Marouf (1993) presented definitions for asymptotically equivalent sequences and asymptotic regular matrices. Patterson (2003), extended those concepts by presenting an asymptotically statistical equivalent analog of these definitions and natural regularity conditions for non-negative sum ability matrices. In Patterson and Savas (2006) extended the definitions presented in (Patterson, 2003) to l...

Let f be a 2π periodic function in L[0, 2π] and ∑∞ k=−∞ f̂(nk)e inkx be its Fourier series with ‘small’ gaps nk+1 − nk ≥ q ≥ 1. Here we have obtained sufficiency conditions for the absolute convergence of such series if f is of ∧ BV (p) locally. We have also obtained a beautiful interconnection between lacunary and non-lacunary Fourier series.

Here, we continued the studies initiated by Vinod K. Bhardwaj and Shweta Dhawan which relate different convergence methods involving classical statistical strong Cesàro convergences means of lacunary sequences measures density in N modulated a modulus function f. A method for constructing non-compatible functions was also included, is related to symmetries with respect y=x.

We introduce the sequence spaces [ ˆ w(M)] and [ ˆ w(M)] θ of strong almost convergence and lacunary strong almost convergence respectively defined by the Orlicz function M. We establish certain inclusion relations and show that the above spaces are same for any bounded sequences. 1. Definitions and notations A sequence x ∈ l ∞ , the space of bounded sequences x = (x k), is said to be almost co...