54083834
Interpolation of operators with change of measures
Introduction. This paper has two purposes. First, it is to give general interpolation theorems which allow one to change measures simultaneously with changing exponents of the Lp classes concerned. These theorems are generalizations of theorems of M. Riesz [7] and Marcinkiewicz [4]. The idea of interpolating between measure seems to have been implicit for some time. The general situation has, however, only recently been investigated. (See e.g. [2] and [8].) Our second purpose is to apply these techniques. In §§3 and 4, we shall prove theorems for unbounded orthonormal systems which generalize known results for bounded orthonormal systems. These theorems are extensions of theorems of Paley, Marcinkiewicz and Zygmund [5], and Pitt [6]. §1 will be devoted to necessary definitions and statements of needed known results. §2 will contain the general interpolation theorems. 1. Definitions and statement of general results. Let (M, 93J, p) and (N, yi, v) be two measure spaces. We will consider only measurable, real or complex-valued functions on these two spaces. In this paper we will establish a generalization of the well known convexity theorem of M. Riesz (Theorem (1.1) below) and give some applications of this new theorem to certain orthonormal series. In order to do this we will need to establish some notation and give a few definitions. Suppose T is a mapping of a class of functions on M into a class of functions on N. T is called a sublinear operator if it satisfies the following properties: (i) If/=/i+/2 and Tft (i=l, 2) are defined then Tf is defined; (ii) | T(fx+f2) | ^ | Tf\\ + | Tf2\ almost everywhere; (iii) For any scalar k, we have | T(kf)\ =\k\\ Tf\ almost everywhere. Let p, g2t 1 be two real numbers. We say that T is of type (p, q) in case T is defined for all functions/ in L"(M, W, ju) and there exists a positive number, K, independent of/such that
118896808
A kernel associated with certain multiply connected domains and its applications to factorization theorems
15925390
A derivation of the main results of the theory of Hp spaces
117069300
On the Picard property of lacunary power series
124018984
Complex Methods in Harmonic Analysis
116350194
Invariant systems of conjugate harmonic functions associated with compact Lie groups
124903617
The metric Fuglede property and normality
122425983
On the Inversion of the Specific-Heat Function
In this paper we present a method for the inversion of specific heat data to find the vibrational frequency spectrum of a solid. Our results are given in terms of a power series in UJ, the frequency. In this series the coefficients depend on C v (T) and its d~rivatives at T=O. The series thus derived is valid only in an asymptotic sense, but represents an improvement over inversions by Montroll and Kroll which require a knowledge of functions formed from the data, and the evaluation of double integrals over these functions, A method for forming the successive values of Cv(n) (0) from experimental data is also presented. One of the most frequently measured physical characteristics of a solid is the specific heat at constant volume Cv (T), where T is the temperature. The behavior of this function has quite accurately been predicted by various characterizations of a lattice of atoms which have harmonic interactions. If it is assumed that the specific heat can be attributed solely to harmonic effects, then it can be written as an integral over the vibrational frequency spectrum 1 ;
8267029
Transfer Functions of Regular Linear Systems. Part I: Characterizations of Regularity
We recall the main facts about the representation of regular linear systems, essentially that they can be described by equations of the form x(t) = Ax(t) + Bu(t), y(t) = Cx(t) + Du(t) , like finite dimensional systems, but now A, B and C are in general unbounded operators. Regular linear systems are a subclass of abstract linear systems. We define transfer functions of abstract linear systems via a generalization of a theorem of Foures and Segal. We prove a formula for the transfer function of a regular linear system, which is similar to the formula in finite dimensions. The main result is a (simple to state but hard to prove) necessary and sufficient condition for an abstract linear system to be regular, in terms of its transfer function. Other conditions equivalent to regularity are also obtained. The main result is a consequence of a new Tauberian theorem, which is of independent interest.
117037404
Two conjectures on the admissibility of control operators
We are searching for necessary and/or sufficient conditions for the admissibility of unbounded control operators for semigroups on Hilbert spaces, with respect to input functions of class L 2. Our first conjecture is that admissibility of an unbounded input element 6 for a semigroup with generator A is equivalent to a certain decay rate of ∥(sI - A)-1 b∥ as Re s → ∞. The second conjecture states that a control operator B defined on a Hilbert space U is admissible if and only if, for any v ∈ U, Bv is an admissible input element. It is proved that both conjectures hold in many important particular cases (e.g., the first conjecture is true if the semigroup is normal).
123105333
On Certain Redundant Systems which Operate at Discrete Times
This paper discusses the effect of using redundant components on the operation of systems which perform a given task repetitively. It is assumed that each component of the system has a constant probability of failure at every cycle. A question which immediately arises is: “What is the probability distribution for the number of the cycle of first system failure?” The solution to this problem generalizes considerations leading to the geometric distribution. The effects of replacements on the failure cycle distribution are analyzed.