用黎曼求和法(R,k)求广义富理埃级数之和时的基卜斯现象

<正> 假设级数,■满足下面两个条件,就是:(i)在原点的某邻域内,对于 h(≠0)的一切值级数

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

We shall say that a series(?)is summable(R,k)to S if the series(?)converges for all values of h(≠O)in some neighborhood of the origin,and(?) be the Fourier series of the function f(x),and let(?)In the present note we establish the following theorems:Theorem 1.For the Cauchy principal value Fowrier seris(?)the set of all limiting values of the ratio(?)is the infinite interval:(?)Theorem 2.For the Complex-integration real generalized Fourier series(?)of the first derivative of the function(?)the set of àll limiting values of the ratiois the infinite interval:(?).Theorem 3.For the Complex-integration real generalized Fourier series(?)of the (k-1)-th derivative of the function 1/2 ctg x/2,x(k=4m+2,4m,4m+1,4m-1,m=1,2…),the set of all limiting values of the ratiois the whole number axis:(?)some integers).These theorems show that {R_h~k(x)} presents Gibbs phenomenon for generalizedFourier series,although it does not present for Lebesgue-integration Fourier series.