Vol. 40, No. 1, 1972

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On Fredholm transformations in Yeh-Wiener space

Chull Park

Vol. 40 (1972), No. 1, 173–195
Abstract

Let CY denote the Yeh-Wiener space, i.e., the space of all real-valued continuous functions f(x,y) on I2 [0,1] × [0,1] such that f(0,y) = f(x,0) 0. Yeh has defined a Gaussian probability measure on CY such that the mean of the process

         ∫
m (x,y) ≡   f(x,y)d f = 0
cY       Y

and the convariance

           ∫
R(s,t,x,y) ≡   f (s,t)f (x,y)dY f = (1∕2)min(s,x)min(t,y).
cY

Consider now a linear transformation of CY onto C1 of the form

                           ∫
T : f(x,y) → g(x,y) = f (x,y)+ 2K (x,y,s,t)f (s,t)dsdt,
I
(1.1)

which is often called a Fredholm transformation. The main purpose of this paper is to find the corresponding RadonNikodym derivative thus showing how the Yeh-Wiener integrals transform under the transformation.

Mathematical Subject Classification
Primary: 28A40
Milestones
Received: 23 October 1970
Published: 1 January 1972
Authors
Chull Park