Local Invertibility of Adapted Shifts on Wiener Space and Related Topics

Malliavin Calculus and Stochastic Analysis - Springer Proceedings in Mathematics & Statistics ◽

10.1007/978-1-4614-5906-4_3 ◽

2013 ◽

pp. 25-76 ◽

Cited By ~ 1

Author(s):

Rémi Lassalle ◽

A. S. Üstünel

Keyword(s):

Wiener Space ◽

Local Invertibility

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Local invertibility of adapted shifts on Wiener space, under finite energy condition

Stochastics ◽

10.1080/17442508.2012.720257 ◽

2012 ◽

Vol 85 (6) ◽

pp. 987-996 ◽

Cited By ~ 1

Author(s):

Rémi Lassalle

Keyword(s):

Energy Condition ◽

Finite Energy ◽

Wiener Space ◽

Local Invertibility

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The Hilbert Space of Double Fourier Coefficients for an Abstract Wiener Space

Mathematics ◽

10.3390/math9040389 ◽

2021 ◽

Vol 9 (4) ◽

pp. 389

Author(s):

Jeong-Gyoo Kim

Keyword(s):

Hilbert Space ◽

Fourier Series ◽

Fourier Coefficients ◽

Double Sequence ◽

Intermediate Stage ◽

Wiener Space ◽

Abstract Wiener Space ◽

Abstract Space ◽

Double Sequences ◽

Two Parameter

Fourier series is a well-established subject and widely applied in various fields. However, there is much less work on double Fourier coefficients in relation to spaces of general double sequences. We understand the space of double Fourier coefficients as an abstract space of sequences and examine relationships to spaces of general double sequences: p-power summable sequences for p = 1, 2, and the Hilbert space of double sequences. Using uniform convergence in the sense of a Cesàro mean, we verify the inclusion relationships between the four spaces of double sequences; they are nested as proper subsets. The completions of two spaces of them are found to be identical and equal to the largest one. We prove that the two-parameter Wiener space is isomorphic to the space of Cesàro means associated with double Fourier coefficients. Furthermore, we establish that the Hilbert space of double sequence is an abstract Wiener space. We think that the relationships of sequence spaces verified at an intermediate stage in this paper will provide a basis for the structures of those spaces and expect to be developed further as in the spaces of single-indexed sequences.

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