Abstract In this research, we combine the Cesàro and backward difference operators of different orders which results in introducing a matrix who has two behaviors includes several matrices. We also investigate Köthe duals inclusion relations associated sequence space new matrix. Moreover, compute norm on some well-known spaces.

Let ν be either ω ∈ C \ {0} or q ∈ C \ {0, 1}, and let Dν be the corresponding difference operator defined in the usual way either by Dωp(x) = p(x+ω)−p(x) ω or Dqp(x) = p(qx)−p(x) (q−1)x . Let U and V be two moment regular linear functionals and let {Pn(x)}n≥0 and {Qn(x)}n≥0 be their corresponding orthogonal polynomial sequences (OPS). We discuss an inverse problem in the theory of discrete ort...

We say that the polynomial sequence (Q (λ) n ) is a semiclassical Sobolev polynomial sequence when it is orthogonal with respect to the inner product 〈p, r〉S = 〈u, p r〉+ λ 〈u,DpDr〉 , where u is a semiclassical linear functional, D is the differential, the difference or the q–difference operator, and λ is a positive constant. In this paper we get algebraic and differential/difference properties ...

We give elementary proofs of the theorems mentioned in the title. Our methods rely on a simple version of Ramsey theory and a martingale difference lemma. They also provide quantitative results: if a Banach space contains l only with a bad constant then every bounded sequence admits a subsequence which is “nearly” a weak Cauchy sequence.