Strong summability in Fréchet spaces with applications to Fourier series

AbstractThis paper examines strong Cesàro summability and strong Cesàro sectional boundedness of order 1 ⩽ r < ∞ in Banach and Fréchet spaces E. The major result shows these topological properties of E to be equivalent to multiplier properties of the form E = (dvr ∩ c0) · E and E = dvr · E, where dvr is the space of sequences of dyadic variation of order r defined in this paper. These multiplier results show that several classical spaces of Fourier series have these properties. This introduces a new form of convergence in norm for Fourier series. The space L2π1, for example, has strong Cesàro summability of all orders 1 ⩽ r < ∞. Fejér's Theorem states that for all ƒ ϵ L2π1, (1(n + 1))∥∑k = 0n skƒ − ƒ ∥L1 = o(1), (n → ∞), where skƒ is the kth partial sum of the Fourier series of ƒ; since the dual of L2π1 is L2π∞, this is equivalent to sup∥g∥L∞ ⩽ 1 (1(n + 1))¦∑k = 0n ∫02π g · (skƒ − ƒ)¦ = o(1), (n → ∞). As a consequence of strong Cesàro summability, the absolute value can be taken inside the summation and raised to any power 1 ⩽ r < ∞. Namely, for all ƒ ϵ L2π1, sup∥g∥L∞ ⩽ 11n+1∑k=0n¦∫02π g · (skƒ − ƒ)¦r = 0(1) (n → ∞) The supremum, however, cannot be taken inside the summation