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UID:c4fb90154ea02090bcb0f9da81d9ece3
CATEGORIES:Applied and Computational Math Seminar
CREATED:20260427T005807
SUMMARY:Wave packet methods for quantitative spectral estimates in space-frequency localization
LOCATION:Hill 705
DESCRIPTION:<div data-olk-copy-source="MessageBody" style="border: 0px; font-style: nor
 mal; font-weight: 400; font-size: 12pt; line-height: inherit; font-family: 
 Aptos, Aptos_EmbeddedFont, Aptos_MSFontService, Calibri, Helvetica, sans-se
 rif, serif, EmojiFont; margin: 1em 0px; padding: 0px; vertical-align: basel
 ine; letter-spacing: normal; orphans: 2; text-align: start; text-indent: 0p
 x; text-transform: none; widows: 2; word-spacing: 0px; white-space: normal;
  background-color: #ffffff;">In this talk, I will discuss eigenvalue estima
 tes for operators related to simultaneous localization in space and frequen
 cy. This problem is connected to phase-space localization, the uncertainty 
 principle, and spectral concentration. It also appears naturally in harmoni
 c analysis, signal representation, approximation theory, and mathematical p
 hysics.</div><div style="border: 0px; font-style: normal; font-weight: 400;
  font-size: 12pt; line-height: inherit; font-family: Aptos, Aptos_EmbeddedF
 ont, Aptos_MSFontService, Calibri, Helvetica, sans-serif, serif, EmojiFont;
  margin: 1em 0px; padding: 0px; vertical-align: baseline; letter-spacing: n
 ormal; orphans: 2; text-align: start; text-indent: 0px; text-transform: non
 e; widows: 2; word-spacing: 0px; white-space: normal; background-color: #ff
 ffff;">The main objects are space-frequency limiting operators. Their eigen
 values measure how well a bandlimited function can be concentrated on a giv
 en spatial region. Usually, most eigenvalues are close to either 0 or 1, an
 d only a smaller number are in the middle transition region, which is often
  called the<span>&nbsp;</span><i>plunge region</i>. Estimating this region 
 gives quantitative information about the effective number of degrees of fre
 edom. This is similar in spirit to counting states in phase space.</div><di
 v style="border: 0px; font-style: normal; font-weight: 400; font-size: 12pt
 ; line-height: inherit; font-family: Aptos, Aptos_EmbeddedFont, Aptos_MSFon
 tService, Calibri, Helvetica, sans-serif, serif, EmojiFont; margin: 1em 0px
 ; padding: 0px; vertical-align: baseline; letter-spacing: normal; orphans: 
 2; text-align: start; text-indent: 0px; text-transform: none; widows: 2; wo
 rd-spacing: 0px; white-space: normal; background-color: #ffffff;">The metho
 d is based on a wave packet decomposition adapted to the geometry of the sp
 atial and frequency domains. These wave packets are constructed using smoot
 h Gevrey cutoffs. This gives strong Fourier localization and also useful sp
 atial control. The wave packets behave like approximate eigenfunctions for 
 the localization operator.</div><div style="border: 0px; font-style: normal
 ; font-weight: 400; font-size: 12pt; line-height: inherit; font-family: Apt
 os, Aptos_EmbeddedFont, Aptos_MSFontService, Calibri, Helvetica, sans-serif
 , serif, EmojiFont; margin: 1em 0px; padding: 0px; vertical-align: baseline
 ; letter-spacing: normal; orphans: 2; text-align: start; text-indent: 0px; 
 text-transform: none; widows: 2; word-spacing: 0px; white-space: normal; ba
 ckground-color: #ffffff;">As a result, we obtain explicit bounds on the num
 ber of eigenvalues in (ε,1−ε). I will explain how phase-space localization 
 and geometric decomposition can lead to quantitative spectral estimates in 
 higher-dimensional settings.</div>
CONTACT:Azita Mayeli (City University of New York)
DTSTAMP:20260826T125514
DTSTART;TZID=America/New_York:20260430T110000
DTEND;TZID=America/New_York:20260430T120000
SEQUENCE:0
TRANSP:OPAQUE
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