GCD矩阵的扩张函数级数和谱范数的收敛性
Convergence of series of dilated functions and spectral norms of GCD matrices
摘要：
建立了一类$α\in (1/2,1)$的$j$阶傅里叶系数为$\mathcal{O}(j^{-α})$的扩张函数和的$L^2$范数与一类最大公因数（GCD）矩阵的谱范数之间的联系.利用这些谱范数的最新界，我们得到了$L^2$收敛和扩张函数级数几乎处处收敛的尖锐条件。
英文摘要：
We establish a connection between the $L^2$ norm of sums of dilated functions whose $j$th Fourier coefficients are $\mathcal{O}(j^{-α})$ for some $α\in (1/2,1)$, and the spectral norms of certain greatest common divisor (GCD) matrices. Utilizing recent bounds for these spectral norms, we obtain sharp conditions for the convergence in $L^2$ and for the almost everywhere convergence of series of dilated functions.
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