Explicit formulas for the distribution of complex zeros of a family of random sums

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• 作者：Andrew Ledoan ^{Andrew-Ledoan@utc.edu" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Cholesky factorization ; Expected number of zeros ; Explicit formula ; Random sum
• 刊名：Journal of Mathematical Analysis and Applications
• 出版年：2016
• 出版时间：15 December 2016
• 年：2016
• 卷：444
• 期：2
• 页码：1304-1320
• 全文大小：6347 K

文摘

The present paper provides an explicit formula for the average intensity of the distribution of complex zeros of a family of random sums of the form inlineImage" height="21" width="155" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022247X16303262-si1.gif">iner hidden">$S_n(z)={\sum }_{j=0}^n{\eta }_j{f}_j(z)$, where z   is the complex variable x+iyiner hidden">$x+iy$, η_j=α_j+iβ_jiner hidden">${\eta }_j={\alpha }_j+i{\beta }_j$ and inlineImage" height="20" width="57" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022247X16303262-si23.gif">iner hidden">${\{{\alpha }_j\}}_{j=0}^n$ and inlineImage" height="20" width="55" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022247X16303262-si24.gif">iner hidden">${\{{\beta }_j\}}_{j=0}^n$ are sequences of standard normal independent random variables, and inlineImage" height="20" width="54" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022247X16303262-si6.gif">iner hidden">${\{f_j\}}_{j=0}^n$ is a sequence of given analytic functions that are real-valued on the real number line. In addition, the numerical computations of the intensity functions and the empirical distributions for the special cases of random Weyl polynomials, random Taylor polynomials and random truncated Fourier cosine series are included as examples.