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 National Science
Foundation Award #0630818 |
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Problems in Harmonic Analysis |
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| Investigator(s): |
Xiaochun Li (PI)
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| Sponsor: |
University of Illinois at Urbana-Champaign, IL 61820 2173332187
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| Start Date/Expiration Date |
2006-03-01 to 2006-05-31 (amended 2006-05-03) |
| Awarded Amount to Date: |
$15,902 |
| Abstract: Proposal Number: DMS-0140376
PI: Xiaochun Li
ABSTRACT
Research will be conducted on a variety of problems in
harmonic analysis arising in the study of multilinea
singular integral operators. These problems are related,
but not limited, to the study of the Carleson operator
and the bilinear Hilbert transform. Recently a significant
breakthrough on the bilinear Hilbert transform was made
by the inspiring work of Lacey and Thiele. It turns out
that the time-frequency analysis is a powerful tool to
solve problems related to the study of multilinear
operators. Moreover, although this is still under
investigation, the analysis of these problems gives us
hopes to solve other important and difficult problems in
the field, such as the Hilbert transform along vector fields,
the Kakeya problem, and Carleson's maximal operator of
the partial sums of Fourier series in two dimensions.
It will also be very interesting to see how
to use this delicate analysis to solve some problems in
other fields such as partial differential equations,
number theory, etc. Actually, multilinear operators
have been used in the study of partial differential
equations, since they naturally appear in series expansions
of solutions of many equations.
Harmonic analysis is not only an area of theoretical
mathematics, but also an applicable area lying at the
heart of the intersection of fields as diverse as optics,
signal processing, meteorology, and music. The main them
in Harmonic analysis is about disassembling and assembling
complicated objects into simpler well-understood pieces,
by analogy to decomposition of intricate musical pieces
into arrangements of a few basic notes. In signal processing,
harmonic analysis is used to detect irregularities of
signals and images. The Fouriertransform is a very useful
tool to locate these irregularities. The appearance of a
nonsmooth symbol inthe study of multiplier problems is
analogous to physical phenomena where the frequencies of
signals are altered by an abrupt operation, such as the
interruption of radio communication or television
transmission by meteorological phenomena. Such a sudden
and unexpected operation causes the loss of information.
To avoid the loss of information or to retrieve the
original data is the main topic of the theoretical research
proposed here. |
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| NSF Org: |
DMS - Division of Mathematical Sciences |
| Award Number: |
0630818 |
| Award Instrument: |
Standard Grant |
| Program Manager: |
Joe W. Jenkins
DMS Division of Mathematical Sciences
MPS Directorate for Mathematical & Physical Sciences
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| NSF Program(s): |
ANALYSIS PROGRAM |
| Field Application(s): |
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| Program Reference Code(s): |
UNASSIGNED, 0000 |
| Program Element Code(s): |
1281 |
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