Class Of Integrable Functions Research Articles

On an approach to the definition of a stochastic integral with respect to a Gaussian measure

We define a stochastic Riemann integral with respect to a Gaussian measure. The class of integrable functions is introduced in which there exists a solution of a stochastic Fredholm integral equation. It is shown by examples how to pass from the integral defined here to the Ito and Stratonovich integrals.

Some remarks on generalized Riemann integral

Because of the desire to calculate the areas of elementary figures, a variety of integrals has been established, popular among which are the Lebesgue integral and the Riemann integral. It is clear that Riemann integral is fundamental in elementary calculus and it can be used to define and calculate many geometric and physical quantities, such as area, volume, and work. However, the Riemann integral has its limitations. The theory of the Lebesgue integral reveals that the Riemann integral is basically used for continuous functions. In fact, f: [a, b] -+ R is Riemann integrable iff S is continuous a.e. on [a, b]. Also, as is known, the convergence theorems for this integral are severely restricted. With the motivation of generalizing the Riemann integral so as to enlarge the class of integrable functions for which the convergence theorems hold, the Lebesgue integral was successfully established. Generalizations of the Lebesgue integral, such as the Perron integral and special Denjoy integral, appeared later. The most interesting generalization is the generalized Riemann integral (GR integral for short), discovered by Kurzweil and Henstock independently, although it is equivalent to Perron and special Denjoy integrals. Contrary to the classical exposition of the Lebesgue integral which needs the concepts of measurable sets and measurable functions, before defining the integral, one can define the GR integral directly based on Riemann sums; therefore, the definition is constructive. Then, if one wishes, one can find all the Lebesgue measurable sets and measurable functions via the definition of GR integral. Furthermore, all the convergence theorems can be proved using the definition of the integral [l-3]. It is known that f is Lebesgue integrable iff both f and If] are GR integrable. Hence, the Lebesgue integral can be introduced through the GR integral, avoiding measure theory. Since the definition of the GR integral is very similar to that of the Riemann integral, one can easily grasp the 515 0022-247X/89 $3.00

Arbitrarily slow decay of correlations in quasiperiodic systems

For diffusive motion in random media it is widely believed that the velocity autocorrelation function c(t) exhibits power law decay as time t ..-->.. infinity. We demonstrate that the decay of c(t) in quasiperiodic media can be arbitrarily slow within the class of integrable functions. For example, in d = 1 with a potential V(x) = cos x + cos kx, there is a dense set of irrational k's such that the decay of c(k, t) is slower than 1/t/sup (1+epsilon/)/ for any epsilon > 0. The irrationals producing such a slow decay of c(k, t) are very well approximated by rationals.

An incompressibility condition for a certain class of integral functionals. II

On the basis of the description of the set of all solutions of the equation del y=1, wherey is a mapping of Ω⊂Rm into Rm. whiley is its Jacobi matrix, in the neighborhood of the given solution one computes the first and the second variation of an integral functional under the conditions of “incompressibility” in the neighborhood of ∂Ω.

An incompressibility condition for a certain class of integral functionals. I

An incompressibility condition for a certain class of integral functionals. I

Lower Semicontinuity, Optimization and Regularizing Extensions of Integral Functionals

We prove the lower semicontinuity, relative to the local convergence topology induced in $W^{1,1} (\Omega )$ by $L_{{\text{loc}}}^1 (\Omega )$, of a large class of integral functionals, under two different sets of hypotheses. We obtain also an optimization theorem and apply these results to the study of some extensions of integral functionals.

Almost everywhere convergence of a class of integrable functions

Almost everywhere convergence of a class of integrable functions

THE ITERATIVE SOLUTION OF TWO-POINT LINEAR DIFFERENTIAL EIGENVALUE PROBLEMS-II

An automatic root-finding algorithm that will systematically yield all the real zeros of a certain class of integral functions in any given interval is described, with particular reference to the problem of finding the eigenvalues of a liner differential operator under two-point boundary conditions by locating the zeros of the characteristic determinant. The method is illustrated by considering the vibrating uniform cantilever beam problem.

Some extremal problems for linear operators in a class of integral functions of finite degree

Some extremal problems for linear operators in a class of integral functions of finite degree

On Fourier Transforms of the Class Wσ

This note is concerned with a theorem of Plancherel for Fourier transforms which corresponds to the Parseval theorem for Fourier series. By Wσ we denote the class of integral functions {f(z)} which are of exponential type σ and such that each of the functions {f(x)} belongs to L 2 (-∞, ∞) for values of z = x on the real axis.

Fourier transforms of certain classes of integrable functions

This theorem characterizes those functions 4 E L. (oo, so) of the form +(x) = j (x) a.e. in terms of a continuity condition on the linear functional defined by F(f)= f f(x) b(x)dx for feL1(o,oo). Given a subset N c V(oo, cc) it is possible to ask whether additional continuity conditions on F can be found which, combined with the above condition, are necessary and sufficient in order that +(x) (x) a.e. for some tie N. The following theorem by A. C. Berry [1] illustrates such a condition when N is the class absolutely continuous functions of bounded variation. Here we define

On a Class of Integral Functions of Zero Order

On a Class of Integral Functions of Zero Order

A Class of Integral Functions Bounded on Certain Curves

A Class of Integral Functions Bounded on Certain Curves

A note on the representation of functions by absolutely convergent fourier integrals

1. We write L for the class of integrable functions in (− ∞, ∞), V for the class of functions of bounded variation, and define A, A to be the classes of functions F(x) which may be expressed in the formsrespectively.

On a Class of Integral Functions

On a Class of Integral Functions