论文标题
nyman-beburing标准中的多项式近似
Polynomial approximations in a generalized Nyman-Beurling criterion
论文作者
论文摘要
Nyman-Beberling标准相当于Riemann假设（RH），是$（0，\ infty）$上的正方形函数空间中的一个近似问题，涉及因子$θ_k\ in（0,1）$，$ k \ ge1 $ abterable部分的扩张。随机化$θ_K$生成新的结构和标准。其中之一是RH的足够条件，它将（i）分成（i）表明指示器函数可以通过与分数部分进行卷积近似，（ii）控制近似值的系数。这份独立的论文概括了涉及（1/2,1）$的$σ_0\的条件（i）和（ii），并暗示$ζ（σ+it）\ neq 0 $ in Strip $ 1/2 <σ\σ\ letσ_0_0<1 $。然后，我们通过多项式近似值识别（i）无条件地保持的功能。这产生了简短的概率证据，证明了维也纳陶伯里亚定理的已知后果。在这种情况下，在（ii）中将证明RH的困难重新分配，这在很大程度上依赖于相应的革兰氏矩阵，为此获得了两个显着的结构。我们表明，近似序列的特定调整导致了第二克矩阵的惊人简化，然后读取为hankel形式。
The Nyman-Beurling criterion, equivalent to the Riemann hypothesis (RH), is an approximation problem in the space of square integrable functions on $(0,\infty)$, involving dilations of the fractional part function by factors $θ_k\in(0,1)$, $k\ge1$. Randomizing the $θ_k$ generates new structures and criteria. One of them is a sufficient condition for RH that splits into (i) showing that the indicator function can be approximated by convolution with the fractional part, (ii) a control on the coefficients of the approximation. This self-contained paper generalizes conditions (i) and (ii) that involve a $σ_0\in(1/2,1)$, and imply $ζ(σ+it)\neq 0$ in the strip $1/2<σ\leσ_0<1$. We then identify functions for which (i) holds unconditionally, by means of polynomial approximations. This yields in passing a short probabilistic proof of a known consequence of Wiener's Tauberian theorem. In this context, the difficulty for proving RH is then reallocated in (ii), which heavily relies on the corresponding Gram matrices, for which two remarkable structures are obtained. We show that a particular tuning of the approximating sequence leads to a striking simplification of the second Gram matrix, then reading as a block Hankel form.
