Notes on Fourier analysis (XXV): Quasi-Tauberian theorem

Abstract

The first general treatment of quasi-Tauberian theorems was given by N. Wiener C16J. In this note the author proves another quasi-Tauberian theorem concerning absolute limit under Wiener's conditions. Wiener derived the Cesaro summability theorem of Fourier series frcm his general theorem.In this paper further applications of his theorem and analogues concerning absolute limit are given. Some of theorems proved in this paper are known and the other are new. It is interesting that theεe theorems are derived fi om two key theorems which are given in § 1. In § 2, the Cesaro and absolute Cesaro summability theorems due to Paley [11J and Boεanquet [lj C2j are derived frcm the key theorems. In § 3, we prove the Cesaro and absolute Cesaro summability theorems of the conjugate Fourier series. These theorems include the essential parts of the results due to Paley [llj and Boεanquet-Hyslop [5^. The Cesaro summability problems of the derived Fourier series are discussed in § 4. The essential part of the results due to Takahashi (13^, Wang [14J (Ί53, Zygmund C173, Bosanquet £3^ £4J and Hyslop [TJ are derived frcm our key theorems. It is well known that these results are interpreted as the relation between Cesaro summation and Riemann summation of the first kind and their a nalogues concerning absolute summation. In § 5, the relations between the generalized jump of a function and its Fourier constants are discussed. This problem was treated early by Zygmund [18J and Szεεz Γ12> Theεe results are known as the relation between Cesaro summation, Riemann summation of the second kind and their analogues concerning absolute summation. As applications of these theorems, we can also prove theorems analogous to the results due to Misra C9D and Moursund C10>In the sequel we shall fully use the notations and theorems of Chapter VII in the Wiener's work Clβ^ without references.

Quasi-Tauberian theorems, applied to the summability of Fourier series by Riesz's logarithmic means

“…there exists η>0 such that I \R(x)\x~ιdx<oo y v4) / \N(x)\χr 1 dx< oo.Then for every £, 0 < £ < ^-, we have Now we shall finish the proof. By the assumption, for any η > 0, there exists δ>0 such that if 0<;y<δ(3,8) We may assume 2£<δ.…”