## Abstract

Since the last decade, motivated by attempts of positive frequency decomposition of signals, complex periodic functions \(s(e^{it})=\rho (t)e^{i\theta (t)}\) satisfying the conditions

have been sought, where *H* is the circular Hilbert transform and the phase derivative \(\theta '(t)\) is suitably defined and interpreted as instantaneous frequency of the signal \(\rho (t)\cos \theta (t)\). Functions satisfying the above conditions are called mono-components. Mono-components have been found to form a large pool and used to decompose and analyze signals. This note in a great extent concludes the study of seeking for mono-components through characterizing two classes of mono-components of which one is phrased as the Blaschke type and the other the starlike type. The Blaschke type mono-components are of the form \(\rho (t)\cos \theta (t)\), where \(\rho (t)\) is a real-valued (generalized) amplitude functions and \(e^{i\theta (t)}\) is the boundary limit of a finite or infinite Blaschke product. For the starlike type mono-components, we assume the condition \(\int _{0}^{2\pi }\theta '(t)dt=n\pi \), where *n* is a positive integer. It shows that such class of mono-components is identical with the class consisting of products between *p*-starlike and boundary \((n-2p) \)-starlike functions. The results of this paper explore connections between harmonic analysis, complex analysis, and signal analysis.

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## References

Kumaresan, R., Rao, A.: Model-based approach to envelope and positive instantaneous frequency estimation of signals with speech applications. J. Acoust. Soc. Am.

**105**(3), 1912–1924 (1999)Xia, X.G., Cohen, L.: On analytic signals with nonnegative instantaneous frequency. IEEE Int. Conf. Acoust. Speech Signal Process.

**3**, 1329–1332 (1999)Doroslovac̆ki, M.I.: On nontrival analytic signals with positive instantaneous frequency. Signal Process.

**83**(3), 655–658 (2003)Qian, T., Wang, Y.B., Dang, P.: Adaptive decomposition into mono-components. Adv. Adapt. Data Anal.

**01**(04), 703–709 (2009)Qian, T., Wang, R., Xu, Y., Zhang, H.: Orthonormal bases with nonlinear phases. Adv. Comput. Math.

**33**(1), 75–95 (2010)Qian, T., Wang, Y.B.: Adaptive decomposition into basic signals of non-negative instantaneous frequencies—a variation and realization of greedy algorithm. Adv. Comput. Math.

**34**(3), 279–293 (2011)Tan, L.H., Yang, L.H., Huang, D.R.: The structure of instantaneous frequencies of periodic analytic signals. Sci. China Math.

**53**(2), 347–355 (2010)Qian, T.: Characterization of boundary values of functions in Hardy spaces with application in signal analysis. J. Integral Equ. Appl.

**17**(2), 159–198 (2005)Qian, T.: Mono-components for decomposition of signals. Math. Method Appl. Sci.

**29**, 1187–1198 (2006)Qian, T.: Phase derivatives of Nevanlinna functions and applications. Math. Method Appl. Sci.

**32**, 253–263 (2009)Picinbono, B.: On instantaneous amplitude and phase of signals. IEEE Trans. Signal Process.

**45**(3), 552–560 (1997)Xu, Y.S., Yan, D.Y.: The Bedrosian identity fot the Hilbert transform of product functions. Proc. Am. Math. Soc.

**134**, 2719–2728 (2006)Yu, B., Zhang, H.Z.: The Bedrosian identity and homoge- neous semi-convolution equations. J. Integral Equ. Appl.

**20**(4), 527–568 (2008)Tan, L.H., Yang, L.H., Huang, D.R.: Construction of periodic analytic signals satisfying the circular Bedrosian identity. IMA J. Appl. Math.

**75**, 246–256 (2010)Bultheel, A.: Orthogonal Rational Functions. Cambridge University Press, Cambridge (1999)

Cima, J., Ross, W.: The Backward Shift on the Hardy Space. American Mathematical Society, Providence (2000)

Garnett, J.B.: Bounded Analytic Function. Academic Press, New York (1987)

Shen, X.C.: Complex Approximation. Science Press, Beijing (1991). (Chinese Version)

Qian, T., Tan, L.H.: Backard shift invariant subspace with applications to band preserving and phase retrieval problem. Math. Methods Appl. Sci (2015). doi:10.1002/mma.3591

Qian, T., Chen, Q.H., Tan, L.H.: Rational orthogonal systems are Schauder bases. Complex Var. Elliptic Equ.

**59**(6), 841–846 (2014)Hummel, J.A.: Multivalent starlike function. J. d’Analyse Math.

**18**, 133–160 (1967)Robertson, M.S.: Univalent functions starlike with respect to a boundary point. J. Math. Anal. Appl.

**81**, 327–345 (1981)Lyzzaik, A.: On a conjecture of M.S. Robertson. Proc. Am. Math. Soc.

**91**, 108–110 (1984)Lecko, A.: On the class of functions starlike with respect to a boundary point. J. Math. Anal. Appl.

**261**, 649–664 (2001)Lecko, A., Lyzzaik, A.: A note on univalent functions starlike with respect to a boundary point. J. Math. Anal. Appl.

**282**, 846–851 (2003)

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Communicated by Irene Sabadini.

T. Qian supported by Research Grant of University of Macau MYRG116(Y1-L3)-FST13-QT, Macao Science and Technology Fund FDCT/098/2012/A3.

L. Tan supported by NSFC (61471132) and Cultivation Program for Outstanding Young College Teachers (Yq2014060) of Guangdong Province.

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Qian, T., Tan, L. Characterizations of Mono-Components: the Blaschke and Starlike Types.
*Complex Anal. Oper. Theory* **12**, 1383–1399 (2018). https://doi.org/10.1007/s11785-015-0491-6

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DOI: https://doi.org/10.1007/s11785-015-0491-6