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It has been proved recently that the two-direction refinement equation of the form $f (x) = \sum_{n \in} c_{n, 1} f (k x - n) + \sum_{n \in ℤ} c_{n, - 1} f (- k x - n)$ 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 $f (x) = \sum_{n \in ℤ} c ₙ f (k x - n)$ , which has been used in many areas of mathematics with important applications. The following continuous extension of the classical refinement equation $f (x) = \int_{ℝ} c (y) f (k x - y) d y$ has also various interesting applications....

Reflections on Value Regions, Limit Regions and Truncation Errors for Continued Fractions.

Egil Rye, Haakon Waadeland (1985)

Numerische Mathematik

Regions of Convergence for Hypergeometric Series in Three Variables.

Per W. Karlsson (1974)

Mathematica Scandinavica

Regular statistical convergence of double sequences

Ferenc Móricz (2005)

Colloquium Mathematicae

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 $x_{j k} : (j, k) \in ℕ ²$ is said to be regularly statistically convergent if (i) the double sequence $x_{j k}$ is statistically convergent to some ξ ∈ ℂ, (ii) the single sequence $x_{j k} : k \in ℕ$ is statistically convergent to some $ξ_{j} \in ℂ$ for each fixed j ∈ ℕ ∖ ₁, (iii) the single sequence $x_{j k} : j \in ℕ$ is statistically convergent to some $η_{k} \in ℂ$ for...

Relations between sequences and selection properties.

Djurčić, D., Kočinac, Lj.D.R., Žižović, M.R. (2007)

Abstract and Applied Analysis

Remark on a formula of P.H. Jackson

Carlitz, L. (1970)

Portugaliae mathematica

Remark on the convergence of the sequences defined by certain difference equations

B. Jovanović, E. Udovičić (1972)

Matematički Vesnik

Remarks on a paper of S. K. Bhattacharyya and B. K. Lahiri

J. A. Guthrie, J. E. Nymann (1986)

Matematički Vesnik

Remarks on duality for $Σ$ -groups. I. Products and coproducts

Denis Higgs (1989)

Cahiers de Topologie et Géométrie Différentielle Catégoriques

Remarks on several types of convergence of bounded sequences

Vladimír Baláž, Oto Strauch, Tibor Šalát (2006)

Acta Mathematica Universitatis Ostraviensis

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

Remarks on statistical and $I$ -convergence of series

Jaroslav Červeňanský, Tibor Šalát, Vladimír Toma (2005)

Mathematica Bohemica

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.

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Rad prevrátených hodnôt všetkých prvočísel a niektoré výsledky o konvergencii čiastočných radov harmonického radu

Ramanujan's evaluations of Rogers-Ramanujan type continued fractions at primitive roots of unity

Ramanujan's harmonic number expansion into negative powers of a triangular number.

Randomly Weighted Series of Contractions in Hilbert Spaces.

Rate of convergence of summation-integral type operators with derivatives of bounded variation.

Rational sequences converging to $π$ .

Refinement type equations: sources and results