It has been proved recently that the two-direction refinement equation of the form $f\left(x\right)={\sum }_{n\in }{c}_{n,1}f(kx-n)+{\sum }_{n\in \mathbb{Z}}{c}_{n,-1}f(-kx-n)$ can be used in wavelet theory for constructing two-direction wavelets, biorthogonal wavelets, wavelet packages, wavelet frames and others. The two-direction refinement equation generalizes the classical refinement equation $f\left(x\right)={\sum }_{n\in \mathbb{Z}}cₙf(kx-n)$, which has been used in many areas of mathematics with important applications. The following continuous extension of the classical refinement equation $f\left(x\right)={\int }_{ℝ}c\left(y\right)f(kx-y)dy$ has also various interesting applications....

The concepts of statistical convergence of single and double sequences of complex numbers were introduced in [1] and [7], respectively. In this paper, we introduce the concept indicated in the title. A double sequence $x_{jk}:(j,k)\in \mathbb{N}²$ is said to be regularly statistically convergent if (i) the double sequence $x_{jk}$ is statistically convergent to some ξ ∈ ℂ, (ii) the single sequence $x_{jk}:k\in \mathbb{N}$ is statistically convergent to some ${\xi }_j\in ℂ$ for each fixed j ∈ ℕ ∖ ₁, (iii) the single sequence $x_{jk}:j\in \mathbb{N}$ is statistically convergent to some ${\eta }_k\in ℂ$ for...

In this paper we analyze relations among several types of convergences of bounded sequences, in particulars among statistical convergence, ${ℐ}_u$-convergence, $\varphi$-convergence, almost convergence, strong $p$-Cesàro convergence and uniformly strong $p$-Cesàro convergence.

In this paper we investigate the relationship between the statistical (or generally $I$-convergence) of a series and the usual convergence of its subseries. We also give a counterexample which shows that Theorem 1 of the paper by B. C. Tripathy “On statistically convergent series”, Punjab. Univ. J. Math. 32 (1999), 1–8, is not correct.