## Abstract

We study decomposition of functions in the Hardy space \(H^2(\mathbb{D} )\) into linear combinations of the basic functions (modified Blaschke products) in the system

where the points *a*
_{
n
}’s in the unit disc \(\mathbb{D}\) are adaptively chosen in relation to the function to be decomposed. The chosen points *a*
_{
n
}’s do not necessarily satisfy the usually assumed hyperbolic non-separability condition

in the traditional studies of the system. Under the proposed procedure functions are decomposed into their intrinsic components of successively increasing non-negative analytic instantaneous frequencies, whilst fast convergence is resumed. The algorithm is considered as a variation and realization of greedy algorithm.

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Communicated by Yuesheng Xu.

The work was supported by Macao FDCT 014/2008/A1 and research grant of the University of Macau No. RG-UL/07-08s/Y1/QT/FSTR.

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Qian, T., Wang, YB. Adaptive Fourier series—a variation of greedy algorithm.
*Adv Comput Math* **34**, 279–293 (2011). https://doi.org/10.1007/s10444-010-9153-4

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DOI: https://doi.org/10.1007/s10444-010-9153-4

### Keywords

- Rational orthonormal system
- Blaschke product
- Complex hardy space
- Analytic signal
- Instantaneous frequency
- Mono-components
- Adaptive decomposition of functions
- Greedy algorithm