doiSerbia

Lacunary strong convergence of difference sequences with respect to a modulus function

Čolak Rifat (Department of Mathematics, Faculty of Science and Arts, Firat University, Elazig, Turkiye)

A sequence Θ = (kr) of positive integers is called lacunary if k0 = 0, 0 < kr < kr+1 and hr = kr – kr-1 → ∞ as r → ∞. The intervals determined by Θ are denoted by Ir = (kr-1, kr]. Let ω be the set of all sequences of complex numbers and f be a modulus function. Then we define NΘ(Δm, f) = {x є ω: lim 1/hr Σ f(|Δm xk -l|)=0 for some l} r kєIr NΘ0(Δm, f) = {x є ω: lim 1/hr Σ f(|Δm xk|)=0} r kєIr NΘ∞(Δm, f) = {x є ω: sup 1/hr Σ f(|Δm xk|)< ∞} r kєIr where Δxk = xk - xk+1, Δmxk = Δm-1xk - Δm-1xk+1 and m is a fixed positive integer. In this study we give various properties and inclusion relations on these sequence spaces.

Keywords: difference sequence, lacunary sequence, modulus function