用黎曼求和法(R,1)求和時的基卜斯現象

<正> 假設級數滿足下面兩個條件,即:(甲)在原點的某一鄰域內,對於h(≠0)的一切值級數收斂;

ON THE GIBBS PHENOMENON FOR THE RIEMANN SUMMATION (R,1) OF FOURIER SERIES

We shall say that a series sum from n=1 to ∞ (u_n) is summable (R, 1) to s if the series converges for all values of h(≠0) in some neighbourhood of the origin, and It is known that (R, 1) is not regulad. However, Hardy and Littlewood proved the following theorem: Suppose that a_n=O(1/n),b_n=O(1/n),(1) then a necessary and sufficient condition that the series should converge, at a point x, to s, is that it should be summable (R,I) to s.Thus under the condition (1), the ordinary convergence and the method (R, 1) are equivalent. Let be the Fourier series of the function f(x), and We define the Riemann-Gibbs set of {R_h (x)}at x_o to be the aggregate of values λ = =lim R_h(x_o+α(h)), where α(h) tends to zero with h and α(h)/h→β(-∞≤β≤+∞).In the present note we establish the following two theorems:Theorem 1. For the series is the closed interval the Riemann-Gibbs set of{R_h (x)}at x=0Theorem 2. If series (2) is the Fourier series of a function f(x) of bounded variation, and if ξ is a point of jump of f(x), then the Riemann-Gibbs set of {R_h(x)}at point ξis the closed interval of length | f(ξ + 0) - f(ξ-0) | centred round 1/2{f(ξ+0) + f(ξ-0)}.In establishing Theorem 2, we need the following lemma.Lemma. If series (2)is the Fourier series of a function f(x) of bounded variation, and x_o is a point of continuity of f(x); then {R_h(x)}is uniformly convergent at x_o.For a series (2) satisfying condition (1), we need not to distinguish the ordinary convergence from (R, 1) summation, so far as the summability problem is concerned. But, when we consider the problem of Gibbs phenomenon, the situation are quite different, as the interval of Riemann and Gibbs is shortened in the case of (R, 1) summability.