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Keywords
(7)
Differential Equation
Fourier Series
Harmonic Measure
Hermite Polynomial
Integral Transforms
poisson summation formula
Sampling Theorem
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Chirp transforms and Chirp series
Chirp transforms and Chirp series,10.1016/j.jmaa.2010.07.037,Journal of Mathematical Analysis and Applications,Guangbin Ren,Qiuhui Chen,Paula Cerejeir
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Chirp transforms and Chirp series
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Guangbin Ren
,
Qiuhui Chen
,
Paula Cerejeiras
,
Uwe Kaehle
Motivated by the recent work on the nonharmonic Fourier atoms initiated by T. Qian and the nonharmonic
Fourier series
which originated from the celebrated work of Paley and Wiener, we introduce an integral version of the nonharmonic Fourier series, called Chirp transform. As an integral transform with kernel eiϕ(t)θ(ω), the Chirp transform is an unitary isometry from L2(R,dϕ) onto L2(R,dθ) and it can be explicitly defined in terms of generalized Hermite polynomials. The corresponding Chirp series take einθ(t) as a basis which in some sense is dual to the theory of nonharmonic
Fourier series
which take eiλnt as a basis. The Chirp version of the Shannon
sampling theorem
and the
Poisson summation formula
are also considered by dealing with sampling points which may nonequally distributed. Since the Chirp transform interchanges weighted derivatives into multiplications, it plays a role in solving certain differential equations with variable coefficients. In addition, we extend T. Qian's theorem on the characterization of a measure to be a linear combination of a number of harmonic measures on the unit disc with positive integer coefficients to that with positive rational coefficients.
Journal:
Journal of Mathematical Analysis and Applications  J MATH ANAL APPL
, vol. 373, no. 2, pp. 356369, 2011
DOI:
10.1016/j.jmaa.2010.07.037
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References
(8)
Nontrivial harmonic waves with positive instantaneous frequency
(
Citations: 4
)
Yingxiong Fu
,
Luoqing Li
Journal:
Nonlinear Analysistheory Methods & Applications  NONLINEAR ANALTHEOR METH APP
, vol. 68, no. 8, pp. 24312444, 2008
Characterization of Boundary Values of Functions in Hardy Spaces with Applications in Signal Analysis
(
Citations: 18
)
Tao Qian
Journal:
Journal of Integral Equations and Applications  J INTEGRAL EQU APPL
, vol. 17, no. 2005, pp. 159198, 2005
Analytic signals and harmonic measures
(
Citations: 10
)
Tao Qian
Journal:
Journal of Mathematical Analysis and Applications  J MATH ANAL APPL
, vol. 314, no. 2, pp. 526536, 2006
Monocomponents for decomposition of signals
(
Citations: 17
)
Tao Qian
Journal:
Mathematical Methods in The Applied Sciences  MATH METH APPL SCI
, vol. 29, no. 10, pp. 11871198, 2006
Analytic unit quadrature signals with nonlinear phase
(
Citations: 23
)
Tao Qian
,
Qiuhui Chen
,
Luoqing Li
Journal:
Physica Dnonlinear Phenomena  PHYSICA D
, vol. 203, no. 1, pp. 8087, 2005
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Citations
(1)
Influence of lubricants on plain bearing performance: Evaluation of bearing performance with polymercontaining oils
Moritsugu Kasai
,
Michel Fillon
,
Jean Bouyer
,
Sebastien Jarny
Journal:
Tribology International  TRIBOL INT