Integrability on the Abstract Wiener Space of Double Sequences and Fernique Theorem

The integrability of a function defined on the abstract Wiener space of double Fourier coefficients is explored. The abstract Wiener space is also a Hilbert space. We define an orthonormal system of the Hilbert space to establish a measure and integration on the abstract Wiener space. We examine the integrability of a function e α · 2 defined on the abstract Wiener space for Fernique theorem. With respect to the abstract Wiener measure, the integral of the function turns out to be convergent for α < 1 / 2 . The result provides a wider choice of the constant α than that of Fernique.

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

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.

A Change of Scale Formula for Wiener Integrals about the First Variation on the Product Abstract Wiener Space

We shall prove the existence of the Wiener integral and the analytic Wiener and Feynman integral and we obtain those relationships and later, we prove the change of scale formula for the Wiener integral about the first variation of a function defined on the product abstract Wiener space. Later, we obtain those relationships in the Fresnel class as it’s corollaries.

An analytic bilateral Laplace–Feynman transform on Hilbert space

In this paper, we examine the analytic bilateral Laplace–Feynman transform (BLFT) for functions on the Hilbert space H. We then proceed to establish a relationship between the analytic BLFT on H and the analytic Fourier–Feynman transform on the abstract Wiener space B.

Change of Scale Formulas for Wiener Integrals Related to Fourier-Feynman Transform and Convolution

Cameron and Storvick discovered change of scale formulas for Wiener integrals of functionals in Banach algebraSon classical Wiener space. Yoo and Skoug extended these results for functionals in the Fresnel classF(B)and in a generalized Fresnel classFA1,A2on abstract Wiener space. We express Fourier-Feynman transform and convolution product of functionals inSas limits of Wiener integrals. Moreover we obtain change of scale formulas for Wiener integrals related to Fourier-Feynman transform and convolution product of these functionals.