It is clear that

pointwise convergence implies convergence almost surely and it implies convergence in measure.

The

pointwise convergence rate estimate is a continuous functional

The main results of this paper are exact constants of the main terms of asymptotic errors and optimal parameters for improved

pointwise convergence of rational approximations.

In fact, Herrero [6] and Chan [7] showed that chaotic linear operators are dense (with respect to

pointwise convergence) in the set of bounded linear operators.

The

pointwise convergence, denoted by [mathematical expression not reproducible], means that

On [R.sup.F] we use the topology of

pointwise convergence. If all f [member of] F are continuous (which they automatically are if we endow F with the discrete topology) and bounded, then the embedding is continuous and its range is a product of bounded intervals, hence compact by Tychonov's theorem.

Remark 2: The above proof actually shows the following : if X is separable and S is any closed subspace of [X.sup.*], then S is contained in NA(X) if and ONLY IF the unit ball of X is compact for the topology [t.sub.p] (S) of

pointwise convergence on S.

The mathematicians investigate the

pointwise convergence of weighted averages linked to averages along cubes, divergent ergodic averages along the squares, the one-sided ergodic Hilbert transform, deterministic walks in Markov environments with constant rigidity, limit theorems for sequential expanding dynamical systems, and random Fourier-Stieltjes transforms.

We will show similar results for the

pointwise convergence. If [gamma] has compact support and c [member of] [s.sub.p,q] then [[sigma].sup.[theta].sub.K,N]c [right arrow] [R.sub.[gamma]]c a.e., whenever [??] [member of] [[??].sub.r'], ([[??].sup.d]), 1 [less than or equal to] r [less than or equal to] p < [infinity], 1/r + 1/r' = 1.

They have proved under suitable conditions on [mu], [L.sup.p]-norm and

pointwise convergence of this integral.

We show that the convergence of Bayesian estimators comes directly from our first result, the

pointwise convergence of the likelihood.

The function from the infinite dimensional vectors with the topology of

pointwise convergence to the sum of moduli of their components is discontinuous, its type of discontinuity being nearer to the intuitive idea of unboundedness than to that of 'jumps' in the graph.