Abubakar Masha & Buhari Alhaji Ali
Department of Mathematics and Computer Science, Kashim Ibrahim University
*Corresponding author’s Email: mashafantami@gmail.com, doi.org/10.55639/607.020100145
ABSTRACT
Statistical convergence and its ideal based generalizations have become an active area of summability theory since their independent introduction by Fast and Schoenberg, with subsequent extensions to double sequences, weighted means, and lacunary sequences by several authors. Building on the concept of weighted lacunary I-statistical convergence recently introduced for single sequences, this paper extends the notion to double sequences by defining weighted lacunary I₂-statistical convergence and the associated weighted lacunary I₂-summability method [R², βₘ, γₙ, θᵣ,ₛ]ᴵ² for double sequences with respect to an admissible ideal I₂ of ℕ × ℕ. Several inclusion relations connecting these two convergence notions are established. It is shown that weighted lacunary I₂-summability implies weighted lacunary I₂-statistical convergence in general, with the converse holding under a boundedness condition on the weighted double sequence. Relations between the weighted and unweighted forms of lacunary I₂-statistical convergence are also obtained, depending on whether the weight sequences (βⱼ) and (γₖ) are bounded above or below by unity. Finally, conditions on the double lacunary ratio qᵣ,ₛ are used to relate weighted lacunary I₂-statistical convergence to weighted (non-lacunary) I₂-statistical convergence, showing that the two notions coincide when liminfᵣ,ₛ qᵣ,ₛ > 1 and when limsupᵣ,ₛ qᵣ,ₛ < ∞. These results generalize several known inclusion theorems for statistical and lacunary statistical convergence of double sequences to the weighted, ideal-based setting.
KEYWORDS
Ideal,
Filter,
Weighted,
Lacunary
II-
statistical
convergence
sequences,
Summability.