Adaptive Fourier decompositions and rational approximations, part I: Theory

Qian, Tao

doi:10.1142/s0219691314610086

Cited by 26 publications

(11 citation statements)

References 19 publications

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“…where B n is chosen among all Blaschke products with n zeroes as the one yielding the largest inner product. Since, in particular, the functions z n are elements of that library, this approach may be understood as a generalization of Fourier series -among their results is also an independent rediscovery of the Guido and Mary Weiss algorithm [17] and of the unwinding series [21]. There are also similarities in spirit with recent work of Mallat [12].…”

Section: A Formal Seriesmentioning

confidence: 88%

“…This paper was brought to our attention after this paper had been completed and we summarize his argument below. Closely related is also another approach developed by Qian, Ho, Leong and Wang [19] (and elaborated in further papers by Qian and collaborators [16,17,20,22]), which they call adaptive Fourier transform. The main idea is to use Blaschke products as a library and proceed by a projection pursuit approach, where at each step one projects onto the element in the library yielding the largest inner product with the function:…”

Section: A Formal Seriesmentioning

confidence: 92%

“…Our main statement is that the Blaschke factorization acts nicely on these spaces. The first part of our statement is known (being ascribed to Digital Signal Processing in [17]) and can be equivalently phrased as follows: given a Blasche decomposition F = B · G and assuming both functions are expanded into a Fourier series…”

Section: 4mentioning

confidence: 99%

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Nonlinear Phase Unwinding of Functions

Coifman

Steinerberger

2016

J Fourier Anal Appl

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We study a natural nonlinear analogue of Fourier series. Iterative Blaschke factorization allows one to formally write any holomorphic function F as a series which successively unravels or unwinds the oscillation of the function where a i ∈ C and B i is a Blaschke product. Numerical experiments point towards rapid convergence of the formal series but the actual mechanism by which this is happening has yet to be explained. We derive a family of inequalities and use them to prove convergence for a large number of function spaces: for example, we have convergence in L 2 for functions in the Dirichlet space D. Furthermore, we present a numerically efficient way to expand a function without explicit calculations of the Blaschke zeroes going back to Guido and Mary Weiss.2010 Mathematics Subject Classification. 30B50 (primary), and 30A10, 65T99 (secondary).

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Section: A Formal Seriesmentioning

confidence: 88%

Section: A Formal Seriesmentioning

confidence: 92%

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Nonlinear Phase Unwinding of Functions

Coifman

Steinerberger

2016

J Fourier Anal Appl

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“…The adaptive Fourier decomposition (AFD), introduced by Qian et. al., is a type of positive frequency expansion algorithm based on a given basis search dictionary [1][2][3]. It offers fast energy decomposition via the adaptive basis, which is different from the conventional Fourier decomposition that is based on the Fourier basis.…”

Section: Introductionmentioning

confidence: 99%

Fast basis search for adaptive Fourier decomposition

Wang

Wan

Wong

et al. 2018

EURASIP J. Adv. Signal Process.

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The adaptive Fourier decomposition (AFD) uses an adaptive basis instead of a fixed basis in the rational analytic function and thus achieves a fast energy convergence rate. At each decomposition level, an important step is to determine a new basis element from a dictionary to maximize the extracted energy. The existing basis searching method, however, is only the exhaustive searching method that is rather inefficient. This paper proposes four methods to accelerate the AFD algorithm based on four typical optimization techniques including the unscented Kalman filter (UKF) method, the Nelder-Mead (NM) algorithm, the genetic algorithm (GA), and the particle swarm optimization (PSO) algorithm. In the simulation of decomposing four representative signals and real ECG signals, compared with the existing exhaustive search method, the proposed schemes can achieve much higher computation speed with a fast energy convergence, that is, in particular, to make the AFD possible for real-time applications.

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“…Additionally, a number of potential applications of Blaschke factorization have been reviewed, such as electrocardiography signal analysis by Tan, Zhang, and Wu [21], study of acoustic underwater scattering by Letelier and Saito in [9], and study of the Doppler Effect by Healy [8]. Furthermore, Qian has extensively studied adaptive Fourier decomposition and its connections to Complex Analysis and Approximation Theory (see [13], [14], [15], [16], [17], [18]).…”

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confidence: 99%

Iterative Variable-Blaschke Factorization

Lukianchikov

Nazarchuk

Xue

2019

Complex Anal. Oper. Theory

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Blaschke factorization allows us to write any holomorphic function F as a formal serieswhere a i ∈ C and B i is a Blaschke product. We introduce a more general variation of the canonical Blaschke product and study the resulting formal series. We prove that the series converges exponentially in the Dirichlet space given a suitable choice of parameters if F is a polynomial and we provide explicit conditions under which this convergence can occur. Finally, we derive analogous properties of Blaschke factorization using our new variable framework.2010 Mathematics Subject Classification. 30A10, 30B50.

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Adaptive Fourier decompositions and rational approximations, part I: Theory

Abstract: In this paper, we will give a survey on adaptive Fourier decompositions (AFDs) in oneand multi-dimensions. Theoretical formulations of three different types of AFDs in onedimension, viz., Core AFD, Cyclic AFD in conjunction with best rational approximation and Unwending AFD are provided.

Cited by 26 publications

References 19 publications

Nonlinear Phase Unwinding of Functions

Nonlinear Phase Unwinding of Functions

Fast basis search for adaptive Fourier decomposition

Iterative Variable-Blaschke Factorization

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