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 L 2 (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.
In recent study adaptive decomposition of functions into basic functions of analytic instantaneous frequencies has been sought. Fourier series is a particular case of such decomposition. Adaptivity addresses certain optimal property of the decomposition. The present paper presents a fast decomposition of functions in the L 2 (∂D) spaces into a series of inner and weighted inner functions of increasing frequencies.Keywords Fourier series · Inner and outer functions · Hardy space · The Nevanlinna factorization theorem · Blaschke product · Analytic signal · Instantaneous frequency and amplitude · Mono-components · Adaptive decomposition of functions Mathematics Subject Classification (2000) 42A50 · 32A30 · 32A35 · 46J15
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