Takenaka-Malmquist System Research Articles

Fejér-type positive operator based on Takenaka–Malmquist system on unit circle

Let φ={φk}k=−∞∞ denote the extended Takenaka–Malmquist system on unit circle T and let σn,φ(f), f∈L1(T), be the Fejér-type operator based on φ, introduced by V. N. Rusak. We give the convergence criteria for σn,φ(f) in Banach space C(T). Also we prove the Voronovskaya-type theorem for σn,φ(f) on class of holomorphic functions representable by Cauchy-type integrals with bounded densities.

Weighted singular value decomposition basis of Szegő kernel and its applications to signal reconstruction and denoising

For any circular analytic signal f+(eit)∈H2(T), if the sequence {ak}k=0∞ in the unit disc D satisfies the hyperbolic non-separability condition ∑k=0∞(1−|ak|)=∞, it is known that the first N+1 term approximation function based on the Takenaka–Malmquist system (SN+1f+)(eit)=∑k=0N〈f+(eit),Bk(eit)〉Bk(eit),can approximate the analytic signal f+(eit) in the L2-norm sense, where Bk(eit)=1−|ak|21−ak¯eit∏l=0k−1eit−al1−al¯eit, k∈N is the Takenaka–Malmquist system which can be seen as a generalization of Fourier series. We find that (SN+1f+)(eit) can also be represented by the standard basis, Lagrange basis and weighted SVD basis. Different bases have different merits. The standard basis is simple. The Lagrange basis has a neat and compact form, which is very convenient for theoretical analysis. With the Takenaka–Malmquist system, the first N+1 term approximation function (SN+1f+)(eit) is able to write in an iterative form. The Takenaka–Malmquist system has the advantage of inheritance, and the point ak∈D can be easily selected step by step through the greedy principle. In this paper, we focus our attention on the weighted SVD basis. We give some orthogonal properties of the weighted SVD basis. It demonstrates that any real signal f(eit)∈L2(T) can be precisely reconstructed within the given error range by the weighted SVD basis decomposition. The experimental results reveal that the algorithm based on weighted SVD basis is more stable and accurate. Furthermore, compared with classical Fourier decomposition and adaptive Fourier decomposition algorithms, the experimental results show that the weighted SVD basis decomposition is more insensitive to noise. This will provide a new method for signal denoising.

Pseudohyperbolic distance and n‐best rational approximation in ℍ2 space

Through reducing the problem to rational orthogonal system (Takenaka–Malmquist system), this note gives a proof for existence of n‐best rational approximation to functions in the Hardy space by using pseudohyperbolic distance.

A remark on Riesz bases of subspaces in Hardy spaces

ABSTRACT In this paper we study Riesz bases of subspaces spanned by Szegö kernels in Hardy spaces of the unit disc and the polydisc where and The study is based on the results of Carleson's interpolation problem and, as well as the Schauder-property of Takenaka-Malmquist systems.

Best Approximations of the Cauchy–Szegö Kernel in the Mean on the Unit Circle

We compute the values of the best approximations of the Cauchy–Szeg¨o kernel in the mean on the unit circle by quasipolynomials with respect to the Takenaka–Malmquist system.

Extracting outer function part from Hardy space function

Any analytic signal f a (eit ) can be written as a product of its minimum-phase signal part (the outer function part) and its all-phase signal part (the inner function part). Due to importance of such decomposition, Kumarasan and Rao (1999), implementing the idea of the Szegő limit theorem (see below), propose an algorithm to obtain approximations of the minimum-phase signal of a polynomial analytic signal $${f_a}\left( {{e^{it}}} \right) = {e^{i{N_0}t}}\sum\limits_{k = 0}^M {{a_k}{e^{ikt}}} ,$$ (0.1) where a 0 ≠ 0, a M ≠ 0. Their method involves minimizing the energy $$E\left( {{f_a},{h_1},{h_2}, \ldots ,{h_H}} \right) = \frac{1}{{2\pi }}\int_0^{2\pi } {{{\left| {1 + \sum\limits_{k = 1}^H {{h_k}{e^{ikt}}} } \right|}^2}{{\left| {{f_a}\left( {{e^{it}}} \right)} \right|}^2}dt} $$ (0.2) with the undetermined complex numbers h k ’s by the least mean square error method. In the limiting procedure H → ∞, one obtains approximate solutions of the minimum-phase signal. What is achieved in the present paper is two-fold. On one hand, we rigorously prove that, if f a (eit ) is a polynomial analytic signal as given in (0.1), then for any integer H ≥ M, and with |f a (eit )|2 in the integrand part of (0.2) being replaced with 1/|f a (eit )|2, the exact solution of the minimum-phase signal of f a (eit ) can be extracted out. On the other hand, we show that the Fourier system e ikt used in the above process may be replaced with the Takenaka-Malmquist (TM) system, \({r_k}\left( {{e^{it}}} \right): = \frac{{\sqrt {1 - {{\left| {{\alpha _k}} \right|}^2}} {e^{it}}}}{{1 - \overline {{\alpha _k}} {e^{it}}}}\Pi _{i = 1}^{k - 1}\frac{{{e^{it}} - {\alpha _j}}}{{1 - \overline {{\alpha _j}} {e^{it}}}},k = 1,2, \ldots ,{r_0}\left( {{e^{it}}} \right) = 1\), i.e., the least mean square error method based on the TM system can also be used to extract out approximate solutions of minimum-phase signals for any functions f a in the Hardy space. The advantage of the TM system method is that the parameters α 1,...,α n ,... determining the system can be adaptively selected in order to increase computational efficiency. In particular, adopting the n-best rational (Blaschke form) approximation selection for the n-tuple {α 1,...,α n }; n ≥ N, where N is the degree of the given rational analytic signal, the minimum-phase part of a rational analytic signal can be accurately and efficiently extracted out.

Adaptive Fourier tester for statistical estimation

Based on Takenaka–Malmquist (TM) system, a new nonparametric estimator for probability density function is proposed. The TM estimation method is completely different from the existent density estimation methods in that the estimator depends on an approximate system with poles in a complex plane. Compared with the classic Fourier estimator, the TM estimator will offer more flexibility and adaptivity for real data due to the poles and nonlinearity of the phase of TM system. We compare the TM estimator with kernel, wavelet, and spline estimators by simulations. It shows that the introduced TM estimator is a more promising method than the existing and commonly used methods. Copyright © 2016 John Wiley & Sons, Ltd.

Best Approximations for the Cauchy Kernel on the Real Axis

We compute the values of the best approximations for the Cauchy kernel on the real axis ℝ by some subspaces from L q (ℝ). This result is applied to the evaluation of the sharp upper bounds for pointwise deviations of certain interpolation operators with interpolation nodes in the upper half plane and certain linear means of the Fourier series in the Takenaka–Malmquist system from the functions lying in the unit ball of the Hardy space H p , 2 ≤ p < ∞.

The AFD methods to compute Hilbert transform

In the literature adaptive Fourier decomposition is abbreviated as AFD that addresses adaptive rational approximation, or alternatively adaptive Takenaka–Malmquist system approximation. The AFD type approximations may be characterized as adaptive approximations by linear combinations of parameterized Szegö and higher order Szegö kernels. This note proposes two kinds of such analytic approximations of which one is called maximal-energy AFDs, including core AFD, Unwending AFD and Cyclic AFD; and the other is again linear combinations of Szegö kernels but generated through SVM methods. The proposed methods are based on the fact that the imaginary part of an analytic signal is the Hilbert transform of its real part. As consequence, when a sequence of rational analytic functions approximates an analytic signal, then the real parts and imaginary parts of the functions in the sequence approximate, respectively, the original real-valued signals and its Hilbert transform. The two approximations have the same errors in the energy sense due to the fact that Hilbert transformation is a unitary operator in the L2 space. This paper for the first time promotes the complex analytic method for computing Hilbert transforms. Experiments show that such computational methods are as effective as the commonly used one based on FFT.

Frequency-domain identification: An algorithm based on an adaptive rational orthogonal system

This paper presents a new adaptive algorithm for frequency-domain identification. The algorithm is related to the rational orthogonal system (Takenaka–Malmquist system). This work is based on an adaptive decomposition algorithm previously proposed for decomposing the Hardy space functions, in which a greedy sequence is obtained according to the maximal selection criterion. We modify the algorithm through necessary changes for system identification.

Adaptive Fourier decomposition of functions in quaternionic Hardy spaces

We study decompositions of functions in the Hardy spaces into linear combinations of the basic functions in the orthogonal rational systems Bn, which are obtained in the respective contexts through Gram–Schmidt orthogonalization process on shifted Cauchy kernels. Those lead to adaptive decompositions of quaternionic‐valued signals of finite energy. This study is a generalization of the main results of the first author's recent research in relation to adaptive Takenaka–Malmquist systems in one complex variable. Copyright © 2011 John Wiley &amp; Sons, Ltd.

A fast adaptive model reduction method based on Takenaka–Malmquist systems

An adaptive model reduction method is proposed for linear time-invariant systems based on the continuous-time rational orthogonal basis (Takenaka–Malmquist basis). The method is to find an adaptive approximation in the energy sense by selecting optimal points for the rational orthogonal basis. The stability of the reduced models holds, and the steady-state values of step responses are kept to be equal. Furthermore, the method automatically ensures the reduced system to be in the Hardy space H 2 . The existence of the best approximation in the Hardy space H 2 by n Blaschke forms is proved in the proposed approach. The effectivity of this method is illustrated through three well-known examples.

Best linear methods of approximation and optimal orthonormal systems of the Hardy space

We construct the best linear methods of approximation for functions of the Hardy space H p on compact subsets of the unit disk. It is shown that the Takenaka-Malmquist systems are optimal systems of functions orthonormal on the unit circle for the construction of the best linear methods of approximation.

Best linear methods for approximation of bounded harmonic functions

We construct a best linear method for the approximation of bounded harmonic functions on compact subsets of the unit disk. We show that a system of functions orthonormal on the unit circle and optimal for the construction of the best linear approximation method is a Takenaka-Malmquist system.

A Tauberian Theorem for Series of Orthogonal Rational Functions

The so-called Takenaka-Malmquist system consists of a sequence of functions obtained from the sequence of rational fractions by applying the Gram-Schmidt orthogonalization procedure with respect to the Lesbegue measure on the unite circle. A Tauberian theorem for series with respect to the Takenaka-Malmquist system is proved. Bibliography: 7 titles.