A NEW CLASS OF MEAN CONVERGENCE: STRUCTURED MEAN CONVERGENCE

Abstract

In this paper, we introduce the concept of \textit{structured mean convergence} (SMC), a generalization of Ces\`{a}ro convergence that allows averaging over selected index subsets rather than the entire sequence. This approach provides a flexible framework for analyzing the asymptotic behavior of sequences under customized selection criteria. We explore the fundamental properties of SMC and establish its relationships with classical, Ces\`{a}ro, and statistical convergence. Additionally, we examine its connection to lacunary mean convergence, proving that lacunary mean convergence is a special case of SMC under certain density conditions. We also present counterexamples demonstrating that SMC does not necessarily imply lacunary mean convergence. These results suggest that SMC provides a broader perspective on summability theory and sequence transformations. Potential applications of SMC extend to functional analysis, ergodic theory, and number theory, where selective summability plays a crucial role.