Fourier-Stieltjes Transforms of Signed Measures

1975 ◽  
pp. 11-49
Author(s):  
ROGER CUPPENS
Keyword(s):  
Stieltjes Transforms

scholarly journals Recent developments on the Stieltjes transform of generalized functions

1987 ◽  
Vol 10 (4) ◽  
pp. 641-670 ◽  
Author(s):  
Ram Sankar Pathak ◽  
Lokenath Debnath
Keyword(s):  
Final Section ◽  
Generalized Functions ◽  
Stieltjes Transform ◽  
Poisson Transform ◽  
Open Questions ◽  
The Laplace Transform ◽  
Recent Developments ◽  
The Subject ◽  
Abelian Theorems ◽  
Stieltjes Transforms

This paper is concerned with recent developments on the Stieltjes transform of generalized functions. Sections 1 and 2 give a very brief introduction to the subject and the Stieltjes transform of ordinary functions with an emphasis to the inversion theorems. The Stieltjes transform of generalized functions is described in section 3 with a special attention to the inversion theorems of this transform. Sections 4 and 5 deal with the adjoint and kernel methods used for the development of the Stieltjes transform of generalized functions. The real and complex inversion theorems are discussed in sections 6 and 7. The Poisson transform of generalized functions, the iteration of the Laplace transform and the iterated Stieltjes transfrom are included in sections 8, 9 and 10. The Stieltjes transforms of different orders and the fractional order integration and further generalizations of the Stieltjes transform are discussed in sections 11 and 12. Sections 13, 14 and 15 are devoted to Abelian theorems, initial-value and final-value results. Some applications of the Stieltjes transforms are discussed in section 16. The final section deals with some open questions and unsolved problems. Many important and recent references are listed at the end.

On the ordered partial sums of real random variables

1977 ◽  
Vol 14 (1) ◽  
pp. 75-88 ◽  
Author(s):  
Lajos Takács
Keyword(s):  
Random Variables ◽  
Partial Sums ◽  
Markov Sequence ◽  
Stieltjes Transforms ◽  
Independent And Identically Distributed

In 1952 Pollaczek discovered a remarkable formula for the Laplace-Stieltjes transforms of the distributions of the ordered partial sums for a sequence of independent and identically distributed real random variables. In this paper Pollaczek's result is proved in a simple way and is extended for a semi-Markov sequence of real random variables.

First-order autoregressive gamma sequences and point processes

1980 ◽  
Vol 12 (3) ◽  
pp. 727-745 ◽  
Author(s):  
D. P. Gaver ◽  
P. A. W. Lewis
Keyword(s):  
Parameter Estimation ◽  
Point Processes ◽  
Random Variables ◽  
Innovation Process ◽  
Sums Of Random Variables ◽  
First Order ◽  
Wide Range ◽  
Serial Correlations ◽  
Generating Sequences ◽  
Stieltjes Transforms

It is shown that there is an innovation process {∊n} such that the sequence of random variables {Xn} generated by the linear, additive first-order autoregressive scheme Xn = pXn-1 + ∊n are marginally distributed as gamma (λ, k) variables if 0 ≦p ≦ 1. This first-order autoregressive gamma sequence is useful for modelling a wide range of observed phenomena. Properties of sums of random variables from this process are studied, as well as Laplace-Stieltjes transforms of adjacent variables and joint moments of variables with different separations. The process is not time-reversible and has a zero-defect which makes parameter estimation straightforward. Other positive-valued variables generated by the first-order autoregressive scheme are studied, as well as extensions of the scheme for generating sequences with given marginal distributions and negative serial correlations.

Real inversion theorems for generalized Stieltjes transforms. II

1982 ◽  
Vol 92 (2) ◽  
pp. 275-291 ◽  
Author(s):  
E. R. Love ◽  
Angelina Byrne
Keyword(s):  
Stieltjes Transforms

This paper is a continuation of (3), in which we proved extensions, to generalized Stieltjes transforms, of some inversion theorems of Widder(l) for ordinary Stieltjes transforms. The numbering of formulae, lemmas and theorems continues on consecutively from that in (3). The last formula in (3) is numbered (32), and the last lemma or theorem is Theorem 18.

First-order autoregressive gamma sequences and point processes

1980 ◽  
Vol 12 (03) ◽  
pp. 727-745 ◽  
Author(s):  
D. P. Gaver ◽  
P. A. W. Lewis
Keyword(s):  
Parameter Estimation ◽  
Point Processes ◽  
Random Variables ◽  
Innovation Process ◽  
Sums Of Random Variables ◽  
First Order ◽  
Wide Range ◽  
Serial Correlations ◽  
Generating Sequences ◽  
Stieltjes Transforms

It is shown that there is an innovation process {∊ n } such that the sequence of random variables {X n } generated by the linear, additive first-order autoregressive scheme X n = pXn-1 + ∊ n are marginally distributed as gamma (λ, k) variables if 0 ≦p ≦ 1. This first-order autoregressive gamma sequence is useful for modelling a wide range of observed phenomena. Properties of sums of random variables from this process are studied, as well as Laplace-Stieltjes transforms of adjacent variables and joint moments of variables with different separations. The process is not time-reversible and has a zero-defect which makes parameter estimation straightforward. Other positive-valued variables generated by the first-order autoregressive scheme are studied, as well as extensions of the scheme for generating sequences with given marginal distributions and negative serial correlations.

scholarly journals Second-Order Differential Operators in the Limit Circle Case

2021 ◽  
Author(s):  
Dmitri R. Yafaev ◽  
◽  
◽  
Keyword(s):  
Differential Equation ◽  
Differential Operators ◽  
Second Order ◽  
Spectral Measures ◽  
Limit Circle ◽  
Jacobi Operators ◽  
Limit Circle Case ◽  
Circle Case ◽  
Stieltjes Transforms

We consider symmetric second-order differential operators with real coefficients such that the corresponding differential equation is in the limit circle case at infinity. Our goal is to construct the theory of self-adjoint realizations of such operators by an analogy with the case of Jacobi operators. We introduce a new object, the quasiresolvent of the maximal operator, and use it to obtain a very explicit formula for the resolvents of all self-adjoint realizations. In particular, this yields a simple representation for the Cauchy-Stieltjes transforms of the spectral measures playing the role of the classical Nevanlinna formula in the theory of Jacobi operators.

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