Sequences of linear operators related to Cesàro-convergent sequences
Given a Cesàro-convergent sequence of real numbers\((a_n)_nN \), a sequence \((φ_n)_nN \) of operators is defined on the Banach space \(R(I,F)\) of regular functions defined on \(I=[0,1]\) and having values in a Banach space \(F\), \[φ_n(f)=1n_k=1^na_kf( kn) .\]It is proved that if, in addition, the sequence \(( |a_1| +…+| a_n| n)_nN \) is bounded, then \(φ_n(f)\) converges to \(a·olimits_0^1f,\) where \(a=_n→∞a_1+…+a_n n.\) The converse of this statement is also true. Another result is that the supplementary condition can be dropped if the operators are considered on the space \(C^1(I,F)\)., number=2, journal=Rev. Anal. Numér. Théor. Approx., author=Anisiu, Mira-Cristiana and Anisiu, Valeriu, year=2002, month=Aug., pages=135–141