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A REMARK ON THE ARCSINE DISTRIBUTION AND THE HILBERT TRANSFORM.
Journal of Fourier Analysis and Applications ( IF 1.2 ) Pub Date : 2019-03-21 , DOI: 10.1007/s00041-019-09678-w
Ronald R Coifman 1 , Stefan Steinerberger 1
Affiliation  

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:

关于Arsine分布和Hilbert变换的说明。

众所周知,如果\((p_n)_ {n \ in \ mathbb {N}} \)\(L ^ 2([-1,1],w(x)dx)\ ),然后根据各种权重wx)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\((-1,1)\)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\(f(x)(1-x ^ 2)^ {1/4} \ in L ^ 2(-1,1)\)\ (f(x)\ sqrt {1-x ^ 2} \)\((-1,1)\)上的平均值为0 ,然后$$ \ begin {aligned} \ int _ {-1} ^ {1 } {(Hf)(x)^ 2 \ sqrt {1-x ^ 2} dx} = \ int _ {-1-1 ^^ {1} {f(x)^ 2 \ sqrt {1-x ^ 2} dx }。\ end {aligned} $$
更新日期:2019-03-21
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