一类广义傅里叶级数的敛散性

Convergence and Divergence of A Class of Generalized Fourier Series

研究由分段线性谱序列生成的广义傅里叶级数的逐点敛散性.估计一类函数的广义傅里叶系数趋于零的速度,给出广义傅里叶级数逐点收敛的判别方法,并证明两个否定的结果,即存在周期为1的连续函数,其广义傅里叶级数在一点发散;存在周期为1的可积函数,其广义傅里叶级数处处发散.

In this paper, the pointwise convergence and divergence of a class of generalized Fourier series generated by piecewise linear spectral sequences are studied. For a class of functions, the rate of decay of generalized Fourier coefficients is estimated. Some criteria for pointwise convergence of generalized Fourier series are given. Two negative results are presented. One is that there exist 1-periodic continuous functions whose generalized Fourier series diverge at some point. The other is that there exist 1-periodic integrable functions whose generalized Fourier series diverge everywhere.