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BEGIN:VEVENT
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:In this talk, I will discuss eigenvalue estimates for operators related to 
 simultaneous localization in space and frequency. This problem is connected
  to phase-space localization, the uncertainty principle, and spectral conce
 ntration. It also appears naturally in harmonic analysis, signal representa
 tion, approximation theory, and mathematical physics.The main objects are s
 pace-frequency limiting operators. Their eigenvalues measure how well a ban
 dlimited function can be concentrated on a given spatial region. Usually, m
 ost eigenvalues are close to either 0 or 1, and only a smaller number are i
 n the middle transition region, which is often called the plunge region. Es
 timating this region gives quantitative information about the effective num
 ber of degrees of freedom. This is similar in spirit to counting states in 
 phase space.The method is based on a wave packet decomposition adapted to t
 he geometry of the spatial and frequency domains. These wave packets are co
 nstructed using smooth Gevrey cutoffs. This gives strong Fourier localizati
 on and also useful spatial control. The wave packets behave like approximat
 e eigenfunctions for the localization operator.As a result, we obtain expli
 cit bounds on the number of eigenvalues in (ε,1−ε). I will explain how phas
 e-space localization and geometric decomposition can lead to quantitative s
 pectral estimates in higher-dimensional settings.
X-ALT-DESC;FMTTYPE=text/html:<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:20260823T042651
DTSTART;TZID=America/New_York:20260430T110000
DTEND;TZID=America/New_York:20260430T120000
SEQUENCE:0
TRANSP:OPAQUE
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