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  1. <!doctype html>
  2. <html lang="en">
  3. <head>
  4. <meta charset="utf-8">
  5. <meta name="author" content="Eemeli Blåsten">
  6. <title>Hermite functions</title>
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  8. <meta name="viewport" content="width=device-width, initial-scale=1.0">
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  11. <!-- CSS style sheet -->
  12. <link rel="stylesheet" href="CSSJS/lettus.css">
  13. <!-- Configure MathJax -->
  14. <script src="CSSJS/lettus_conf_mj.js"></script>
  15. <!-- Typeset MathJax formulas and number theorems etc. -->
  16. <script src="CSSJS/lettus_mj.js"></script>
  17. <!-- Latex macros -->
  18. \(
  19. \newcommand{\abs}[1]{{\left\lvert #1 \right\rvert}}
  20. \newcommand{\norm}[1]{{\left\lVert #1 \right\rVert}}
  21. \newcommand{\supp}{\operatorname{supp}}
  22. \newcommand{\C}{\mathbb{C}}
  23. \newcommand{\R}{\mathbb{R}}
  24. \newcommand{\Z}{\mathbb{Z}}
  25. \newcommand{\N}{\mathbb{N}}
  26. \newcommand{\D}{\mathscr{D}}
  27. \newcommand{\S}{\mathscr{S}}
  28. \newcommand{\F}{\mathscr{F}}
  29. \)
  30. </head>
  31. <body class="content">
  32. <p>Test MJ scaling and font: <i>u$u$i$i$f$f$</i></p>
  33. <p>Test ümlauts, first in the current font (normal, umlaut), then using
  34. in mathmode normal, using <code>\ddot</code> and <code>\"</code>:
  35. <i>
  36. uü$u\ddot u\"u$,
  37. aä$a\ddot a\"a$,
  38. oö$o\ddot o\"o$,
  39. aå$a\phantom{a}\r{a}$
  40. </i></p>
  41. <a href="http://www.gnu.org">GNU project <span class="referenceNbr">[3]</span></a>
  42. <div class="theorem" id="ttt">
  43. <span class="theorem"><a class="addTxtForbidden" href="#ttt">Theorem A.</a></span>
  44. <span>a span</span>
  45. </div>
  46. <div class="theorem">
  47. <span class="theorem">Theorem 0.1.</span>
  48. <p>a paragraph</p>
  49. </div>
  50. <div class="theorem">
  51. <span class="theorem">Theorem.</span>
  52. No enclosing tags!<p>a paragraph</p>
  53. </div>
  54. <div class="theorem">
  55. <span class="theorem">Theorem B.</span>
  56. <span>Enclosed in span.</span><p>a paragraph</p>
  57. </div>
  58. <div class="theorem">
  59. <span class="theorem">Theorem.</span>
  60. <span></span>
  61. <p>an empty span before this &lt;p&gt;</p>
  62. </div>
  63. <div class="theorem">
  64. <span class="theorem">Theorem 0.2.</span>
  65. </div>
  66. <div class="lemma">
  67. <span class="lemma">Lemma.</span>
  68. The thm above is completely empty
  69. </div>
  70. <div class="corollary">
  71. <span class="corollary">Corollary.</span>
  72. <span>span</span>
  73. </div>
  74. <div class="proposition">
  75. <span class="proposition">Proposition 0.3.</span>
  76. <p>paragraph</p>
  77. </div>
  78. <div class="conjecture">
  79. <span class="conjecture">Conjecture.</span>
  80. <p>paragraph</p>
  81. </div>
  82. <div class="definition">
  83. <span class="definition">Definition 0.4.</span>
  84. <p>paragraph</p>
  85. </div>
  86. <div class="remark">
  87. <span class="remark">Remark.</span>
  88. <p>paragraph</p>
  89. </div>
  90. <div class="proof">
  91. <span class="proof">Proof.</span>
  92. <p>
  93. These three sentences are a paragraph. According to <a
  94. href="#ttt">Theorem A</a> we have it. Or simply by <a href="#ttt">A</a>
  95. </p>
  96. </div>
  97. <h1>Basics of the Hermite functions and transform</h1>
  98. <footer>
  99. <small>
  100. <pre> Copyright (C) 2015-2018 Eemeli Blåsten.
  101. Permission is granted to copy, distribute and/or modify this document
  102. under the terms of the GNU Free Documentation License, Version 1.3
  103. or any later version published by the Free Software Foundation;
  104. with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts.
  105. A copy of the license is included in the section entitled &quot;GNU
  106. Free Documentation License&quot;.
  107. </pre>
  108. </small>
  109. </footer>
  110. <div class="motivation">
  111. <span class="motivation">Motivation.</span>
  112. <p>
  113. The reason to have this CRAZY document is just to try all the
  114. different HTML, CSS and JavaScript features.
  115. </p>
  116. </div>
  117. <p>
  118. Here is a try for citing an offline reference: <a
  119. href="#articleTry"><span class="referenceNbr">[1]</span></a>, and with
  120. something more descriptive than a simple number: Blåsten,
  121. Meikäläinen<a href="#articleTry"><span class="referenceNr">
  122. [1]</span></a>. In particular see <a href="#articleTry"><span
  123. class="referenceNbr"> [1, Chapter II]</span></a>. Note that above
  124. <q>Blåsten, Meikäläinen</q> is currently not part of the anchor. If it
  125. was, then it would display like <q>Chapter II</q> above.
  126. </p>
  127. <section>
  128. <h2>Hermite functions</h2>
  129. <p>
  130. Lorem ipsum dolor sit amet, consectetur adipiscing elit. Morbi ligula
  131. lorem, rhoncus vel hendrerit a, varius a turpis. Praesent in mauris
  132. non ex vehicula vestibulum sit amet in nulla. Morbi elementum justo
  133. vestibulum commodo elementum. Aenean sed gravida nulla. Fusce
  134. faucibus, justo sed porta elementum, turpis nunc tempus urna, a
  135. tincidunt elit magna non lacus. Fusce ut arcu congue, molestie erat
  136. sed, ultricies elit. Fusce a interdum elit. Etiam pharetra vestibulum
  137. diam malesuada euismod. Nulla non vestibulum purus. Fusce semper lorem
  138. enim, sed lobortis ex interdum blandit. Proin a dapibus augue, non
  139. venenatis magna. Maecenas est diam, laoreet eu porta ut, vulputate nec
  140. purus.
  141. </p>
  142. <p>
  143. Vestibulum at iaculis risus. Etiam consectetur ornare ligula quis
  144. tincidunt. Nulla ultricies felis a posuere interdum. Quisque interdum
  145. neque eu auctor ornare. Curabitur commodo, mi non tincidunt dignissim,
  146. ligula lectus ultrices elit, in consectetur ex augue a nibh. Aenean
  147. ornare enim pulvinar semper suscipit. Mauris facilisis nisl ligula,
  148. eget tempus nisl egestas eget. Aliquam erat volutpat. Aliquam aliquet
  149. fermentum tincidunt. Nulla lorem nisl, semper in imperdiet nec,
  150. gravida quis arcu. Duis euismod interdum odio, eget fermentum felis
  151. eleifend eu. Integer cursus posuere pulvinar.
  152. </p>
  153. <p>
  154. Ut porta dignissim mi ac rhoncus. Ut in turpis tortor. Quisque vel
  155. aliquet mi, sed elementum metus. Lorem ipsum dolor sit amet,
  156. consectetur adipiscing elit. Etiam mauris neque, vestibulum at posuere
  157. eu, dignissim nec nisl. Nullam ullamcorper ipsum et risus pretium, ac
  158. cursus mi efficitur. Aenean vel fringilla justo, nec dignissim justo.
  159. </p>
  160. <p>
  161. An unordered list:
  162. <ul>
  163. <li>asdf</li>
  164. <li>faeifs</li>
  165. <li>fewafwa</li>
  166. </ul>
  167. </p>
  168. <p>
  169. An ordered list:
  170. <ol>
  171. <li>asdf</li>
  172. <li>faeifs</li>
  173. <li>fewafwa</li>
  174. </ol>
  175. </p>
  176. <p>
  177. A description list:
  178. <dl>
  179. <dt>aasdaasdadssdf</dt>
  180. <dd>adaw dad wad wad wdwa daw dwaw dad wa</dd>
  181. <dt>faeidawdadsfs</dt>
  182. <dd>lijfes s fesi fhslkjadf wijasd<dd>
  183. <dt>fewafwadwwda</dt>
  184. <dd>fdwalidaw lidaw ijalw dwa ldwijald ja</dd>
  185. <dt>adaliwjd</dt>
  186. <dt>ijwefoiewf</dt>
  187. </dl>
  188. </p>
  189. <p>
  190. The following theorem, <a href="#basics">Theorem 1.1</a>, is of utmost
  191. importance. But don't forget the other stuff in the second <a
  192. href="example2.html">file<span class="referenceNbr"> [2]</span></a>.
  193. Namely <a href="example2.html#corolla">AA<span class="referenceNbr">
  194. [2, #corolla]</span></a> and for example <a
  195. href="example2.html#eq1">AA<span class="referenceNbr"> [2,
  196. #eq1]</span></a>, <a href="example2.html#cEquation">AA<span
  197. class="referenceNbr"> [2, #cEquation]</span></a> and <a
  198. href="example2.html#thm1">AA<span class="referenceNbr"> [2,
  199. #thm1]</span></a>. Here is a link pretending to be in this text: <a
  200. href="#nothere">link</a>.
  201. </p>
  202. <div class="theorem" title="Basic identities" id="basics">
  203. <span class="theorem">
  204. <a href="#basics">Theorem 1.1 <span class="theoremName">(Basic identities)</span>.</a>
  205. </span>
  206. <p>
  207. Let $\psi_\alpha$, $\alpha\in\N$, be Hermite functions. Then
  208. \[
  209. \psi_\alpha'(x) = \sqrt{\frac{\alpha}{2}}\psi_{\alpha-1}(x) -
  210. \sqrt{\frac{\alpha+1}{2}}\psi_{\alpha+1}(x)
  211. \]
  212. and
  213. \[
  214. x\psi_\alpha(x) = \sqrt{\frac{\alpha}{2}}\psi_{\alpha-1}(x) +
  215. \sqrt{\frac{\alpha+1}{2}}\psi_{\alpha+1}(x).
  216. \]
  217. Moreover the Hermite functions are eigenfunctions to the quantum
  218. mechanical oscillator
  219. \begin{equation}
  220. \label{eigenFunction}
  221. (x^2 - \partial_x^2) \psi_\alpha(x) = (2\alpha + 1)\psi_\alpha(x).
  222. \end{equation}
  223. </p>
  224. </div>
  225. </section>
  226. <section>
  227. <h2>Hermite transform</h2>
  228. <p>
  229. In this section we will first show that the Hermite functions form an
  230. orthonormal set in $L^2(\R)$. After that we will show that the set is
  231. complete and that leads to the Hermite transform.
  232. </p>
  233. <div class="lemma" id="orthonormalSequenceLemma">
  234. <span class="lemma">
  235. <a href="#orthonormalSequenceLemma">Lemma 2.1.</a>
  236. </span>
  237. <p>
  238. Let $\psi_\alpha:\R\to\R$ be Hermite functions. Then
  239. \[
  240. \int_{-\infty}^\infty \psi_\alpha(x)\psi_\beta(x) dx =
  241. \delta_{\alpha\beta}
  242. \]
  243. i.e. the sequence $(\psi_\alpha)_{\alpha=0}^\infty$ is orthonormal in
  244. $L^2(\R)$.
  245. </p>
  246. </div>
  247. <div class="proof">
  248. <span class="proof">Proof.</span>
  249. <p>
  250. Note first the identity $(x+\partial_x)(x-\partial_x) =
  251. (x^2-\partial_x^2) + 1$. Combine it with the fact that $\psi_n$ is
  252. an eigenfunction for the quantum oscillator \eqref{eigenFunction} to
  253. get
  254. \[
  255. (x+\partial_x)(x-\partial_x)\psi_n = 2(n+1)\psi_n
  256. \]
  257. for any $n\in\N$. Note also that the transpose of $x-\partial_x$ in
  258. the $L^2$ inner product is $x+\partial_x$.
  259. </p>
  260. <p>
  261. Hence we get
  262. \begin{align*}
  263. &\int \psi_\alpha \psi_\beta dx = \frac{1}{2\sqrt{\alpha\beta}}
  264. \int (x-\partial_x)\psi_{\alpha-1}(x+\partial_x)\psi_{\beta-1} dx
  265. \\ &\qquad = \frac{1}{2\sqrt{\alpha\beta}} \int \psi_{\alpha-1}
  266. (x+\partial_x)(x-\partial_x)\psi_{\beta-1} \\ &\qquad =
  267. \sqrt{\frac{\beta}{\alpha}} \int \psi_{\alpha-1}\psi_{\beta-1} dx.
  268. \end{align*}
  269. </p>
  270. <p>
  271. Using the previous equation we see that
  272. \[
  273. \int \psi_\alpha \psi_\alpha dx = \int \psi_0^2 dx =
  274. \pi^{-1/2}\int_{-\infty}^\infty e^{-x^2} dx = 1.
  275. \]
  276. </p>
  277. <p>
  278. If, on the other hand for example $\alpha\lt\beta$, then
  279. \[
  280. \int\psi_\alpha\psi_\beta dx = \ldots = c_{\alpha,\beta} \int
  281. \psi_0\psi_{\beta-\alpha} dx = c'_{\alpha,\beta}
  282. \int_{-\infty}^\infty \partial_x^{\beta-\alpha} e^{-x^2} dx = 0
  283. \]
  284. since all the derivatives of Gaussians vanish at infinity.
  285. </p>
  286. </div>
  287. <div class="corollary">
  288. <span class="corollary">Corollary 2.2.</span>
  289. <p>
  290. Let $\mathscr{O}=\{x, \partial_x\}$ be the set of operators containing
  291. the multiplication by $x$ and differentiation by $x$ operators. Let
  292. $L^0_\alpha = \{\psi_\alpha\}$ and
  293. \[
  294. L^{k+1}_\alpha = \{ Sf \mid S\in\mathscr{O}, \, f\in L^k_\alpha \}.
  295. \]
  296. Then
  297. \[
  298. \norm{f}_{L^2(\R)} \leq 2^{k/2} \prod_{\ell=1}^k \sqrt{\alpha+\ell}
  299. = \sqrt{ 2^k \frac{(a+k)!}{a!} }
  300. \]
  301. for any $f \in L^k_\alpha$.
  302. </p>
  303. </div>
  304. <div class="proof">
  305. <span class="proof">Proof.</span>
  306. <p>
  307. The sequence $(\psi_\alpha)_\alpha$ is orthonormal by <a
  308. href=#orthonormalSequenceLemma>Lemma 2.1</a>. Hence we have
  309. $\norm{\psi_\alpha}_2=1$. Assume that the claim is true for
  310. $L^k_\alpha$ for any $\alpha\in\N$. We will prove it for
  311. $L^{k+1}_\alpha$. Let that $f\in L^k_\alpha$. Then there are
  312. operators $S_1,\ldots,S_k \in \mathscr{O}$ such that $f =
  313. S_1S_2\cdots S_k \psi_\alpha$.
  314. </p>
  315. <p>
  316. Consider the case $S_k = \partial_x$. Let $O\in\mathscr{O}$ and use
  317. one of the basic identities to get
  318. \begin{align*}
  319. &\norm{Of}_2 = \norm{OS_1\cdots S_{k-1} \partial_x \psi_\alpha}_2
  320. \\ &\qquad = \norm{OS_1\cdots S_{k-1} \left(
  321. \sqrt{\tfrac{\alpha}{2}}\psi_{\alpha-1} -
  322. \sqrt{\tfrac{\alpha+1}{2}}\psi_{\alpha+1}\right)}_2 \\ &\qquad
  323. \leq \sqrt{\tfrac{\alpha}{2}} \norm{OS_1\cdots S_{k-1}
  324. \psi_{\alpha-1}}_2 + \sqrt{\tfrac{\alpha+1}{2}} \norm{OS_1\cdots
  325. S_{k-1} \psi_{\alpha+1}}_2 \\ &\qquad \leq
  326. \sqrt{\tfrac{\alpha+1}{2}} 2^{k/2} \left( \prod_{\ell=1}^k
  327. \sqrt{\alpha-1+\ell} + \prod_{\ell=1}^k \sqrt{\alpha+1+\ell}
  328. \right) \\ &\qquad \leq 2^{(k-1)/2} \sqrt{\alpha+1} \cdot 2
  329. \prod_{\ell=1}^k\sqrt{\alpha+1+\ell} \\ &\qquad \leq 2^{(k+1)/2}
  330. \prod_{\ell=1}^{k+1} \sqrt{\alpha+\ell}
  331. \end{align*}
  332. since $OS_1\cdots S_{k-1} \psi_{\alpha-1}$ and $OS_1\cdots S_{k-1}
  333. \psi_{\alpha+1}$ are in $L^k_{\alpha-1}$ and $L^k_{\alpha+1}$,
  334. respectively. The same kind of deduction works in the case when
  335. $S_k$ is multiplication by $x$. Hence the claim follows by
  336. induction.
  337. </p>
  338. </div>
  339. </section>
  340. <footer id="bibliography">
  341. <h2>References</h2>
  342. <h3>Offline sources</h3>
  343. <ul class="referenceList">
  344. <li id="articleTry" class="reference">
  345. <span class="ref-nbr">[1]</span>
  346. <span class="ref-authors">E. Blåsten</span> and
  347. <span class="ref-authors">M. Meikäläinen:</span>
  348. <span class="ref-title">A proof of the Riemann conjecture,</span>
  349. <span class="ref-journal">Annals of Mathematics,</span>
  350. <span class="ref-volume">23,</span>
  351. <span class="ref-issue">2</span>
  352. <span class="ref-year">(2069),</span>
  353. <span class="ref-pages">1&ndash;503.</span>
  354. </li>
  355. </ul>
  356. <h3>Links</h3>
  357. <ul class="referenceList">
  358. <li class="reference">
  359. <span class="referenceNbr">[2]</span>
  360. <a href="example2.html" class="addTxtForbidden">example2.html</a>
  361. </li>
  362. <li class="reference">
  363. <span class="referenceNbr">[3]</span>
  364. <a href="http://www.gnu.org" class="addTxtForbidden">http://www.gnu.org</a>
  365. </li>
  366. </ul>
  367. </footer>
  368. </body>