An Extension of Kakutani's Theorem on Infinite Product Measures to the Tensor Product of Semifinite w ∗ -Algebras

1969 ◽  
Vol 135 ◽  
pp. 199 ◽  
Author(s):  
Donald Bures
Keyword(s):  
Tensor Product ◽  
Infinite Product ◽  
Product Measures

On Equivalence of Infinite Product Measures

1948 ◽  
Vol 49 (1) ◽  
pp. 214 ◽  
Author(s):  
Shizuo Kakutani
Keyword(s):  
Infinite Product ◽  
Product Measures

scholarly journals Γ-convergence of Onsager–Machlup functionals: II. Infinite product measures on Banach spaces

2021 ◽  
Vol 38 (2) ◽  
pp. 025006 ◽  
Author(s):  
Birzhan Ayanbayev ◽  
Ilja Klebanov ◽  
Han Cheng Lie ◽  
T J Sullivan
Keyword(s):  
Hilbert Spaces ◽  
Infinite Product ◽  
A Posteriori ◽  
Transition Path Theory ◽  
Bayesian Inverse Problems ◽  
Countable Product ◽  
Convergence Of Sequences ◽  
Product Measures ◽  
Γ Convergence ◽  
Convergent Sequences

Abstract We derive Onsager–Machlup functionals for countable product measures on weighted ℓ p subspaces of the sequence space R N . Each measure in the product is a shifted and scaled copy of a reference probability measure on R that admits a sufficiently regular Lebesgue density. We study the equicoercivity and Γ-convergence of sequences of Onsager–Machlup functionals associated to convergent sequences of measures within this class. We use these results to establish analogous results for probability measures on separable Banach or Hilbert spaces, including Gaussian, Cauchy, and Besov measures with summability parameter 1 ⩽ p ⩽ 2. Together with part I of this paper, this provides a basis for analysis of the convergence of maximum a posteriori estimators in Bayesian inverse problems and most likely paths in transition path theory.

A Class of Singular Functions

1968 ◽  
Vol 20 ◽  
pp. 1425-1431 ◽  
Author(s):  
Harold S. Shapiro
Keyword(s):  
Infinite Product ◽  
General Theorem ◽  
Sufficient Conditions ◽  
Measurable Space ◽  
Absolutely Continuous ◽  
Borel Field ◽  
Singular Functions ◽  
Product Measures ◽  
Necessary And Sufficient ◽  
Set Function

Kakutani (2) has proved a very general theorem, giving necessary and sufficient conditions for two infinite product measures to be mutually absolutely continuous. To formulate Kakutani's result, let us first recall that a measurable space is a pair (E, B), where B denotes a Borel field (also called σ-ring) of subsets of E, and a measure m on this space is a countably additive set function on B (see Halmos (1)).

Reference Signal Based Tensor Product Expansion for EOG-Related Artifact Separation in EEG

2017 ◽  
Vol E100.A (11) ◽  
pp. 2230-2237
Author(s):  
Akitoshi ITAI ◽  
Arao FUNASE ◽  
Andrzej CICHOCKI ◽  
Hiroshi YASUKAWA
Keyword(s):  
Tensor Product ◽  
Reference Signal

On degeneracy problem of NFSRs via semi-tensor product

2020 ◽  
Author(s):  
Xinyu Zhao ◽  
Biao Wang ◽  
Shuqian Zhu ◽  
Jun-e Feng
Keyword(s):  
Tensor Product ◽  
Degeneracy Problem

On the Buchstaber Subring in MSp∗

1998 ◽  
Vol 5 (5) ◽  
pp. 401-414
Author(s):  
M. Bakuradze
Keyword(s):  
Tensor Product ◽  
Characteristic Classes ◽  
Generalized Quaternion ◽  
Symplectic Cobordisms

Abstract A formula is given to calculate the last n number of symplectic characteristic classes of the tensor product of the vector Spin(3)- and Sp(n)-bundles through its first 2n number of characteristic classes and through characteristic classes of Sp(n)-bundle. An application of this formula is given in symplectic cobordisms and in rings of symplectic cobordisms of generalized quaternion groups.

scholarly journals Fractal Frames of Functions on the Rectangle

2021 ◽  
Vol 5 (2) ◽  
pp. 42
Author(s):  
María A. Navascués ◽  
Ram Mohapatra ◽  
Md. Nasim Akhtar
Keyword(s):  
Tensor Product ◽  
Product Space ◽  
Integrable Function ◽  
Two Dimensional ◽  
Tensor Product Space ◽  
Fractal Functions

In this paper, we define fractal bases and fractal frames of L2(I×J), where I and J are real compact intervals, in order to approximate two-dimensional square-integrable maps whose domain is a rectangle, using the identification of L2(I×J) with the tensor product space L2(I)⨂L2(J). First, we recall the procedure of constructing a fractal perturbation of a continuous or integrable function. Then, we define fractal frames and bases of L2(I×J) composed of product of such fractal functions. We also obtain weaker families as Bessel, Riesz and Schauder sequences for the same space. Additionally, we study some properties of the tensor product of the fractal operators associated with the maps corresponding to each variable.

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