A New Class of Mean Convergence : Structured Mean Convergence

In this paper, we introduce the concept of structured mean convergence (SMC), a generalization of Cesàro 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, Cesàro, 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 include functional analysis, ergodic theory, and number theory, where selective summability plays a crucial role.