STRONG SUMMABILITY IN FRECHET SPACES WITH APPLICATIONS TO FOURIER-SERIES

This 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 f{hook} ε{lunate} L2π1, ( 1 (n + 1))∥∑k = 0n skf{hook} - f{hook} ∥L = o(1), (n → ∞), where skf{hook} is the kth partial sum of the Fourier series of f{hook}; since the dual of L2π1 is L2π∞, this is equivalent to sup∥g∥ ( 1 (n + 1))|∑k = 0n ∫02π g · (skf{hook} - f{hook})| = 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 f{hook} ε{lunate} L2π1, sup∥g∥ 1 n+1 ∑ k=0 n|∫02π g · (skf{hook} - f{hook})|r = 0(1) (n → ∞) The supremum, however, cannot be taken inside the summation. © 1992.