Rational orthogonal bases satisfying the Bedrosian identity

Lihui Tan; Lixin Shen; Lihua Yang

doi:10.1007/s10444-009-9133-8

Rational orthogonal bases satisfying the Bedrosian identity

Lihui Tan, Lixin Shen, Lihua Yang

Research output: Contribution to journal › Article › peer-review

25 Scopus citations

Abstract

We develop a necessary and sufficient condition for the Bedrosian identity in terms of the boundary values of functions in the Hardy spaces. This condition allows us to construct a family of functions such that each of which has non-negative instantaneous frequency and is the product of two functions satisfying the Bedrosian identity. We then provide an efficient way to construct orthogonal bases of L²(ℝ) directly from this family. Moreover, the linear span of the constructed basis is norm dense in L^p(ℝ), 1 < p < ∞. Finally, a concrete example of the constructed basis is presented.

Original language	English (US)
Pages (from-to)	285-303
Number of pages	19
Journal	Advances in Computational Mathematics
Volume	33
Issue number	3
DOIs	https://doi.org/10.1007/s10444-009-9133-8
State	Published - 2010

Keywords

Analytic signal
Bedrosian identity
Hardy space
Hilbert transform
Positive instantaneous frequency

ASJC Scopus subject areas

Computational Mathematics
Applied Mathematics

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10.1007/s10444-009-9133-8

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title = "Rational orthogonal bases satisfying the Bedrosian identity",

abstract = "We develop a necessary and sufficient condition for the Bedrosian identity in terms of the boundary values of functions in the Hardy spaces. This condition allows us to construct a family of functions such that each of which has non-negative instantaneous frequency and is the product of two functions satisfying the Bedrosian identity. We then provide an efficient way to construct orthogonal bases of L2(ℝ) directly from this family. Moreover, the linear span of the constructed basis is norm dense in Lp(ℝ), 1 < p < ∞. Finally, a concrete example of the constructed basis is presented.",

keywords = "Analytic signal, Bedrosian identity, Hardy space, Hilbert transform, Positive instantaneous frequency",

author = "Lihui Tan and Lixin Shen and Lihua Yang",

note = "Funding Information: Supported by National Science Foundation of China under grants 60475042 and 10631080.",

year = "2010",

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T1 - Rational orthogonal bases satisfying the Bedrosian identity

AU - Tan, Lihui

AU - Shen, Lixin

AU - Yang, Lihua

N1 - Funding Information: Supported by National Science Foundation of China under grants 60475042 and 10631080.

PY - 2010

Y1 - 2010

N2 - We develop a necessary and sufficient condition for the Bedrosian identity in terms of the boundary values of functions in the Hardy spaces. This condition allows us to construct a family of functions such that each of which has non-negative instantaneous frequency and is the product of two functions satisfying the Bedrosian identity. We then provide an efficient way to construct orthogonal bases of L2(ℝ) directly from this family. Moreover, the linear span of the constructed basis is norm dense in Lp(ℝ), 1 < p < ∞. Finally, a concrete example of the constructed basis is presented.

AB - We develop a necessary and sufficient condition for the Bedrosian identity in terms of the boundary values of functions in the Hardy spaces. This condition allows us to construct a family of functions such that each of which has non-negative instantaneous frequency and is the product of two functions satisfying the Bedrosian identity. We then provide an efficient way to construct orthogonal bases of L2(ℝ) directly from this family. Moreover, the linear span of the constructed basis is norm dense in Lp(ℝ), 1 < p < ∞. Finally, a concrete example of the constructed basis is presented.

KW - Analytic signal

KW - Bedrosian identity

KW - Hardy space

KW - Hilbert transform

KW - Positive instantaneous frequency

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JO - Advances in Computational Mathematics

JF - Advances in Computational Mathematics

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ER -

Rational orthogonal bases satisfying the Bedrosian identity

Abstract

Keywords

ASJC Scopus subject areas

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