Lacunary Strong Riesz Summability

In this paper, we introduce a new class of sequence spaces N-theta,R((k)) by combining lacunary block structures with Riesz-type weighted means of order k. This construction extends the classical notion of lacunary strong convergence to a higher-order weighted setting. We establish several inclusion relations between the classical strong Riesz summability method of order k and the corresponding lacunary space N-theta,R((k)) in the spirit of Freedman-type comparisons. The sharpness of the obtained conditions is illustrated by appropriate counterexamples. Basic topological properties of the new spaces are also discussed.