Fourier-Feynman transform, convolution and first variation

We examine several interesting relationships and expressions involving Fourier-Feynman transform, convolution product and first variation for functionals in the Fresnel class F(B) of an abstract Wiener space B. We also prove a translation theorem and Parseval's identity for the analytic Feynman integral. This revised version was published online in June 2006 with corrections to the Cover Date.     

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.     

Analytic Feynman Integrals of Functionals in a Banach Algebra Involving the First Variation

This paper deals with the analytic Feynman integral of functionals on a Wiener space. First the authors establish the existence of the analytic Feynman integrals of functionals in a Banach algebra S_α. The authors then obtain a formula for the first variation of integrals. Finally, various analytic Feynman integration formulas involving the first variation are established.     

On the Feynman integral in the space of continuous functions on a compact set. I

The Feynman integral in the space of continuous functions defined on a compact set is defined as the analytic continuation of the generalized Wiener integral in the sense of J. Kuelbs. We prove a theorem on the computation of Feynman integrals of functions that depent on linear functionals. Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, No. 37, 1994, pp. 32–36.     

A New Concept of the Analytic Operator-Valued Feynman Integral on Wiener Space

In this article we introduce a new concept of an analytic operator-valued Feynman integral for functionals on Wiener space, which we then use to explain various physical phenomena. We then establish the existence of some analytic operator-valued Feynman integrals that prove useful in establishing various applications in quantum mechanics.     

On the feynman integral in the space of continuous functions on a compact set. II

For the Feynman integral, defined as the analytic continuation of the generalized Wiener integral in the sense of J. Kuelbs in the space of continuous functions on a compact set, we give formulas for exact and approximate computations of the integral of functionals that are functions of quadratic functionals. Translated fromMatematichni Metodi ta Fiziko-Mekhanichni Polya, Vol. 40, No. 2, 1997, pp. 25–30.     

Some relationships for the double modified generalized analytic function space Fourier–Feynman transform and its applications

In this paper we first introduce the concept of a double modified analytic function space Fourier–Feynman transform using the double modified analytic function space integral. We then proceed to establish the existence of the modified analytic function space Fourier–Feynman transform for all functionals in the Banach algebra. Finally we use this double modified analytic function space transform to explain various physical phenomenon.     

Operator-valued Feynman integral via conditional Feynman integrals on a function space

Let C ₀ ^r [0; t] denote the analogue of the r-dimensional Wiener space, define X _t : C ^r [0; t] → ?^2r by X _t (x) = (x(0); x(t)). In this paper, we introduce a simple formula for the conditional expectations with the conditioning function X _t . Using this formula, we evaluate the conditional analytic Feynman integral for the functional $$ \Gamma _t \left( x \right) = exp \left\{ {\int_0^t {\theta \left( {s,x\left( s \right)} \right)d\eta \left( s \right)} } \right\}\varphi \left( {x\left( t \right)} \right) x \in C^r \left[ {0,t} \right] $$ , where η is a complex Borel measure on [0, t], and θ(s, ·) and φ are the Fourier-Stieltjes transforms of the complex Borel measures on ? ^r . We then introduce an integral transform as an analytic operator-valued Feynman integral over C ^r [0, t], and evaluate the integral transform for the function Γ _t via the conditional analytic Feynman integral as a kernel.     

A Modified Analytic Function Space Feynman Integral of Functionals on Function Space

In this paper, the authors introduce a class of functionals. This class forms a Banach algebra for the special cases. The main purpose of this paper is to investigate some properties of the modified analytic function space Feynman integral of functionals in the class. Those properties contain various results and formulas which were not obtained in previous papers. Also, the authors establish some relationships involving the first variation via the translation theorem on function space. In partic...     

A Banach algebra of Feynman integrable functionals with application to an integral equation formally equivalent to Schroedinger's equation

A Banach algebra A of functionals on C[a, b] is introduced and it is proved that the operator-valued Feynman integral recently defined by Cameron and Storvick exists for functionals in A. Two existence theorems of Cameron and Storvick are seen to be special cases of this result; in fact, even in these cases, the present theorem gives improved results.Cameron and Storvick have used their function space integral to give a solution to an integral equation formally equivalent to Schroedinger's equation; using our existence theorem, we give a relatively brief and transparent proof of this result.     

Evaluation Formulas for a Conditional Feynman Integral Over Wiener Paths in Abstract Wiener Space

In this paper, we introduce a simple formula for conditional Wiener integrals over , the space of abstract Wiener space valued continuous functions. Using this formula, we establish various formulas for a conditional Wiener integral and a conditional Feynman integral of functionals on in certain classes which correspond to the classes of functionals on the classical Wiener space introduced by Cameron and Storvick. We also evaluate the conditional Wiener integral and conditional Feynman integral for functionals of the form which are of interest in Feynman integration theories and quantum mechanics.     

Computation of Feynman integrals of functionals that are functions of quadratic functionals

If the arguments of a function G: ^{d 1} are taken as quadratic functional defined on a space C of continuous functions, we obtain a functional G: C ¹.We give a formula for computing analytic Feynman integrals of such functionals. We also propose a method of approximate computation of sequential Feynman integrals based on replacing the kernel of an integral operator by a degenerate kernel.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 32, 1990, pp. 58–61.     

Generalized Brownian functionals and the Feynman integral

To obtain a sufficiently rich class of nonlinear functionals of white noise, resp. the Wiener process, we study riggings of the L² space with the white noise measure. Particular examples are local functionals such as e.g. the ‘square of white noise’ and its exponential with applications in the theory of Feynman Integral.     

Effect of drift of the generalized Brownian motion process: an example for the analytic Feynman integral

In the theory of the analytic Feynman integral, the integrand is a functional of the standard Brownian motion process. In this note, we present an example of a bounded functional which is not Feynman integrable. The bounded functionals discussed in this note are defined in sample paths of the generalized Brownian motion process.     

Quadratic functionals of Brownian motion

Functionals of Brownian motion can be dealt with by realizing them as functionals of white noise. Specifically, for quadratic functionals of Brownian motion, such a realization is a powerful tool to investigate them. There is a one-to-one correspondence between a quadratic functional of white noise and a symmetric L²(R²)-function which is considered as an integral kernel. By using well-known results on the integral operator we can study probabilistic properties of quadratic or certain exponential functionals of white noise. Two examples will illustrate their significance.     

Smooth functional derivatives in Feynman path integrals by time slicing approximation

We give a fairly general class of functionals on a path space so that Feynman path integral has a mathematically rigorous meaning. More precisely, for any functional belonging to our class, the time slicing approximation of Feynman path integral converges uniformly on compact subsets of the configuration space. Our class of functionals is closed under addition, multiplication, functional differentiation, translation and real linear transformation. The integration by parts and Taylor's expansion formula with respect to functional differentiation holds in Feynman path integral. Feynman path integral is invariant under translation and orthogonal transformation. The interchange of the order with Riemann-Stieltjes integrals, the interchange of the order with a limit, the semiclassical approximation and the fundamental theorem of calculus in Feynman path integral stay valid as well as N. Kumano-go [Bull. Sci. Math. 128 (3) (2004) 197-251].     

On the Feynman integral in the space of continuous functions on a compact set. III

Some results of Cameron-Storvick on the structure of the Banach algebra S of functionals on the space C^p[a, b] of p-dimensional vector-valued functions x(t), t [a, b], x(a)=0, are generalized to a Banach algebra containing functionals defined on the set C^p(Y), where Y is an infinite-dimensional compact set. In particular it is shown that analytic Feynman integrals of such functionals over the space C^p(Y) exist.Translated fromMatematichni Metodi ta Fiziko-Mekhanichni Polya, Vol. 40, No. 3, 1997, pp. 45–49.     

Path integrals as analysis on path space by time slicing approximation

This is a survey of our papers [3, 4]. We give a fairly general class of functionals on a path space so that Feynman path integral has a mathematically rigorous meaning. More precisely, for any functional belonging to our class, the time slicing approximation of Feynman path integral converges uniformly on compact subsets of the configuration space. Our class of functionals is closed under addition, multiplication, translation, real linear transformation and functional differentiation. The invariance under translation and orthogonal transformation, the interchange of the order with Riemann-Stieltjes integrals and some limits, the semiclassical approximation, the integration by parts and the Taylor expansion formula with respect to functional differentiation, and the fundamental theorem of calculus hold in Feynman path integral. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Regularity results for minimizers of (2, q)-growth functionals in the Heisenberg Group

We consider integral functionals in the Heisenberg group, whose convex C ²-integrand has quadratic growth from below, and growth of order q > 2 from above. We prove Hölder regularity for the full gradient of minimizers under the condition that q is less than an explicitly calculated dimension-dependent bound.     

Numerical integration based on trivariate C 2 quartic spline quasi-interpolants

In this paper we consider the space generated by the scaled translates of the trivariate C ² quartic box spline B defined by a set X of seven directions, that forms a regular partition of the space into tetrahedra. Then, we construct new cubature rules for 3D integrals, based on spline quasi-interpolants expressed as linear combinations of scaled translates of B and local linear functionals. We give weights and nodes of the above rules and we analyse their properties. Finally, some numerical tests and comparisons with other known integration formulas are presented.