Eigenvalues of a Membrane of Given Area.- (19542) On Certain Mean Values of Polynomials (with A. Zygmund).- (19543) On the Singularities of Zonal Harmonic Expansions.- (19544) On a Theorem of C.Carathéodory.- (19545) Otto Szász.- (19546) Review of Higher Transcendental Functions, Vols. 1 and 2.- (19551) On Algebraic Equations with Integral Coefficients Whose Roots Belong to a Given Point Set (with M. Fekete).- (19552) On a Certain Kind of Symmetrization and Its Applications.- (19561) Relations between Different Capacity Concepts.- (19581) Note to My Paper On Membranes and Plates.- (19591) Recent Advances and Open Questions on the Asymptotic Expansions of Orthogonal Polynomials.- (19601) On Certain Differential-Integral Equations (with S. Karlin).- (19602) Concerning the Fourier Coefficients of a Nonnegative Function.- (19603) A Problem in Prediction Theory (with Henry Helson).- (19604) Leopold Fejér: In Memoriam 18801959.- (19605) Emlékezés Fejér Lipótra.- (19606) An Extremum Problem for Polynomials (with I. J. Schoenberg).- (19607) On the Gradient of Solid Harmonic Polynomials.- (19608) On Certain Determinants Whose Elements are Orthogonal Polynomials (with S. Karlin).- (19611) On the Monotone Convergence of Certain Riemann Sums (with P. Turán).- (19621) On Positive Harmonic Polynomials.- (19622) An Inequality for Jacobi Polynomials.- (19631) On Bi-Orthogonal Systems of Trigonometric Polynomials.- (19632) On the Positivity of Certain Sums of Ultraspherical Polynomials, by Ervin Feldheim with a note by G. Szegö.- (19641) On a Problem of the Best Approximation.- (19642) On Some Problems of Approximations.- (19681) An Outline of the History of Orthogonal Polynomials.- (19721) The Contributions of L. Fejér to the Constructive Function Theory.