Strong approximation of non-periodic functions (Q1902323)

The author examines the existence and structural properties of the complex-valued functions \[ S_\lambda[f](z):= {1\over \pi} \int^{\to + \infty}_{\to - \infty} f(u) D_\lambda(z- u) du, \] where \(f\in L_{\text{loc}}\) and \(\lambda\in [0, \infty)\). He proves two interesting and very general theorems, but without giving a lot of notations it is not possible for the reviewer to recall them succinctly, so a corollary of the theorems will be presented here. If \(\omega_r(\delta; f)< \infty\) for some \(r\in N\) and if \(rq< \beta\), then for a.e. \(x\in R\) and all \(n\), \[ \Biggl\{ {1\over n^\beta} \int^n_0 \lambda^{\beta- 1}|S_\lambda[f](x)- f(x)|^q d\lambda\Biggr\}^{1/q}\leq K(\beta, q, r) \omega_r \Biggl({1\over n}; f\Biggr). \]