關於有一無限極限的一個函數的窣锇Ъ墧祵墩笖悼偤托

<正> 1.我們已經證明開於有一無限極限的一個單調函數的福里哀級數對於负指數(c,r)總和性的情形的定理,很自然地,人們還要問起:對於正指數的情形是怎麼樣?現在進行討論如下.

ON THE SUMMABILITY FOR POSITIVE INDICES OF THE FOURIER SERIES OF A FUNCTION WITH AN INFINITE LIMIT

In a previous paper [Ⅰ] we proved the theorem, concerning the behaviour of the sumnability (c, r) for negative indices of the Fourier series of a monotonic function with an infinite limit. It is natural to inquire what happens when the indices are positive.Let be the Fourier series of an L-integrable function f(θ), and σ_n~r(θ) the r-th Cesaro mean of the series at the point θ.Define φ(t) = 1/2 [f(θ + t) + f(θ - t)]. We shall be concerned with behaviour at a single point θ, which we may suppose to be θ=0. And then σ_n~r(O) is the r-th Cesaro mean of the Fourier series of the even function φ(t) at the point t=0.Theorem (i) If r≥1 and limφ(t) = + ∞, then (ii) If0