emily
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Lettus-CSS-MathJax


			
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<head>
  <meta charset="utf-8">
  <meta name="author" content="Eemeli Blåsten">
  <title>Hermite functions</title>
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  \(
    \newcommand{\abs}[1]{{\left\lvert #1 \right\rvert}}
    \newcommand{\norm}[1]{{\left\lVert #1 \right\rVert}}
    \newcommand{\supp}{\operatorname{supp}}
    \newcommand{\C}{\mathbb{C}}
    \newcommand{\R}{\mathbb{R}}
    \newcommand{\Z}{\mathbb{Z}}
    \newcommand{\N}{\mathbb{N}}
    \newcommand{\D}{\mathscr{D}}
    \newcommand{\S}{\mathscr{S}}
    \newcommand{\F}{\mathscr{F}}
  \)
</head>


<body class="content">

<p>Test MJ scaling and font: <i>u$u$i$i$f$f$</i></p>

<p>Test ümlauts, first in the current font (normal, umlaut), then using 
in mathmode normal, using <code>\ddot</code> and <code>\"</code>:
<i>
  uü$u\ddot u\"u$,
  aä$a\ddot a\"a$,
  oö$o\ddot o\"o$,
  aå$a\phantom{a}\r{a}$
</i></p>

<a href="http://www.gnu.org">GNU project</a>

<div class="theorem" data-nbr="A" id="ttt"><span>a span</span></div>
<div class="theorem" data-counters="2"><p>a paragraph</p></div>
<div class="theorem">No enclosing tags!<p>a paragraph</p></div>
<div class="theorem" data-nbr="B"><span>Enclosed in span.</span><p>a paragraph</p></div>
<div class="theorem"><span></span><p>an empty span before this &lt;p&gt;</p></div>
<div class="theorem" data-counters="2"></div>
<div class="lemma">The thm above is completely empty</div>
<div class="corollary"><span>span</span></div>
<div class="proposition" data-counters="2"><p>paragraph</p></div>
<div class="conjecture"><p>paragraph</p></div>
<div class="definition" data-counters="2"><p>paragraph</p></div>
<div class="remark"><p>paragraph</p></div>
<div class="proof">
  <p>
    These three sentences are a paragraph. According to <a 
    href="#ttt">Theorem</a> we have it. Or simply by <a href="#ttt"></a>
  </p>
</div>

<h1>Basics of the Hermite functions and transform</h1>

<footer>
<small>
<pre>    Copyright (C)  2015-2018  Eemeli Blåsten.
    Permission is granted to copy, distribute and/or modify this document
    under the terms of the GNU Free Documentation License, Version 1.3
    or any later version published by the Free Software Foundation;
    with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts.
    A copy of the license is included in the section entitled &quot;GNU
    Free Documentation License&quot;.
</pre>
</small>
</footer>

<div class="motivation">
<p>
  The reason to have this CRAZY document is just to try all the 
  different HTML, CSS and JavaScript features.
</p>
</div>

<span id="focusHere"></span>
<p>
  Here is a try for citing an offline reference: <a 
  href="#articleTry"></a>, and with something more descriptive than a 
  simple number: Blåsten, Meikäläinen<a href="#articleTry"></a>. In 
  particular see <a href="#articleTry">Chapter II</a>. Note that above 
  <q>Blåsten, Meikäläinen</q> is currently not part of the anchor. If it 
  was, then it would display like <q>Chapter II</q> above.
</p>


<section data-counters="1">
<h2>Hermite functions</h2>

<p>
  Lorem ipsum dolor sit amet, consectetur adipiscing elit. Morbi ligula 
  lorem, rhoncus vel hendrerit a, varius a turpis. Praesent in mauris 
  non ex vehicula vestibulum sit amet in nulla. Morbi elementum justo 
  vestibulum commodo elementum. Aenean sed gravida nulla. Fusce 
  faucibus, justo sed porta elementum, turpis nunc tempus urna, a 
  tincidunt elit magna non lacus. Fusce ut arcu congue, molestie erat 
  sed, ultricies elit. Fusce a interdum elit. Etiam pharetra vestibulum 
  diam malesuada euismod. Nulla non vestibulum purus. Fusce semper lorem 
  enim, sed lobortis ex interdum blandit. Proin a dapibus augue, non 
  venenatis magna. Maecenas est diam, laoreet eu porta ut, vulputate nec 
  purus.
</p>
<p>
  Vestibulum at iaculis risus. Etiam consectetur ornare ligula quis 
  tincidunt. Nulla ultricies felis a posuere interdum. Quisque interdum 
  neque eu auctor ornare. Curabitur commodo, mi non tincidunt dignissim, 
  ligula lectus ultrices elit, in consectetur ex augue a nibh. Aenean 
  ornare enim pulvinar semper suscipit. Mauris facilisis nisl ligula, 
  eget tempus nisl egestas eget. Aliquam erat volutpat. Aliquam aliquet 
  fermentum tincidunt. Nulla lorem nisl, semper in imperdiet nec, 
  gravida quis arcu. Duis euismod interdum odio, eget fermentum felis 
  eleifend eu. Integer cursus posuere pulvinar.
</p>
<p>
  Ut porta dignissim mi ac rhoncus. Ut in turpis tortor. Quisque vel 
  aliquet mi, sed elementum metus. Lorem ipsum dolor sit amet, 
  consectetur adipiscing elit. Etiam mauris neque, vestibulum at posuere 
  eu, dignissim nec nisl. Nullam ullamcorper ipsum et risus pretium, ac 
  cursus mi efficitur. Aenean vel fringilla justo, nec dignissim justo.
</p>

<p>
  An unordered list:
  <ul>
    <li>asdf</li>
    <li>faeifs</li>
    <li>fewafwa</li>
  </ul>
</p>
<p>
  An ordered list:
  <ol>
    <li>asdf</li>
    <li>faeifs</li>
    <li>fewafwa</li>
  </ol>
</p>
<p>
  A description list:
  <dl>
    <dt>aasdaasdadssdf</dt>
      <dd>adaw dad wad wad wdwa daw dwaw dad wa</dd>
    <dt>faeidawdadsfs</dt>
      <dd>lijfes s fesi fhslkjadf wijasd<dd>
    <dt>fewafwadwwda</dt>
      <dd>fdwalidaw lidaw ijalw dwa ldwijald ja</dd>
    <dt>adaliwjd</dt>
    <dt>ijwefoiewf</dt>
</dl>
</p>

<p>
  The following theorem, <a href="#basics">Theorem</a>, is of utmost 
  importance. But don't forget the other stuff in the second <a 
  href="example2.html">file</a>. Namely <a 
  href="example2.html#corolla">AA</a> and for example <a 
  href="example2.html#eq1">AA</a>, <a 
  href="example2.html#cEquation">AA</a> and <a 
  href="example2.html#thm1">AA</a>. Here is a link pretending to be in 
  this text: <a href="#nothere">link</a>.
</p>

<div class="theorem" title="Basic identities" id="basics" data-counters="2">
<p>
  Let $\psi_\alpha$, $\alpha\in\N$, be Hermite functions. Then
  
  \[
    \psi_\alpha'(x) = \sqrt{\frac{\alpha}{2}}\psi_{\alpha-1}(x) - 
    \sqrt{\frac{\alpha+1}{2}}\psi_{\alpha+1}(x)
  \]
  
  and
  
  \[
    x\psi_\alpha(x) = \sqrt{\frac{\alpha}{2}}\psi_{\alpha-1}(x) + 
    \sqrt{\frac{\alpha+1}{2}}\psi_{\alpha+1}(x).
  \]
  
  Moreover the Hermite functions are eigenfunctions to the quantum 
  mechanical oscillator
  
  \begin{equation}
    \label{eigenFunction}
    (x^2 - \partial_x^2) \psi_\alpha(x) = (2\alpha + 1)\psi_\alpha(x).
  \end{equation}
</p>
</div>
</section>


<section data-counters="1">
<h2>Hermite transform</h2>

<p>
  In this section we will first show that the Hermite functions form an 
  orthonormal set in $L^2(\R)$. After that we will show that the set is 
  complete and that leads to the Hermite transform.
</p>

<div class="lemma" data-counters="2" id="orthonormalSequenceLemma">
<p>
  Let $\psi_\alpha:\R\to\R$ be Hermite functions. Then
  
  \[
    \int_{-\infty}^\infty \psi_\alpha(x)\psi_\beta(x) dx = 
    \delta_{\alpha\beta}
  \]
  
  i.e. the sequence $(\psi_\alpha)_{\alpha=0}^\infty$ is orthonormal in 
  $L^2(\R)$.
</p>
</div>
<div class="proof">
  <p>
    Note first the identity $(x+\partial_x)(x-\partial_x) = 
    (x^2-\partial_x^2) + 1$. Combine it with the fact that $\psi_n$ is 
    an eigenfunction for the quantum oscillator \eqref{eigenFunction} to 
    get
    
    \[
      (x+\partial_x)(x-\partial_x)\psi_n = 2(n+1)\psi_n
    \]
    
    for any $n\in\N$. Note also that the transpose of $x-\partial_x$ in 
    the $L^2$ inner product is $x+\partial_x$.
  </p>
  <p>
    Hence we get
    
    \begin{align*}
      &\int \psi_\alpha \psi_\beta dx = \frac{1}{2\sqrt{\alpha\beta}} 
      \int (x-\partial_x)\psi_{\alpha-1}(x+\partial_x)\psi_{\beta-1} dx 
      \\ &\qquad = \frac{1}{2\sqrt{\alpha\beta}} \int \psi_{\alpha-1} 
      (x+\partial_x)(x-\partial_x)\psi_{\beta-1} \\ &\qquad = 
      \sqrt{\frac{\beta}{\alpha}} \int \psi_{\alpha-1}\psi_{\beta-1} dx.
    \end{align*}
  </p>
  <p>
    Using the previous equation we see that
    
    \[
      \int \psi_\alpha \psi_\alpha dx = \int \psi_0^2 dx = 
      \pi^{-1/2}\int_{-\infty}^\infty e^{-x^2} dx = 1.
    \]
  </p>
  <p>
    If, on the other hand for example $\alpha\lt\beta$, then
    
    \[
      \int\psi_\alpha\psi_\beta dx = \ldots = c_{\alpha,\beta} \int 
      \psi_0\psi_{\beta-\alpha} dx = c'_{\alpha,\beta} 
      \int_{-\infty}^\infty \partial_x^{\beta-\alpha} e^{-x^2} dx = 0
    \]
    
    since all the derivatives of Gaussians vanish at infinity.
  </p>
</div>


<div class="corollary" data-counters="2">
<p>
  Let $\mathscr{O}=\{x, \partial_x\}$ be the set of operators containing 
  the multiplication by $x$ and differentiation by $x$ operators. Let 
  $L^0_\alpha = \{\psi_\alpha\}$ and
  
  \[
    L^{k+1}_\alpha = \{ Sf \mid S\in\mathscr{O}, \, f\in L^k_\alpha \}.
  \]
  
  Then 
  
  \[
    \norm{f}_{L^2(\R)} \leq 2^{k/2} \prod_{\ell=1}^k \sqrt{\alpha+\ell} 
    = \sqrt{ 2^k \frac{(a+k)!}{a!} }
  \]
  
  for any $f \in L^k_\alpha$.
</p>
</div>
<div class="proof">
  <p>
    The sequence $(\psi_\alpha)_\alpha$ is orthonormal by <a 
    href=#orthonormalSequenceLemma>Lemma </a>. Hence we have 
    $\norm{\psi_\alpha}_2=1$. Assume that the claim is true for 
    $L^k_\alpha$ for any $\alpha\in\N$. We will prove it for 
    $L^{k+1}_\alpha$. Let that $f\in L^k_\alpha$. Then there are 
    operators $S_1,\ldots,S_k \in \mathscr{O}$ such that $f = 
    S_1S_2\cdots S_k \psi_\alpha$.
  </p>
  <p>
    Consider the case $S_k = \partial_x$. Let $O\in\mathscr{O}$ and use 
    one of the basic identities to get
    
    \begin{align*}
      &\norm{Of}_2 = \norm{OS_1\cdots S_{k-1} \partial_x \psi_\alpha}_2 
      \\ &\qquad = \norm{OS_1\cdots S_{k-1} \left( 
      \sqrt{\tfrac{\alpha}{2}}\psi_{\alpha-1} - 
      \sqrt{\tfrac{\alpha+1}{2}}\psi_{\alpha+1}\right)}_2 \\ &\qquad 
      \leq \sqrt{\tfrac{\alpha}{2}} \norm{OS_1\cdots S_{k-1} 
      \psi_{\alpha-1}}_2 + \sqrt{\tfrac{\alpha+1}{2}} \norm{OS_1\cdots 
      S_{k-1} \psi_{\alpha+1}}_2 \\ &\qquad \leq 
      \sqrt{\tfrac{\alpha+1}{2}} 2^{k/2} \left( \prod_{\ell=1}^k 
      \sqrt{\alpha-1+\ell} + \prod_{\ell=1}^k \sqrt{\alpha+1+\ell} 
      \right) \\ &\qquad \leq 2^{(k-1)/2} \sqrt{\alpha+1} \cdot 2 
      \prod_{\ell=1}^k\sqrt{\alpha+1+\ell} \\ &\qquad \leq 2^{(k+1)/2} 
      \prod_{\ell=1}^{k+1} \sqrt{\alpha+\ell}
    \end{align*}
    
    since $OS_1\cdots S_{k-1} \psi_{\alpha-1}$ and $OS_1\cdots S_{k-1} 
    \psi_{\alpha+1}$ are in $L^k_{\alpha-1}$ and $L^k_{\alpha+1}$, 
    respectively. The same kind of deduction works in the case when 
    $S_k$ is multiplication by $x$. Hence the claim follows by 
    induction.
  </p>
</div>
</section>


<footer id="bibliography">
<h2>References</h2>
<h3>Offline sources</h3>
<ul class="referenceList">
  <li id="articleTry" class="reference">
    <span class="ref-authors">E. Blåsten</span> and 
    <span class="ref-authors">M. Meikäläinen:</span>
    <span class="ref-title">A proof of the Riemann conjecture,</span>
    <span class="ref-journal">Annals of Mathematics,</span>
    <span class="ref-volume">23,</span>
    <span class="ref-issue">2</span>
    <span class="ref-year">(2069),</span> 
    <span class="ref-pages">1&ndash;503.</span>
  </li>
</ul>

</footer>


</body>