These proceedings contain the lectures presented at the Conference on Linear Operators and Approximation held at the Oberwolfach Mathematical Research In­ stitute, August 14-22, 1971. There were thirty-eight such lectures while four addi­ tional papers, subsequently submitted in writing, are also included in this volume. Two of the three lectures presented by Russian mathematicians are rendered in English, the third in Russian. Furthermore, there is areport on new and unsolved problems based upon special problem sessions, with later communications from the participants. In fact, two of the papers inc1uded are devoted to solutions of some of the problems posed. The papers have been classified according to subject matter into five chapters, but it needs little emphasis that such thematic groupings are necessarily somewhat arbitrary. Thus Chapter I on Operator Theory is concerned with linear and non­ linear semi-groups, structure of single operators, unitary operators, spectral and ergodic theory. Chapter Il on Topics in Functional Analysis inc1udes papers on Riesz spaces, boundedness theorems, generalized limits, and distributions. Chapter III, entitled "Approximation in Abstract Spaces", ranges from characterizations of c1asses of functions in approximation theory to approximation-theoretical topics connected with extensions to Banach (or more general) spaces. Chapter IV contains papers on harmonic analysis in connection with approximation and, finally, Chapter V is devoted to approximation by splines, algebraic polynomials, rational functions, and to Pade approximation. A large part of the general editorial work connected with these proceedings was competently handled by Miss F. Feber, while G.

Inhalt

I. Operator Theory.- Generalizations of Landau's inequality.- Semi-groups of nonlinear transformations.- Singular perturbations of semi-group generators.- Cyclic vectors and commutants.- Some applications of operator valued analytic functions of two complex variables.- Weyl's theorems.- Some approximation problems in the theory of stationary processes.- Ein Operatorenkalkül für das approximationstheoretische Verhalten des Ergodensatzes im Mittel..- A note on the Landau-Kallmann-Rota-Hille inequality.- II. Topics in Functional Analysis.- Representation theorems for Riesz spaces.- Boundedness from measure theory.- Exchange of infinite summation with limits in Banach spaces.- Finite parts of distributions.- Courants et courants cylindriques sur des variétés de dimension infinie.- Subdefinite functions.- Eine Bemerkung zum Banachschen Fixpunktsatz..- On a certain converse of Hölder's inequality.- III. Approximation in Abstract Spaces.- On some geometric properties of the unit sphere.- Diskrete Konvergenz linearer Operatoren III.- On set-valued metric projections.- On the comparison of approximation processes in Hubert spaces.- Sätze vom Jackson-Typ auf Darstellungsräumen kompakter, zusammenhängender Liegruppen.- Approximation d'espaces de fonctions C? et interpolation.- Nonlinear transformations with the conservation of differential properties of functions.- Positive and monotone approximation.- Sätze vom Bohmann-Korowkin-Typ für Banachsche Funktionenräume.- IV. Harmonic Analysis and Approximation.- Projection métrique de L1 (T) sur des sous-espaces fermés invariants par translation.- Bases dans H1 et bases de sous-espaces de dimension finie dans A.- Logarithmic and exponential variants of Bernstein's inequality and generalized derivatives.- Fourier multipliers whose multiplier norm is an attained value.- Nikolskü constants for positive singular integrals of perturbed Fejér-type.- A pointwise "o" saturation theorem for positive convolution operators.- Convolution operators for Fourier-Jacobi expansions.- V. Spline- and Algebraic Approximation.- Cardinal interpolation and spline functions IV. The exponential Euler splines.- ?????????? ???????? ? ?????? ??????????? (Anwendungen von Splines in der Approximationstheorie) 419.- The fundamental theorem of algebra for monosplines with multiplicities.- A contribution to the problem of weighted polynomial approximation.- Zur Konvergenz der Stufenpolynome über den Nullstellen der Legendre-Polynome.- Chebyshev semi-discrete approximations for linear parabolic problems.- On the convergence theory of Padé approximants.- Singular self-adjoint multipoint boundary value problems: Solutions and approximations.- Über die Konvergenz von natürlichen interpolierenden Splines.- New and Unsolved Problems.- Alphabetical List of Papers.