The arithmetic of certain semigroups of positive operators

1968 ◽  
Vol 64 (1) ◽  
pp. 161-166 ◽  
Author(s):  
John Lamperti
Keyword(s):  
Finite Number ◽  
Positive Definite ◽  
Positive Operators ◽  
Linear Operators ◽  
Real Numbers ◽  
Ultraspherical Polynomials ◽  
Real Polynomials ◽  
Fixed Parameter ◽  
Summation Sign ◽  
Image Position

Some time ago, S. Bochner gave an interesting analysis of certain positive operators which are associated with the ultraspherical polynomials (1,2). Let {Pn(x)} denote these polynomials, which are orthogonal on [ − 1, 1 ] with respect to the measureand which are normalized by settigng Pn(1) = 1. (The fixed parameter γ will not be explicitly shown.) A sequence t = {tn} of real numbers is said to be ‘positive definite’, which we will indicate by writing , provided thatHere the coefficients an are real, and the prime on the summation sign means that only a finite number of terms are different from 0. This condition can be rephrased by considering the set of linear operators on the space of real polynomials which have diagonal matrices with respect to the basis {Pn(x)}, and noting that

Unbounded Positive Definite Functions

1969 ◽  
Vol 21 ◽  
pp. 1309-1318 ◽  
Author(s):  
James Stewart
Keyword(s):  
Abelian Group ◽  
Positive Measure ◽  
Positive Definite Function ◽  
Positive Definite ◽  
Locally Compact Abelian Group ◽  
Definite Function ◽  
Complex Numbers ◽  
Stieltjes Transform ◽  
Real Numbers ◽  
Image Position

Let G be an abelian group, written additively. A complexvalued function ƒ, defined on G, is said to be positive definite if the inequality1holds for every choice of complex numbers C1, …, cn and S1, …, sn in G. It follows directly from (1) that every positive definite function is bounded. Weil (9, p. 122) and Raïkov (5) proved that every continuous positive definite function on a locally compact abelian group is the Fourier-Stieltjes transform of a bounded positive measure, thus generalizing theorems of Herglotz (4) (G = Z, the integers) and Bochner (1) (G = R, the real numbers).If ƒ is a continuous function, then condition (1) is equivalent to the condition that2

scholarly journals Application of the Ritz method to non-standard eigenvalue problems

1969 ◽  
Vol 10 (3-4) ◽  
pp. 367-384 ◽  
Author(s):  
A. L. Andrew
Keyword(s):  
Eigenvalue Problems ◽  
Ritz Method ◽  
Positive Definite ◽  
Linear Operators ◽  
Extensive Literature ◽  
Maximum Eigenvalue ◽  
Special Cases ◽  
Image Position ◽  
Linear Manner

There is an extensive literature on application of the Ritz method to eigenvalue problems of the type where L1, L2 are positive definite linear operators in a Hilbert space (see for example [1]). The classical theory concerns the case in which there exists a minimum (or maximum) eigenvalue, and subsequent eigenvalues can be located by a well-known mini-max principle [2; p. 405]. This paper considers the possibility of application of the Ritz method to eigenvalue problems of the type (1) where the linear operators L1L2 are not necessarily positive definite and a minimum (or maximum) eigenvalue may not exist. The special cases considered may be written with the eigenvalue occurring in a non-linear manner.

On diagonal values of probability vectors of infinitely many components

1961 ◽  
Vol 57 (4) ◽  
pp. 759-766
Author(s):  
A. S. Besicovitch
Keyword(s):  
Finite Number ◽  
Full Range ◽  
Dimensional Vector ◽  
Probability Vector ◽  
Real Numbers ◽  
Borel Sets ◽  
Image Position ◽  
Borel Class ◽  
Vector Valued ◽  
Range Of Values

Given † probability vector μ(X) = (μ1(X), … μk (X)) of a finite number of components on a Borel class of sets X, we say that μ(X0) has a diagonal value α if μi(X0) = α for all i = 1, 2,…,K. J. Neyman(l), (2), (3) has proved that in the class of Borel sets of real numbers any non-atomic vector μ(X) takes all diagonal values. A. Liapounoff has studied the full range of values of k-dimensional vector-valued measures and in two papers (4) he has proved that the range is closed and in the case of non-atomic measures the range is also convex. He also gave an example showing that neither of these results holds in the case of vectors of infinitely many components. A simplified proof of Liapounoff's results has been given by P. R. Halmos (5). In the present paper I study the range of values of probability vectors of infinitely many components. Various types of conditions are studied which are sufficient to imply that, for each ε > 0, 0 ≤ α ≤ 1, it is possible to find a set X such that

Cosine Representations of Abelian *-Semigroups and Generalized Cosine Operator Functions

1978 ◽  
Vol 30 (03) ◽  
pp. 474-482 ◽  
Author(s):  
G. D. Faulkner ◽  
R. W. Shonkwiler
Keyword(s):  
Functional Equation ◽  
Linear Operators ◽  
Bounded Operators ◽  
Real Numbers ◽  
Bounded Linear ◽  
Cosine Operator ◽  
D’Alembert’S Functional Equation ◽  
Operator Functions ◽  
Cosine Operator Functions ◽  
Image Position

In the following R will denote the real numbers, for a Hilbert space H, B(H) and L(H) will denote the collections of bounded linear operators on H and linear, but not necessarily bounded, operators on H respectively. Cosine Operator Functions, namely functions C:R ⟶ B(H) which satisfy D'Alembert's functional equation (1) and (2)

Orthogonal Recurrence Polynomials and Hamburger Moments

1971 ◽  
Vol 14 (1) ◽  
pp. 53-56 ◽  
Author(s):  
A. G. Law
Keyword(s):  
Real Numbers ◽  
Real Polynomials ◽  
Image Position

A three-term recurrence1where An (n ≥ 0), Bn (n ≥ 0) and Cn (n ≥ 0) are real numbers for which AnCn+1 ≠0 (n ≥ 0), generates a sequence {Pn} of real polynomials in which Pn has degree exactly n.

On calculating extinction probabilities for branching processes in random environments

1969 ◽  
Vol 6 (03) ◽  
pp. 478-492 ◽  
Author(s):  
William E. Wilkinson
Keyword(s):  
Generating Functions ◽  
Infinite Sequence ◽  
Branching Processes ◽  
Transition Probability ◽  
Transition Probability Matrix ◽  
Random Environments ◽  
Real Numbers ◽  
Discrete Time Markov Chain ◽  
Extinction Probabilities ◽  
Image Position

Consider a discrete time Markov chain {Zn } whose state space is the non-negative integers and whose transition probability matrix ║Pij ║ possesses the representation where {Pr }, r = 1,2,…, is a finite or denumerably infinite sequence of non-negative real numbers satisfying , and , is a corresponding sequence of probability generating functions. It is assumed that Z 0 = k, a finite positive integer.

scholarly journals An Inhomogeneous Minimum For Non-Convex Star-Regions With Hexagonal Symmetry

1955 ◽  
Vol 7 ◽  
pp. 337-346 ◽  
Author(s):  
R. P. Bambah ◽  
K. Rogers
Keyword(s):  
Quadratic Form ◽  
Binary Quadratic Form ◽  
Elementary Geometry ◽  
Hexagonal Symmetry ◽  
Real Numbers ◽  
Inhomogeneous Minimum ◽  
Image Position ◽  
Indefinite Binary Quadratic Form

1. Introduction. Several authors have proved theorems of the following type:Let x0, y0 be any real numbers. Then for certain functions f(x, y), there exist numbers x, y such that1.1 x ≡ x0, y ≡ y0 (mod 1),and1.2 .The first result of this type, but with replaced by min , was given by Barnes (3) for the case when the function is an indefinite binary quadratic form. A generalisation of this was proved by elementary geometry by K. Rogers (6).

Inequalities related to Hardy's and Heinig's

1984 ◽  
Vol 96 (1) ◽  
pp. 1-7 ◽  
Author(s):  
James A. Cochran ◽  
Cheng-Shyong Lee
Keyword(s):  
Real Numbers ◽  
Image Position

In a 1975 paper [8], Heinig established the following three inequalities:where A = p/(p + s − λ) with p, s, λ real numbers satisfying p + s > λ,p > 0;where B = p/(2p + sp − λ −1) with p, s, λ real numbers satisfying 2p +sp > λ, + 1, p > 0;where is a sequence of nonnegative real numbers,and C = p[l + l/(p + s−λ)] with p, s, λ real numbers satisfying s > 0, p ≥ 1, and p +s > λ 0.

Analogue of a theorem of Khintchine in fields of formal power series

1966 ◽  
Vol 62 (4) ◽  
pp. 637-642 ◽  
Author(s):  
T. W. Cusick
Keyword(s):  
Power Series ◽  
Positive Constant ◽  
Classical Result ◽  
Real Numbers ◽  
Linear Forms ◽  
Absolute Value ◽  
The Difference ◽  
Image Position ◽  
Formal Power ◽  
Positive Integers

For a real number λ, ‖λ‖ is the absolute value of the difference between λ and the nearest integer. Let X represent the m-tuple (x1, x2, … xm) and letbe any n linear forms in m variables, where the Θij are real numbers. The following is a classical result of Khintchine (1):For all pairs of positive integers m, n there is a positive constant Г(m, n) with the property that for any forms Lj(X) there exist real numbers α1, α2, …, αn such thatfor all integers x1, x2, …, xm not all zero.

scholarly journals Minimal theta functions

1988 ◽  
Vol 30 (1) ◽  
pp. 75-85 ◽  
Author(s):  
Hugh L. Montgomery
Keyword(s):  
Functional Equation ◽  
Quadratic Form ◽  
Zeta Function ◽  
Theta Functions ◽  
Binary Quadratic Form ◽  
Positive Definite ◽  
Epstein Zeta Function ◽  
Image Position

Let be a positive definite binary quadratic form with real coefficients and discriminant b2 − 4ac = −1.Among such forms, let . The Epstein zeta function of f is denned to beRankin [7], Cassels [1], Ennola [5], and Diananda [4] between them proved that for every real s > 0,We prove a corresponding result for theta functions. For real α > 0, letThis function satisfies the functional equation(This may be proved by using the formula (4) below, and then twice applying the identity (8).)

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