Prediction theory in Banach spaces

1975 ◽  
pp. 207-228 ◽  
Author(s):  
A. Weron
Keyword(s):  
Banach Spaces ◽  
Prediction Theory

Operator Geometry in Statistics

2018 ◽  
Author(s):  
Karl Gustafson
Keyword(s):  
Banach Spaces ◽  
Functional Data ◽  
Matrix Inequalities ◽  
Statistical Efficiency ◽  
Prediction Theory ◽  
Geometrical Meaning ◽  
Association Measures ◽  
Canonical Correlations ◽  
Infinite Dimensional ◽  
Two Component

This article discusses the essentials of operator trigonometry developed by the author as it applies to statistics, with emphasis on key elements such as operator antieigenvalues, operator antieigenvectors, and operator turning angles. Operator trigonometry started out infinite dimensional, and remains infinite dimensional, even for Banach spaces. Thus, it is in principle applicable not only to infinite-dimensional statistics but also to cases involving functional data. The article first considers how operator trigonometry gives new geometrical meaning to statistical efficiency before formalizing it in a more deductive manner. It then explains the essentials of operator trigonometry and summarizes the ensuing developments. It also describes two lemmas that are implicit and essential to operator trigonometry, Antieigenvector Reconstruction Lemma and General Two-Component Lemma, and how operator trigonometry provides new geometry to statistics matrix inequalities and canonical correlations. Finally, it presents new results applying operator trigonometry to prediction theory and to association measures.

Controllability for some nonlinear impulsive partial functional integrodifferential systems with infinite delay in Banach spaces

2020 ◽  
Vol 4 (2) ◽  
pp. 104-115
Author(s):  
Khalil Ezzinbi ◽  
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Banach Spaces ◽  
Integrodifferential Equation ◽  
Resolvent Operator ◽  
Measure Of Noncompactness ◽  
Infinite Delay ◽  
Sufficient Conditions ◽  
Mönch Fixed Point Theorem ◽  
The Resolvent Operator

This work concerns the study of the controllability for some impulsive partial functional integrodifferential equation with infinite delay in Banach spaces. We give sufficient conditions that ensure the controllability of the system by supposing that its undelayed part admits a resolvent operator in the sense of Grimmer, and by making use of the measure of noncompactness and the Mönch fixed-point Theorem. As a result, we obtain a generalization of the work of K. Balachandran and R. Sakthivel (Journal of Mathematical Analysis and Applications, 255, 447-457, (2001)) and a host of important results in the literature, without assuming the compactness of the resolvent operator. An example is given for illustration.

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