For a function f continuous on a closed interval, its modulus of fractality ν(f, ε) is defined as the function that maps any ε > 0 to the smallest number of squares of size ε that cover the graph of f. The following condition for the uniform convergence of the Fourier series of f is obtained in terms of the modulus of fractality and the modulus of continuity ω(f, δ): if$$\begin{array}{*{20}{c}} {\omega (f,\pi /n)\ln \left( {\frac{{v(f,\pi /n)}}{n}} \right) \to 0}&{\text{as}}&{n \to + \infty ,} \end{array}$$then the Fourier series of f converges uniformly. This condition refines the known Dini–Lipschitz test. In addition, for the growth order of the partial sums S_n(f, x) of a continuous function f, we derive an estimate that is uniform in x ∈ [0, 2π]:$${S_n}(f,x) = o\left( {\ln \left( {\frac{{v(f,\pi /n)}}{n}} \right)} \right).$$.The optimality of this estimate is shown.

中文翻译：

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函数三角图傅里叶级数的收敛性

对于在闭合区间上连续的函数f，其分形模数ν（f，ε）定义为将ε > 0映射到覆盖f的图的最小大小为ε的平方的函数。根据分形模量和连续模量ω（f，δ），得到f的傅里叶级数的均匀收敛的以下条件：if $$ \ begin {array} {* {20} {c} } {\ omega（f，\ pi / n）\ ln \ left（{\ frac {{v（f，\ pi / n）}} {n}} \ right）\ to 0}＆{\ text {as }}＆{n \ to + \ infty，} \ end {array} $$然后是f的傅立叶级数均匀收敛。此条件完善了已知的Dini–Lipschitz检验。另外，对于部分和的生长顺序小号_Ñ（F，X的连续函数的）˚F，我们导出的估计是在均匀的X ∈[0，2 π ]：$$ {S_N}（F，X ）= o \ left（{\ ln \ left（{\ frac {{v（f，\ pi / n）}} {n}} \ right）} \ right）。$$表示此估计的最优性。