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## Optimal Approximation by Blaschke Forms and Rational Functions ∗ (2011)

Citations: | 1 - 1 self |

### Citations

982 |
Orthogonal Polynomials
- Szego
- 1975
(Show Context)
Citation Context ... remaining energy representation under an n-Blaschke form will be needed. We will adopt the approach developed in [15] and [14]. A Technical alternative is the classical ”Cristoffel-Darboux formula” (=-=[18]-=-). Let F ∈ H2 and F1 = F. Under the usual inner 5 product 〈F,G〉 = ∫ 2pi 0 F (eit)G(eit)dt, we have F (z) = F1(z) = ( F1(z)− 〈F1, e{b1}〉e{b1}(z) ) + 〈F1, e{b1}〉e{b1}(z) = F2(z) z − b1 1− b1z + 〈F1, B{b... |

711 |
Bounded analytic functions
- Garnett
- 2007
(Show Context)
Citation Context ...− 2 )‖RlI‖ − ‖RlI‖ ∑ k∈B ‖Pr ∗B{bl1,...,blk}‖, if r is sufficiently close to 1. Let r be fixed from now on in the rest of the proof. Now, since B{bl1,...,blk} ∈ H ∞(D), we have (Corollary 3.2, p58, =-=[7]-=-) Pr ∗B{bl1,...,blk}(e it) = B{bl1,...,blk}(re it), and thus ‖Pr ∗B{bl1,...,blk}‖ 2 = ‖B{bl1,...,blk}(re i(·))‖2 ≤ 1 2pi ∫ 2pi 0 1− |blk|2 |1− blkreit|2 dt = 1− |blk|2 1− r2|blk|2 . For the fixed r, s... |

256 |
Interpolation and Approximation by Rational Functions
- WALSH
(Show Context)
Citation Context ...ial fraction minimizer of F. Below H2(D) will be abbreviated as H2. The existence and algorithm for best rational approximation to H2-functions is a classical issue. The early references include [6], =-=[20]-=-, [8], [5], [3], [2]. Although the existence has been proved to exist ([20]), from the constructive point of view no practical algorithm for the minimizers have been established. The proofs of our the... |

65 | On instantaneous amplitude and phase of signals,”
- Picinbono
- 1997
(Show Context)
Citation Context ...ent study. Our study on adaptive mono-component decomposition was motivated by the engineering algorithm Empirical Mode Decomposition (EMD) ([9]), as well as some related studies on analytic signals (=-=[10]-=-, [4]). We call a complex-valued function F in Lp(∂D), 1 ≤ p ≤ ∞, a complex-mono-component if with F (eit) = ρ(t)eiθ(t), there hold (i) ρ ≥ 0; (ii) H(ρeiθ) = −iρeiθ, where H is the Hilbert transformat... |

58 |
Time-Frequency Analysis: Theory and Applications
- Cohen
- 1995
(Show Context)
Citation Context ...udy. Our study on adaptive mono-component decomposition was motivated by the engineering algorithm Empirical Mode Decomposition (EMD) ([9]), as well as some related studies on analytic signals ([10], =-=[4]-=-). We call a complex-valued function F in Lp(∂D), 1 ≤ p ≤ ∞, a complex-mono-component if with F (eit) = ρ(t)eiθ(t), there hold (i) ρ ≥ 0; (ii) H(ρeiθ) = −iρeiθ, where H is the Hilbert transformation o... |

15 |
Mono-components for decomposition of signals,”
- Qian
- 2006
(Show Context)
Citation Context ...ytic signal F + iHF is a complex mono-component, then F itself is called a real-mono-component. Being of no ambiguity we use the terminology mono-component for both complex- and real-mono-components (=-=[12]-=-). A function F (eit) is called a pre-complex-mono-component if eiMtF (eit) is a complex moo-component for some M > 0. If F is a real-valued and if its associated analytic signal is a pre-complex-mono... |

14 | Asymptotic uniqueness of best rational approximations to complex Cauchy transforms
- Baratchart, Yattselev
- 2010
(Show Context)
Citation Context ...imizer of F. Below H2(D) will be abbreviated as H2. The existence and algorithm for best rational approximation to H2-functions is a classical issue. The early references include [6], [20], [8], [5], =-=[3]-=-, [2]. Although the existence has been proved to exist ([20]), from the constructive point of view no practical algorithm for the minimizers have been established. The proofs of our theorems, in turn,... |

14 |
Intrinsic mono-component decomposition of functions: An advance of Fourier theory
- Qian
- 1002
(Show Context)
Citation Context ... in any of the mentioned Banach spaces implies the condition (1.2) ([1]). There are parallel theories in the region outside the unit disc, and in the upper- and lower-half complex planes. In [15] and =-=[14]-=- we propose an adaptive algorithm of which the points ak defining the system are consecutively chosen according to the function F ∈ H2(D) to be decomposed. The selections of each ak is based on the ma... |

13 |
and Z.Zhang, “Matching pursuits with time-frequency dictionaries,” Signal Processing
- Mallat
- 1993
(Show Context)
Citation Context ...efining the system are consecutively chosen according to the function F ∈ H2(D) to be decomposed. The selections of each ak is based on the maximal projection principle in spirit of greedy algorithm (=-=[11]-=-). The consecutive selection may not gives rise to the best approximation (See (v), (vi) below), but has a simple algorithm ([15], [14]). The formed system {Bk}∞k=1 using the selected ak’s may not be ... |

9 |
The distribution of poles of rational functions of best approximation, and related
- Levin
- 1969
(Show Context)
Citation Context ...action minimizer of F. Below H2(D) will be abbreviated as H2. The existence and algorithm for best rational approximation to H2-functions is a classical issue. The early references include [6], [20], =-=[8]-=-, [5], [3], [2]. Although the existence has been proved to exist ([20]), from the constructive point of view no practical algorithm for the minimizers have been established. The proofs of our theorems... |

8 |
Boundary derivatives of the phases of inner and outer functions and applications
- Qian
(Show Context)
Citation Context ... (eit) = ρ(t)eiθ(t), there hold (i) ρ ≥ 0; (ii) H(ρeiθ) = −iρeiθ, where H is the Hilbert transformation of the context; and (iii) θ′ ≥ 0. Note that phase derivative θ′ has to be suitably interpreted (=-=[13]-=-), and if and only if θ′ ≥ 0, then θ′ is called instantaneous frequency. If F is a real-valued signal and its (associated) analytic signal F + iHF is a complex mono-component, then F itself is called ... |

8 |
Rational orthogonal bases satisfying the Bedrosian identity,”
- Tan, Shen, et al.
- 2010
(Show Context)
Citation Context ...e study of signals of minimum phase and all-pass filters. Finding out non-unimodular mono-components from the unimodular ones leads to a trend of recent studies on the Bedrosian identity ([17], [16], =-=[19]-=-, [22], [21]). Starlike and p-starlike functions give rise to more general mono-components that do not rely on the unimodular types, nor on the Bedrosian identity. The modified Blaschke products studi... |

7 |
On the best approximation of analytic functions by rational functions with free poles, Dokl. Akad. Nauk SSSR
- Erohin
- 1959
(Show Context)
Citation Context ...-partial fraction minimizer of F. Below H2(D) will be abbreviated as H2. The existence and algorithm for best rational approximation to H2-functions is a classical issue. The early references include =-=[6]-=-, [20], [8], [5], [3], [2]. Although the existence has been proved to exist ([20]), from the constructive point of view no practical algorithm for the minimizers have been established. The proofs of o... |

6 |
Contribution h l'approximation de fonctions de la variable complex au sens de Hermite-Pad6 et de Hardy, PhD thesis, l'Universit~ Scientifique t Mtdicale de Grenoble
- Dora
- 1980
(Show Context)
Citation Context ...n minimizer of F. Below H2(D) will be abbreviated as H2. The existence and algorithm for best rational approximation to H2-functions is a classical issue. The early references include [6], [20], [8], =-=[5]-=-, [3], [2]. Although the existence has been proved to exist ([20]), from the constructive point of view no practical algorithm for the minimizers have been established. The proofs of our theorems, in ... |

6 | Orthonormal bases with nonlinear phases,”
- Qian, Wang, et al.
- 2010
(Show Context)
Citation Context ... to the study of signals of minimum phase and all-pass filters. Finding out non-unimodular mono-components from the unimodular ones leads to a trend of recent studies on the Bedrosian identity ([17], =-=[16]-=-, [19], [22], [21]). Starlike and p-starlike functions give rise to more general mono-components that do not rely on the unimodular types, nor on the Bedrosian identity. The modified Blaschke products... |

5 |
The Bedrosian identity for Hp functions
- Yang, Zhang
- 2008
(Show Context)
Citation Context ...y of signals of minimum phase and all-pass filters. Finding out non-unimodular mono-components from the unimodular ones leads to a trend of recent studies on the Bedrosian identity ([17], [16], [19], =-=[22]-=-, [21]). Starlike and p-starlike functions give rise to more general mono-components that do not rely on the unimodular types, nor on the Bedrosian identity. The modified Blaschke products studied in ... |

4 |
On a rational approximation problem
- Baratchart, Olivi, et al.
(Show Context)
Citation Context ...r of F. Below H2(D) will be abbreviated as H2. The existence and algorithm for best rational approximation to H2-functions is a classical issue. The early references include [6], [20], [8], [5], [3], =-=[2]-=-. Although the existence has been proved to exist ([20]), from the constructive point of view no practical algorithm for the minimizers have been established. The proofs of our theorems, in turn, prov... |

3 |
and D.Y.Yan, The Bedrosian identity fot the Hilbert transform of product functions
- Xu
(Show Context)
Citation Context ...ignals of minimum phase and all-pass filters. Finding out non-unimodular mono-components from the unimodular ones leads to a trend of recent studies on the Bedrosian identity ([17], [16], [19], [22], =-=[21]-=-). Starlike and p-starlike functions give rise to more general mono-components that do not rely on the unimodular types, nor on the Bedrosian identity. The modified Blaschke products studied in this p... |

2 | Adaptive Decomposition Into Basic Signals of Non-negative Instantaneous Frequencies - A Variation and Realization of Greedy Algorithm - Qian, Wang |

1 |
Huang at al, The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time series analysis
- E
- 1998
(Show Context)
Citation Context ...nents (see below for definition) that we merge to this current study. Our study on adaptive mono-component decomposition was motivated by the engineering algorithm Empirical Mode Decomposition (EMD) (=-=[9]-=-), as well as some related studies on analytic signals ([10], [4]). We call a complex-valued function F in Lp(∂D), 1 ≤ p ≤ ∞, a complex-mono-component if with F (eit) = ρ(t)eiθ(t), there hold (i) ρ ≥ ... |

1 |
D.Y.Yan, L.X.Yan and B.Yu,Fourier Spectrum Charaterization of Hardy Spaces and Appliacations
- Qian
(Show Context)
Citation Context ...impact to the study of signals of minimum phase and all-pass filters. Finding out non-unimodular mono-components from the unimodular ones leads to a trend of recent studies on the Bedrosian identity (=-=[17]-=-, [16], [19], [22], [21]). Starlike and p-starlike functions give rise to more general mono-components that do not rely on the unimodular types, nor on the Bedrosian identity. The modified Blaschke pr... |