Abstract
From the change of variable formula on the Wiener space, we calculate various integral transforms for functionals on the Wiener space. However, not all functionals can be obtained by using this formula. In the process of calculating the integral transform introduced by Lee, this formula is also used, but it is also not possible to calculate for all the functionals. In this paper, we define a generalized integral transform. We then introduce a new method to evaluate the generalized integral transform for functionals using series expressions. Our method can be used to evaluate various functionals that cannot be calculated by conventional methods.
Highlights
For a positive real number T, let C0 [0, T ] be the space of all real-valued continuous functions x on [0, T ] with x (0) = 0
In a unifying paper [1], Lee introduced an integral transform of analytic functionals
For certain complex numbers γ and β and for certain classes of functionals, the Fourier-Wiener transform, the Fourier-Feynman transform and the Gauss transform are special cases of Lee’s integral transform Fγ,β defined by the formula
Summary
For certain complex numbers γ and β and for certain classes of functionals, the (modified) Fourier-Wiener transform, the Fourier-Feynman transform and the Gauss transform are special cases of Lee’s integral transform Fγ,β defined by the formula. Mathematics 2020, 8, 539 where hw, x i is the Paley-Wiener-Zygmund (PWZ) stochastic integral and f is a complex measure on the Borel σ-algebra B( L2 [0, T ]), a class E0 of functionals [5] of the form. For many functionals it is difficult or impossible to calculate the generalized integral transform via the change of variable formula on the Wiener space. The following Wiener integral, which appears in calculation of generalized integral transform, sin( x ( T ))dm( x ). By providing a calculation method, more functionals can be calculated
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