It is known that if \((p_n)_{n \in \mathbb {N}}\) is a sequence of orthogonal polynomials in \(L^2([-1,1], w(x)dx)\), then the roots are distributed according to an arcsine distribution \(\pi ^{-1} (1-x^2)^{-1}dx\) for a wide variety of weights w(x). We connect this to a result of the Hilbert transform due to Tricomi: if \(f(x)(1-x^2)^{1/4} \in L^2(-1,1)\) and its Hilbert transform Hf vanishes on \((-1,1)\), then the function f is a multiple of the arcsine distribution$$\begin{aligned} f(x) = \frac{c}{\sqrt{1-x^2}}\chi _{(-1,1)} \qquad \text{ where }~c~\in \mathbb {R}. \end{aligned}$$We also prove a localized Parseval-type identity that seems to be new: if \(f(x)(1-x^2)^{1/4} \in L^2(-1,1)\) and \(f(x) \sqrt{1-x^2}\) has mean value 0 on \((-1,1)\), then$$\begin{aligned} \int _{-1}^{1}{ (Hf)(x)^2 \sqrt{1-x^2} dx} = \int _{-1}^{1}{ f(x)^2 \sqrt{1-x^2} dx}. \end{aligned}$$

中文翻译：

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关于Arsine分布和Hilbert变换的说明。

众所周知，如果\（（p_n）_ {n \ in \ mathbb {N}} \）是\（L ^ 2（[-1,1]，w（x）dx）\ ），然后根据各种权重w（x）根据反正弦分布\（\ pi ^ {-1}（1-x ^ 2）^ {-1} dx \）分布根。我们将此连接到Tricomi导致的希尔伯特变换的结果：if \（f（x）（1-x ^ 2）^ {1/4} \ in L ^ 2（-1,1）\）及其希尔伯特变换Hf在\（（-1,1）\）上消失，则函数f是反正弦分布$$ \ begin {aligned} f（x）= \ frac {c} {\ sqrt {1-x ^ 2}} \ chi _ {（-1,1）} \ qquad \ text {其中}〜c〜\在\ mathbb {R}中。\ end {aligned} $$我们还证明了本地化的Parseval类型身份似乎是新的：如果\（f（x）（1-x ^ 2）^ {1/4} \ in L ^ 2（-1,1）\）和\ （f（x）\ sqrt {1-x ^ 2} \）在\（（-1,1）\）上的平均值为0 ，然后$$ \ begin {aligned} \ int _ {-1} ^ {1 } {（Hf）（x）^ 2 \ sqrt {1-x ^ 2} dx} = \ int _ {-1-1 ^^ {1} {f（x）^ 2 \ sqrt {1-x ^ 2} dx }。\ end {aligned} $$