Hilbert matrix

Abstract: In an earlier paper (Rao, 1959), the author discussed the method of least squares when the observations are dependent and the dispersion matrix is unknown but an independent estimate is available. The unknown dispersion matrix was, however, considered as an arbitrary positive definite matrix. In the present paper we shall consider a class of problems where the dispersion matrix has a known structure and discuss the appropriate statistical methods. More specifically the structure of the dispersion matrix results from considering the parameters in the well-known Gauss-Markoff linear model as random variables. Let Y be a vector random variable with the structure

Abstract: A famous inequality of D. Hilbert [70], [36] asserts that the matrix commonly known as Hilbert's matrix, determines a bounded linear operator on the Hilbert space of square summable complex sequences. Infinite matrices which possess a similar form to H, namely those that are 'one way infinite' and have identical entries in cross diagonals, are called Hankel matrices, and when these matrices determine bounded operators we have Hankel operators, the subject of this article. The formal companions to the Hankel operators are the Toeplitz operators which have representing matrices possessing a constancy along the long diagonals. Since the stimulating paper of A. Brown and P. R. Halmos [9], the properties of these operators have been extensively developed, culminating in a substantial and sophisticated theory for their spectral, algebraic and C*-algebraic aspects. See, for example, [22], [23] and [64; Chapter 10]. The fact that Hankel operators have not enjoyed such attention is partly due to their dearth of algebraic properties, to the rather mysterious relationship that exists between a Hankel operator and its defining symbol function and to the fact that, in some senses, they are not so natural.. However, there are excellent reasons for studying them, over and above the fact that they form a new and curious class for the attention of operator theorists, and perhaps we should affirm some of these reasons in order to provide an outlook for the reader. It would be desirable to have a context for Hilbert's operator H and to be able to perceive its properties (bounded, not compact, positive, spectrum equal to [0,7r] etc.) as instances of more general theorems concerning the Hankel form. Hankel operators are not as special as one might initially think, at least if one allows unitary equivalence. An integral operator on L 2 (0, oo) whose kernel is of the form k{x + y) is equivalent to a Hankel operator. A prime example is the singular integral operator considered by T. Carleman [12], and which may be viewed as the continuous variant of H. This operator is also the square of the Laplace transform, considered as an Received 28 March. 1980.

Abstract: We study the structure of the set of numerically effective divisor classes on a rational surface and apply this to study hilbert functions of the homogeneous coordinate rings of 0-cycles on curves of low degree in P 2 For example, for a nonnegative 0-cycle mipi + � � � + mnpn of points p1,,pn on an irreducible conic in P 2 , we show that the hilbert function depends only on the coefficients mi, as conjectured by Davis and Geramita (DG) We also determine for each such 0-cycle the degree at which the hilbert function stabilizes (first equals its hilbert polynomial) and we characterize all such 0-cycles having a generic hilbert function, a generic hilbert function being one which is equal to the hilbert function of the ring of P 2 up to the point at which it stabilizes

Abstract: We show that the Euclidean condition number of any positive definite Hankel matrix of order $n\geq 3$ may be bounded from below by $\gamma^{n-1}/(16n)$ with $\gamma=\exp(4 \cdot{\it Catalan}/\pi) \approx 3.210$ , and that this bound may be improved at most by a factor $8 \gamma n$ . Similar estimates are given for the class of real Vandermonde matrices, the class of row-scaled real Vandermonde matrices, and the class of Krylov matrices with Hermitian argument. Improved bounds are derived for the case where the abscissae or eigenvalues are included in a given real interval. Our findings confirm that all such matrices – including for instance the famous Hilbert matrix – are ill-conditioned already for “moderate” order. As application, we describe implications of our results for the numerical condition of various tasks in Numerical Analysis such as polynomial and rational i nterpolation at real nodes, determination of real roots of polynomials, computation of coefficients of orthogonal polynomials, or the iterative solution of linear systems of equations.