Let C[0, t] denote the function space of all real-valued continuous paths on [0, t]. Define X n+1 : C[0, t] → R n+2 by X n+1 (x) = (x(t 0), x(t 1), • • • , x(t n), x(t n+1)), where 0 = t 0 < t 1 < • • • < t n < t n+1 = t is a partition of [0, t]. In the present paper, using a simple formula for the conditional expectation with the conditioning function X n+1 , we evaluate the L p (1 ≤ p ≤ ∞)-analytic conditional Fourier-Feynman transforms and the conditional convolution products of the cylinder functions which have the form f ((v 1 , x), • • • , (v r , x)) for x ∈ C[0, t], where {v 1 , • • • , v r } is an orthonormal subset of L 2 [0, t] and f ∈ L p (R r). We then investigate several relationships between the conditional Fourier-Feynman transforms and the conditional convolution products of the cylinder functions.
In the present paper, we evaluate the analytic conditional FourierFeynman transforms and convolution products of bounded functions which are important in Feynman integration theories and quantum mechanics. We then investigate the inverse transforms of the functions with their relationships and finally that the conditional analytic FourierFeynman transforms of the conditional convolution products for the functions, can be expressed in terms of the products of the conditional Fourier-Feynman transform of each function.
Mathematics Subject Classification: 28C20
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