Chull Park

Analytic Fourier-Feynman transforms and convolution

In this paper we develop an Lp Fourier-Feynman theory for a class of functionals on Wiener space of the form F(x) = f(J0 axdx, ... , /0 a„dx). We then define a convolution product for functionals on Wiener space and show that the Fourier-Feynman transform of the convolution product is a product of Fourier-Feynman transforms.

Generalized transforms and convolutions

In this paper, using the concept of a generalized Feynman integral, we define a generalized Fourier-Feynman transform and a generalized convolution product. Then for two classes of functionals on Wiener space we obtain several results involving and relating these generalized transforms and convolutions. In particular we show that the generalized transform of the convolution product is a product of transforms. In addition we establish a Parsevals identity for functionals in each of these classes.

Relationships among transforms, convolutions, and first variations

In this paper, we establish several interesting relationships involving the Fourier-Feynman transform, the convolution product, and the first variation for functionals F on Wiener space of the form F(x)=f(〈α1,x〉,…,〈αn,x〉),                                                      (*) where 〈αj,x〉 denotes the Paley-Wiener-Zygmund stochastic integral ∫0Tαj(t)dx(t).

Fourier-Feynman transforms and the first variation

In this paper we complete the following four objectives: 1. We obtain an integration by parts formula for analytic Feynman integrals. 2. We obtain an integration by parts formula for Fourier-Feynman transforms. 3. We find the Fourier-Feynman transform of a functionalF from a Banach algebra after it has been multiplied byn linear factors. 4. We evaluate the analytic Feynman integral of functionals like those described in 3 above. A very fundamental result by Cameron and Storvick [5, Theorem 1], in which they express the analytic Feynman integral of the first variation of a functionalF in terms of the analytic Feynman integral ofF multiplied by a linear factor, plays a key role throughout this paper.

Normative, descriptive and formative approaches to describing normal work area

This paper examines normative and formative approaches toward describing normal work area. Squires’ model of normal work area is an example of a normative approach, which prescribes the shape of normal work area. Squires’ model has been taken as the starting point for more recent attempts to delineate reach boundaries in the workplace. However, his original paper does not present the derivation for his model or his claim that the distal boundary of his normal work area can be described by a prolate epicycloid. This paper presents the derivation for Squires’ model and shows that the resulting curve is not a prolate epicycloid, but a related form, a hypotrochoid. It then identifies shortcomings of normative models, like Squires’, notably the viability of the reach assumptions on which such normative models are based and their inability to deal with context-conditioned variability in the workplace. Finally, it examines the application of formative models, which attempt to identify constraints on the types of reach actions that people use. Data on how people actually reach may constitute a more appropriate foundation for constructing work area models needed to develop workplace standards.

Multiple path-valued conditional Yeh-Wiener integrals

In this paper we establish various results involving parallel linevalued conditional Yeh-Wiener integrals of the type E(F(x) lx(sj, ) = m ( ), j = 1,... Irn) where 0 < si < < sn. We then develop a formula for converting these multiple path-valued conditional Yeh-Wiener integrals into ordinary Yeh-Wiener integrals. Next, conditional Yeh-Wiener integrals for functionals F of the form F(x) = exp { j f (s, t, x(s, t)) dt ds } are evaluated by solving an appropriate Wiener integral equation. Finally, a Cameron-Martin translation theorem is obtained for these multiple pathvalued conditional Yeh-Wiener integrals.

Sample path-valued conditional Yeh-Wiener integrals and a Wiener integral equation

In this paper we evaluate the conditional Yeh-Wiener integral E(F(x)|x(s,t)=ζ) for functions F of the form F(x)=exp{∫ 0 t ∫ 0 s o(σ,τ,x(σ,τ))dσdτ}. The method we use to evaluate this conditional integral is to first define a sample path-valued conditional Yeh-Wiener integral of the type E(F(x)|x(s,.)=ψ(.)) and show that it satisfies a Wiener integral equation. We next obtain a series solution for E(F(x)|x(s,.)=ψ(.)) by solving this Wiener integral equation. Finally, we integrate this series solution appropriately in order to evaluate E(F(x)|x(s,t)=ζ)

Fundamental theorem of Wiener calculus

In this paper we define and develop a theory of differentiation in Wiener space C[0,T]. We then proceed to establish a fundamental theorem of the integral calculus for C[0,T]. First of all, we show that the derivative of the indefinite Wiener integral exists and equals the integrand functional. Secondly, we show that certain functionals defined on C[0,T] are equal to the indefinite integral of their Wiener derivative.

Linear transformations of Wiener integrals

In this paper we obtain a linear transformation theorem in which the Radon-Nikodym derivative is very closely related to the transformation. We also obtain a vector-valued conditional version of this linear transformation theorem

Fundamental theorem of Yeh – Wiener calculus

In this paper we introduce a theory of differentiation in Yeh–Wiener space We then establish the fundamental theorem of Integral calculus on C(Q). The main tools we employ here are conditional Yeh–Wiener integrals and the martingale convergence theorem