Hille and Tamarkin have proved
a result for the Nörlund summability of the Fourier series of f(t) at t = x, under the
hypothesis (i) φ(t) = {f(x + t) + f(x−t) − 2f(x)}∕2 = o(1),t → 0, which includes as
a special case the corresponding result for the Cesàro summability. However,
under the lighter condition (ii) ∫0tφ(u)du = o(t),t → 0, Astrachan has
proved a theorem for the Nörlund summability which does not cover the
corresponding Cesàro case. The object of the present paper is to prove
theorems for the Nörlund summability and another triangular matrix method
of summability which are subtler than Astrachan’s theorem in the sense
that they include as a special case the corresponding result for the Cesàro
summability.
