[{"@context":"http:\/\/schema.org\/","@type":"BlogPosting","@id":"https:\/\/wiki.edu.vn\/en\/wiki41\/hermite-class-wikipedia\/#BlogPosting","mainEntityOfPage":"https:\/\/wiki.edu.vn\/en\/wiki41\/hermite-class-wikipedia\/","headline":"Hermite class – Wikipedia","name":"Hermite class – Wikipedia","description":"before-content-x4 From Wikipedia, the free encyclopedia after-content-x4 The Hermite or P\u00f3lya class is a set of entire functions satisfying the","datePublished":"2016-06-21","dateModified":"2016-06-21","author":{"@type":"Person","@id":"https:\/\/wiki.edu.vn\/en\/wiki41\/author\/lordneo\/#Person","name":"lordneo","url":"https:\/\/wiki.edu.vn\/en\/wiki41\/author\/lordneo\/","image":{"@type":"ImageObject","@id":"https:\/\/secure.gravatar.com\/avatar\/c9645c498c9701c88b89b8537773dd7c?s=96&d=mm&r=g","url":"https:\/\/secure.gravatar.com\/avatar\/c9645c498c9701c88b89b8537773dd7c?s=96&d=mm&r=g","height":96,"width":96}},"publisher":{"@type":"Organization","name":"Enzyklop\u00e4die","logo":{"@type":"ImageObject","@id":"https:\/\/wiki.edu.vn\/wiki4\/wp-content\/uploads\/2023\/08\/download.jpg","url":"https:\/\/wiki.edu.vn\/wiki4\/wp-content\/uploads\/2023\/08\/download.jpg","width":600,"height":60}},"image":{"@type":"ImageObject","@id":"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/31fda477abe04f8406330533054ac92fb5d3e867","url":"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/31fda477abe04f8406330533054ac92fb5d3e867","height":"","width":""},"url":"https:\/\/wiki.edu.vn\/en\/wiki41\/hermite-class-wikipedia\/","wordCount":3008,"articleBody":"     (adsbygoogle = window.adsbygoogle || []).push({});before-content-x4From Wikipedia, the free encyclopedia       (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4The Hermite or P\u00f3lya class is a set of entire functions satisfying the requirement that if E(z) is in the class, then:[1][2]E(z) has no zero (root) in the upper half-plane.|E(x+iy)|\u2265|E(x\u2212iy)|{displaystyle |E(x+iy)|geq |E(x-iy)|} for x and y real and y positive.|E(x+iy)|{displaystyle |E(x+iy)|} is a non-decreasing function of y for positive y.The first condition (no root in the upper half plane) can be derived from the third plus a condition that the function not be identically zero. The second condition is not implied by the third, as demonstrated by the function        (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4exp\u2061(\u2212iz+eiz).{displaystyle exp(-iz+e^{iz}).} In at least one publication of Louis de Branges, the second condition is replaced by a strict inequality, which modifies some of the properties given below.[3]Every entire function of Hermite class can be expressed as the limit of a series of polynomials having no zeros in the upper half-plane.[4]The product of two functions of Hermite class is also of Hermite class, so the class constitutes a monoid under the operation of multiplication of functions.The class arises from investigations by Georg P\u00f3lya in 1913[5] but some prefer to call it the Hermite class after Charles Hermite.[6]A de Branges space can be defined on the basis of some “weight function” of Hermite class, but with the additional stipulation that the inequality be strict \u2013 that is,        (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4"},{"@context":"http:\/\/schema.org\/","@type":"BreadcrumbList","itemListElement":[{"@type":"ListItem","position":1,"item":{"@id":"https:\/\/wiki.edu.vn\/en\/wiki41\/#breadcrumbitem","name":"Enzyklop\u00e4die"}},{"@type":"ListItem","position":2,"item":{"@id":"https:\/\/wiki.edu.vn\/en\/wiki41\/hermite-class-wikipedia\/#breadcrumbitem","name":"Hermite class – Wikipedia"}}]}]