This paper is a continuation of the so far performed studies on the concept of uniform statistical
convergence. We first characterize two inequalities concerning the uniform
statistical limit superior that lead to two core inclusion results for bounded
real sequences. Using the inverse Fourier transformation, we also give a criterion
on uniform statistical convergence, and study some factorization results of
the space of uniformly statistically convergent sequences. These results are
used to give a Korovkin-type approximation theorem.
A = (ank) be a regular summability matrix. In the present paper we deal with
subspaces of the space of A?statistically convergent sequences obtained by
the rate at which the A?statistical limit tends to zero. We prove that a
sequence is the A?strongly convergent if and only if it is the A?statistically
convergent and the A?uniformly integrable with the rate of o (an) where a =
(an) is a positive nonincreasing sequence. We also make a link between the
A?strong convergence and the A?distributional convergence with the rate of o
(an). Finally, as an application we present an approximation theorem of
Korovkin type.
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