Multipliers in weighted settings and strong convergence of associated operator-valued Fourier series

This note describes the pleasant features that accrue in weighted settings when the partial sums of the operator-valued Fourier series corresponding to a multiplier function ψ : T → C are uniformly bounded in operator norm. This circle of ideas also includes a Tauberiantype condition on the multiplier function ψ sufficient to insure such uniform boundedness of partial sums. These considerations are shown to apply to Riemann’s continuous, “sparsely differentiable,” periodic function. In a larger sense, our considerations aim at showing how pillars of functional analysis and real-varable methods in Fourier analysis can be combined with “bread-and-butter” techniques from these subjects so as to reveal hitherto unnoticed useful tools in multiplier theory for weighted Lebesgue spaces.