Suppose that there exists a nonnegative rd-continuous

bounded function p(t) such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], for all t [member of] T and some positive number [[lambda].sub.0] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for all t [member of] T and some positive number [epsilon].

Finally, the model used in the simulation corresponds to the choice of the identity function for [PSI] in (1), where ([Y.sub.t]) is an unbounded process and r(*) is not a

bounded function. However, r(*) is bounded on Im(r) and so (H4) is fulfilled.

Also, replacing above [phi](t) = 1/2 x [sinc.sup.2](t/2) (the Fejer-type kernel), in [11], Theorem 2.4, we get the same Jackson-type order [[omega].sub.1][(f; 1/W).sub.R] in the approximation of a continuous, positive and

bounded function f on R.

G [2] evaluated multiple integrals of a

bounded function of n-real variables where the rule has been determined for n = 2,3 numerically and an asymptotic error for n = 2 is verified.

If [x.sub.i] (t) is the

bounded function at the interval [J.sub.0] that is called [x.sub.i]--from above ([x.sub.i]--from underarm) non-bounded, i = 1,2 is the certain point, if [x.sub.i] (t) is the from above (from underarm) non-bounded function at the interval [J.sub.0].

Note that p (v, [gamma]) and p ([w.sub.2]; [gamma]) are

bounded functions. Assumption 1 implies that |dy (v, [w.sub.2]; [gamma])| is also a

bounded function.

The function [lambda.sub.v](X,.) is uniquely determined by the above property up to the addition of a

bounded function and is called the local height function associated to X.

As part of their work in [7,11], Coifman and Leeb introduce a family of multiscale diffusion distances and establish quantitative results about the equivalence of a

bounded function f being Lipschitz, and the rate of convergence of [T.sub.t]f to f, as t [right arrow] [0.sup.+] (we are discussing some of their results using a continuous time t for convenience; most of Coifman's and Leeb's derivations are done for dyadically discretized times.

In [41], by using the Aubry-Mather theorem generalized by Pei [37], the present author [41] studied the existence of Aubry-Mather sets and quasiperiodic solutions of (3), under the condition that w [member of] [R.sup.+] in (4) and [psi](t,x) [member of] [C.sup.0,1](SP x R) can be allowed to be either a

bounded function or an unbounded function, which differs from above existing results.

We say that a

bounded function [alpha] [member of] [BB.sup.+] (I, R) is a generalized lower function and write [alpha] [member of] AG(I, R) if, for any interval [c, d] [subset] I on which [alpha] satisfies the Lipschitz condition, the inequality

Combining (29) and (30) yields that M is a

bounded function in [[??].sup.N].

for every essentially

bounded function f and real c.

The classical result says that the problem (4) and (2) is solvable if f is a

bounded function. Otherwise various cases are possible.

which implies that the right side of (4) is also bounded, and y'(t) is a

bounded function on (-[[tau].sup.+], + [infinity]).

Lemma 2.1 Let f is of

bounded function on [a, b] and let T: [[a, b].sup.2] [right arrow] R be given by