arXiv subjects
Publications and source records attributed to Bruce A. Watson.
For a conditional expectation operator $T$ on a Dedekind complete Riesz space, we give representations of the $T$-strong duals of $L^1(T)$ and $L^\infty(T)$. The representation for the $T$-strong dual of $L^1(T)$ follows from the known result for $L^2(T)$. To describe the $T$-strong dual of $L^\infty(T)$, we introduce charges on components of weak order units and develop a corresponding integration theory.
Martingales, Markov processes and Laws of Large Numbers have been well studied in the Riesz space (vector lattice) setting. There has, however, been no attention given in the Riesz space setting to Laws of Small Numbers or to the so called Stein-Chen method. Here we adapt the Stein-Chen method to the Riesz space setting and hence give a conditional Laws of Small Numbers for Bernoulli processes in Riesz spaces. This requires extensive use of functional calculus and the associated f-algebra structure.
We link conditional weak mixing and ergodicity of the tensor product in Riesz spaces. In particular, we characterise conditional weak mixing of a conditional expectation preserving system by the ergodicity of its tensor product with itself or other ergodic systems. In order to achieve this we characterise the components of the weak order units in the tensor product of two Dedekind complete Riesz spaces with weak order units.
Riesz space (non-pointwise) generalizations for iterative processes are given for the concepts of recurrence, first recurrence and conditional ergodicity. Riesz space conditional versions of the Poincaré Recurrence Theorem and the Kac formula are developed. Under mild assumptions, it is shown that every conditional expectation preserving process is conditionally ergodic with respect to the conditional expectation generated by the Cesàro mean associated with the iterates of the process. Applied to processes in $L^1(Ω,{\mathcal A},μ)$, where $μ$ is a probability measure, new conditional versions of the above theorems are obtained.
We extend the Koopman-von Neumann convergence condition on the Cesàro mean to the context of a Dedekind complete Riesz space with weak order unit. As a consequence, a characterisation of conditional weak mixing is given in the Riesz space setting. The results are applied to convergence in $L^1$.
The ergodic theorems of Hopf, Wiener and Birkhoff were extended to the context of Riesz spaces with a weak order unit and conditional expectation operator by Kuo, Labuschagne and Watson in [Ergodic Theory and the Strong Law of Large Numbers on Riesz Spaces. Journal of Mathematical Analysis and Applications, 325,(2007), 422-437.]. However, the precise concept of what constitutes ergodicity in Riesz spaces was not considered. In this short paper we fill in this omission and give some explanations of the choices made. In addition, we consider the interplay between mixing and ergodicity in the Riesz space setting.
In a Dedekind complete Riesz space, $E$, we show that if $(P_n)$ is a sequence of band projections in $E$ then $$\limsup\limits_{n\to \infty} P_n - \liminf\limits_{n\to \infty} P_n = \limsup\limits_{n\to \infty} P_n(I-P_{n+1}).$$ This identity is used to obtain conditional extensions in a Dedekind complete Riesz spaces with weak order unit and conditional expectation operator of the Barndorff-Nielsen and Balakrishnan-Stepanov generalizations of the First Borel-Cantelli Theorem.
In this paper we generalize the Hájek-Rényi-Chow maximal inequality for submartingales to $L^p$ type Riesz spaces with conditional expectation operators. As applications we obtain a submartingale convergence theorem and a strong law of large numbers in Riesz spaces. Along the way we develop a Riesz space variant of the Clarkson's inequality for $1\le p\le 2$.
We consider a Sturm-Liouville equation $\ell y:=-y'' + qy = λy$ on the intervals $(-a,0)$ and $(0,b)$ with $a,b>0$ and $q \in L^2(-a,b)$. We impose boundary conditions $y(-a)\cosα= y'(-a)\sinα$, $y(b)\cosβ= y'(b)\sinβ$, where $α\in [0,π)$ and $β\in (0,π]$, together with transmission conditions rationally-dependent on the eigenparameter via \begin{align*} -y(0^+)\left(λη-ξ-\sum\limits_{i=1}^{N} \frac{b_i^2}{λ-c_i}\right) &= y'(0^+) - y'(0^-),\\ y'(0^-)\left(λκ+ζ-\sum\limits_{j=1}^{M}\frac{a_j^2}{λ-d_j}\right) &= y(0^+) - y(0^-), \end{align*} with $b_i, a_j>0$ for $i=1,\dots,N,$ and $j=1,\dots,M$. Here we take $η, κ\ge 0$ and $N,M\in \N_0$. The geometric multiplicity of the eigenvalues is considered and the cases in which the multiplicity can be $2$ are characterized. An example is given to illustrate the cases. A Hilbert space formulation of the above eigenvalue problem as a self-adjoint operator eigenvalue problem in $L^2(-a,b)\bigoplus \C^{N^*} \bigoplus \C^{M^*}$, for suitable $N^*,M^*$, is given. The Green's function and the resolvent of the related Hilbert space operator are expressed explicitly.
We consider a Sturm-Liouville operator on a finite interval as well as a scattering problem on the real line both with transfer conditions at the origin. On a finite interval we show that the the Titchmarsh-Weyl $m$-function can be uniquely determined from two spectra for the same equation but with varied boundary conditions at one end of the interval. In addition, we prove that the $m$-function can also be uniquely reconstructed from one spectrum and the corresponding norming constants. For the scattering problem on the real line we assume that the potential has compact essential support. For a given symmetric finite intervals containing the essential-support of the potential and a pair of separated boundary conditions imposed at the ends of the interval, the spectrum and corresponding norming constants can be uniquely recoverable from the scattering data on $\R$. Consequently the potential and transfer matrix can be determined.
Strong convergence and convergence in probability were generalized to the setting of a Riesz space with conditional expectation operator, $T$, in [{{\sc Y. Azouzi, W.-C. Kuo, K. Ramdane, B. A. Watson}, {Convergence in Riesz spaces with conditional expectation operators}, {\em Positivity}, {\bf 19} {(2015), 647-657}}] as $T$-strong convergence and convergence in $T$-conditional probability, respectively. Generalized $L^{p}$ spaces for the cases of $p=1,2,\infty$, were discussed in the setting of Riesz spaces as $\mathcal{L}^{p}(T)$ spaces in [{{\sc C. C. A. Labuschagne, B. A. Watson}, {Discrete stochastic integration in Riesz spaces}, {\em Positivity}, {\bf 14} {(2010), 859-875}}]. An $R(T)$ valued norm, for the cases of $p=1,\infty,$ was introduced on these spaces in [{{\sc W. Kuo, M. Rogans, B.A. Watson}, {Mixing processes in Riesz spaces}, {\em Journal of Mathematical Analysis and Application}, {\bf 456} {(2017), 992-1004}}] where it was also shown that $R(T)$ is a universally complete $f$-algebra and that these spaces are $R(T)$-modules. In [{{\sc Y. Azouzi, M. Trabelsi}, {$L^p$-spaces with respect to conditional expectation on Riesz spaces}, {\em Journal of Mathematical Analysis and Application}, {\bf 447} {(2017), 798-816}}] functional calculus was used to consider $\mathcal{L}^{p}(T)$ for $p\in (1,\infty)$. In this paper we prove the strong sequential completeness of the space $\mathcal{L}^{1}(T)$, the natural domain of the conditional expectation operator $T$, and the strong completeness of $\mathcal{L}^{\infty}(T)$.
We consider the model equations for the Timoshenko beam as a first order system in the framework of evolutionary equations. The focus is on boundary damping, which is implemented as a dynamic boundary condition. A change of material laws allows to include a large class of cases of boundary damping. By choosing a particular material law, it is shown that the first order approach to Sturm-Liouville problems with boundary damping is also covered.