### Citations

256 |
Interpolation and Approximation by Rational Functions
- WALSH
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Citation Context ...tion In the one-dimensional cases by functional decomposition of the Fourier type we refer to approximations by the rational orthogonal systems, or alternatively, the Takenaka-Malmquist (TM) systems (=-=[21]-=-). The real-line case and the unit circle case are analogous. In the latter, a TM system is a collection of consecutively parameterized rational functions Bk(z) = √ 1− |ak|2 1− akz k−1∏ l=1 z − al 1− ... |

14 |
Intrinsic mono-component decomposition of functions: An advance of Fourier theory
- Qian
- 1002
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Citation Context ...eters are selected consecutively according to the given signal. The fast decomposition is based on a maximal selection principle (see §1) together with a generalization of backward shift operator. In =-=[9]-=- the author propose an improvement of AFD, or Core AFD, called Unwending AFD. It incorporates at each recursive step a factorization process based on Nevanlinna’s Factorization Theorem. Not only Unwen... |

12 |
Identification and rational L2 approximation: a gradient algorithm
- Baratchart, Cardelli, et al.
- 1991
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Citation Context ...when there is only one critical point for the problem ([10]). We call such a solution a conditional solution. Compared with the existing RARL2 algorithm, that offers also a conditional solution ([1], =-=[2]-=-, [3]), Cyclic AFD is more explicit, and can directly find the poles of the approximating rational function. We will call n∑ k=1 ckBk(z) an n-Blaschke form where the parameters a1, ..., an of Bn are a... |

11 |
Simultaneous rational approximants
- Stahl
- 1995
(Show Context)
Citation Context ...there is only one critical point for the problem ([10]). We call such a solution a conditional solution. Compared with the existing RARL2 algorithm, that offers also a conditional solution ([1], [2], =-=[3]-=-), Cyclic AFD is more explicit, and can directly find the poles of the approximating rational function. We will call n∑ k=1 ckBk(z) an n-Blaschke form where the parameters a1, ..., an of Bn are arbitr... |

10 |
Adaptive Fourier series—a variation of greedy algorithm,”
- Qian, Wang
- 2011
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Citation Context ...eat general functions as boundary limits of Hardy space functions that can be non-smooth functions. We did obtain the convergence rate O(1/n) under conditions that do not directly address smoothness (=-=[14]-=-) which, therefore, does not look nice. Some further study on this topic has been carrying out under the frame work of statistical learning theory. The last point to make in the introduction part is r... |

8 | A Fast Adaptive Model Reduction Method Based on Takenaka-Malmquist Systems, by
- Mi, Qian, et al.
- 2012
(Show Context)
Citation Context ...ximation (see [10] and §3). The AFD formulation is also related to compressed sensing and learning theory. In the application aspect we accomplished some studies in relation to system identification (=-=[8]-=-, [7]), time-frequency distribution in signal analysis, speech analysis and distortion reversing in image processing in relation to harmonic mappings, etc. In approximation one concerns convergence ra... |

7 | Adaptive decomposition by weighted inner functions: a generalization of Fourier series
- Qian, Tan, et al.
(Show Context)
Citation Context ...ant examples especially on singular inner functions ([13]). Remark 6 There are other AFD-variations that first extract factor signals of high frequencies. Those include Double-sequence unwending AFD (=-=[16]-=- and one using what we call high-order Szegö kernels ([18]). The algorithm of double-sequence Unwending AFD is more complicated than that of Unwending AFD but with a similar performance as the latter... |

6 | Adaptive Fourier decomposition of functions in quaternionic Hardy spaces
- Qian, Sproessig, et al.
(Show Context)
Citation Context ...〈 Rk−1(f)‖Rk−1(eak)‖ , eak〉 = n(ak) Rk−1(f)(ak) ‖Rk−1(eak)‖ , (4) 4 where ‖Rk−1(eak)‖2 = 1− k−1∑ l=1 |〈eak , Bl〉|2. The generalization to quaternions of the theory for complex numbers is based on (4) =-=[15]-=-. There is an obstacle for the theory being generalized to Clifford algebra due to the fact that for Clifford algebra-valued functions f the usually defined inner product 〈f, f〉 is not necessarily sca... |

5 |
Analytic phase derivatives, all-pass filters, and signals of minimum phase
- Dang, Qian
(Show Context)
Citation Context ...es are the monomials zk, but there are many more. A rigorous definition of analytic phase derivative involves non-tangential boundary limits of holomorphic functions in the relevant domain [12], [5], =-=[4]-=-. Through a combined effort a large pool of mono-components is found. Next, one wishes to seek for apropriate decompositions of a signal into mono-components. The Fourier decomposition of Hardy space ... |

5 |
Frequency Domain Identification: An Algorithm Based On Adaptive
- Mi, Qian
(Show Context)
Citation Context ...ion (see [10] and §3). The AFD formulation is also related to compressed sensing and learning theory. In the application aspect we accomplished some studies in relation to system identification ([8], =-=[7]-=-), time-frequency distribution in signal analysis, speech analysis and distortion reversing in image processing in relation to harmonic mappings, etc. In approximation one concerns convergence rates. ... |

5 |
Optimal approximation by Blaschke forms
- Qian, Wegert
(Show Context)
Citation Context ...s to n = 2 the k-spherical harmonics decomposition is cos kt = 1 2 (eikt + e−ikt). The projection functions Pl(f) are given by integral operators against the lmultiple Szegö kernels at the zero (see =-=[17]-=-). In the Clifford algebra setting of the Euclidean space things follow the same philosophy. The decomposition (5) under (6) becomes f(x) = ∑ k 6=−1,...,−n+2 Pk(f)(x) that is the corresponding Fourier... |

4 |
Bounded Analyic Functions
- Garnett
- 1981
(Show Context)
Citation Context ... H2 = span{Bk} ⊕ φH2, where span{Bk} is a backward-shift invariant subspace, and φH2 is a shift invariant subspace of the H2 space. These invariant spaces are of the Beurlingor the Beurling-Lax type (=-=[6]-=-), respectively, in the unit disc or the half complex plane contexts. A TM system consists of rational functions in the Hardy space that can approximate functions in the same Hardy space. Given the re... |

4 |
Cyclic AFD Algorithm for Best Approximation by Rational Functions of Given Order
- Qian
(Show Context)
Citation Context ...urier decompositions have strong backgrounds in both the theoretical and application aspects. Apart from the “shift analysis” aspect it also has close relation to rational function approximation (see =-=[10]-=- and §3). The AFD formulation is also related to compressed sensing and learning theory. In the application aspect we accomplished some studies in relation to system identification ([8], [7]), time-fr... |

4 |
Comparison of adaptive mono-component decompositions
- Qian, Li, et al.
(Show Context)
Citation Context ...e is of high frequency, one should “unwending” it but not try first to get maximal portions corresponding to the lowest frequencies. When this idea is implemented the AFD is amended as follows ([10], =-=[13]-=-). First we do the factorization f = f1 = I1O1, where I1 and O1 are, respectively the inner and outer function parts of f. The factorization is based on Nevanlinna’s factorization theorem and the oute... |

4 |
Adaptive Decomposition of Functions by Higher Order Szegö Kernels I: A Method for Mono-component Decomposition, accepted by Acta Applicanda Mathematicae
- Qian, Wang
(Show Context)
Citation Context ... Remark 6 There are other AFD-variations that first extract factor signals of high frequencies. Those include Double-sequence unwending AFD ([16] and one using what we call high-order Szegö kernels (=-=[18]-=-). The algorithm of double-sequence Unwending AFD is more complicated than that of Unwending AFD but with a similar performance as the latter. The high-order Szegö kernel method does the selection fo... |

4 |
Matching Pursuits among Shifted Cauchy Kernels
- Qian, Wang, et al.
(Show Context)
Citation Context ...r product being scalar-valued, however, is crucial in the Gram-Schmidt process. Nevertheless, an analogous theory in the Clifford algebra setting can be established under the idea of matching pursuit =-=[19]-=-. The traditional spherical harmonics expansion, viz., the Fourier-Laplace series expansion on the sphere falls into the same theoretical frame under the Clifford algebra setting. It is a realization ... |

3 |
A remark on uniqueness of best rational approximation of degree 1
- Baratchart
- 2006
(Show Context)
Citation Context ...tion when there is only one critical point for the problem ([10]). We call such a solution a conditional solution. Compared with the existing RARL2 algorithm, that offers also a conditional solution (=-=[1]-=-, [2], [3]), Cyclic AFD is more explicit, and can directly find the poles of the approximating rational function. We will call n∑ k=1 ckBk(z) an n-Blaschke form where the parameters a1, ..., an of Bn ... |

3 |
Hardy-Sobolev spaces decomposition and applications in signal analysis
- Dang, Qian, et al.
(Show Context)
Citation Context ...xamples are the monomials zk, but there are many more. A rigorous definition of analytic phase derivative involves non-tangential boundary limits of holomorphic functions in the relevant domain [12], =-=[5]-=-, [4]. Through a combined effort a large pool of mono-components is found. Next, one wishes to seek for apropriate decompositions of a signal into mono-components. The Fourier decomposition of Hardy s... |

2 | Liming Zhang and Zhi-Xiong Li, Algorithm of Adaptive Fourier Decomposition - Qian |

1 |
Adaptive Fourier decomposition on torus, preprint
- Qian
(Show Context)
Citation Context ...ak ⊗Bbl 〉Bak ⊗Bbl , Dn(f) = Sn(f)− Sn−1(f). Note that Dn(f) has 2n− 1 entries. Based on such setting a maximal selection theorem is available, and a related adaptive decomposition can be established (=-=[11]-=-). We have been explaining the theory of adaptive Fourier decomposition on the circles, the unit spheres and the n-torus. An analogous theory is available in the real-line and the half spaces that rep... |

1 |
Qian,Mono-components for decomposition of signals
- unknown authors
(Show Context)
Citation Context ...asic examples are the monomials zk, but there are many more. A rigorous definition of analytic phase derivative involves non-tangential boundary limits of holomorphic functions in the relevant domain =-=[12]-=-, [5], [4]. Through a combined effort a large pool of mono-components is found. Next, one wishes to seek for apropriate decompositions of a signal into mono-components. The Fourier decomposition of Ha... |