Harmonic analysis with a real function of frequency — II. periodic and bounded cases

A real representation, which is different from but equivalent to the Fourier complex representation, for a periodic function by trigonometric series is described. A generalized series expression of a function defined on a finite interval is formed and its properties are investigated.     

Convergence and formal manipulation in the theory of series from 1730 to 1815

In the early calculus mathematicians used convergent series to represent geometrical quantities and solve geometrical problems. However, series were also manipulated formally using procedures that were the infinitary extension of finite procedures. By the 1720s results were being published that could not be reduced to the original conceptions of convergence and geometrical representation. This situation led Euler to develop explicitly a more formal approach which generalized the early theory. Formal analysis, which was predominant during the second half of the 18th century despite criticisms of it by some researchers, contributed to the enlargement of mathematics and even led to a new branch of analysis: the calculus of operations. However, formal methods could not give an adequate treatment of trigonometric series and series that were not the expansions of elementary functions. The need to use trigonometric series and introduce nonelementary functions led Fourier and Gauss to reject the formal concept of series and adopt a different, purely quantitative notion of series.     

付氏变换在三角级数求和中的应用

本文建立了用付氏变换在三角级数求和中的新的重要定理,并用付氏变换的已知结果,解决了不少困难和复杂的三角级数求和问题.这是三角级数求和的新方法,作者曾用以编著了数以万计的三角级数之和的大表.许多结果都是新的.     

Signal modeling using the gradient search

This letter presents a new method for continuous signal modeling. Firstly, the continuous signal can be represented as a function of the trigonometric functional extension (Fourier series). Fourier series of the signal are parameterized by the fundamental frequency and unknown parameters. Then, the gradient-based iterative identification algorithm is derived, for estimating parameters of the signal model with known and unknown frequencies, separately. Finally, the simulation results indicate that the proposed algorithm is effective.     

Maharam-type kernel representation for operators with a trigonometric domination

Consider a linear and continuous operator T between Banach function spaces. We prove that under certain requirements an integral inequality for T is equivalent to a factorization of T through a specific kernel operator: in other words, the operator T has what we call a Maharam-type kernel representation. In the case that the inequality provides a domination involving trigonometric functions, a special factorization through the Fourier operator is given. We apply this result to study the problem that motivates the paper: the approximation of functions in \(L^{2}[0,1]\) by means of trigonometric series whose Fourier coefficients are given by weighted trigonometric integrals.     

Consecutive minimum phase expansion of physically realizable signals with applications

In digital signal processing, it is a well know fact that a causal signal of finite energy is front loaded if and only if the corresponding analytic signal, or the physically realizable signal, is a minimum phase signal, or an outer function in the complex analysis terminology. Based on this fact, a series expansion method, called unwinding adaptive Fourier decomposition (AFD), to give rise to positive frequency representations with rapid convergence was proposed several years ago. It appears to be a promising positive frequency representation with great potential of applications. The corresponding algorithm, however, is complicated due to consecutive extractions of outer functions involving computation of Hilbert transforms. This paper is to propose a practical algorithm for unwinding AFD that does not depend on computation of Hilbert transform, but, instead, factorizes out the Blaschke product type of inner functions. The proposed method significantly improves applicability of unwinding AFD. As an application, we give the associated Dirac‐type time‐frequency distribution of physically realizable signals. Copyright © 2015 John Wiley & Sons, Ltd.     

An Interplay of Multidimensional Variations in Fourier Analysis

We compare the Fourier integral of a function of bounded variation and the corresponding trigonometric series, generated by that function, in the multidimensional case. Several known notions of bounded variation are used and a new one is introduced. The obtained results are applied to integrability of multidimensional trigonometric series.     

Fourier Series and Approximation on Hexagonal and Triangular Domains

Several problems on Fourier series and trigonometric approximation on regular hexagonal and triangular domains are studied. The results include Abel and Cesàro summability of Fourier series, degree of approximation, and best approximation by trigonometric functions with both direct and inverse theorems. One of the objectives of this study is to demonstrate that Fourier series on spectral sets enjoy a rich structure that permits an extensive theory for Fourier series and approximation.     

Analysis of the energetic radiation intensity of a linear antenna array radiating nonsinusoidal signals

We present the energetic radiation intensity (ERI) as the quadratic form of the family of integral operators on a finite interval. The kernel of each operator is the autocorrelation function of the signal, which is radiated in the given direction. Spectral representation of the operators gives a fast-converging series representation of the ERI. For the signals, whose Fourier transforms are rational functions of the frequency, spectral analysis of the operators is reduced to finite-dimensional linear systems. Moreover, for such signals we express the ERI as the linear combination of the monochromatic directivity diagrams, evaluated in the complex poles of the signal’s Fourier transform. For the isotropic array elements and the most important amplitude distributions the ERI is obtained explicitly. We consider in detail a signal given by a truncated decaying exponent. Bibliography: 32 titles. Dedicated to Vasilii Mikhailovich Babich with high respect and gratitude __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 332, 2006, pp. 239–267.     

On curvilinear distributions expressed by double Fourier series

Summary The paper deals with two problems concerning the theory of distributions expressed by Fourier series. In the first part of the work the formulae for the double trigonometric series coefficients of the curilinear distributions are derived, in the second the differentiation of such series representing the function of two variables with the curvilinear jump is considered. Some numerical examples are given as an illustration of the application of the formulae derived.

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Zusammenfassung Zwei Probleme die die Theorie der mit der Fourier-Reihe abgebildeten Distributionen betreffen, sind in dieser Arbeit dargestellt. Im ersten Teil werden die Formeln für die Reihenentwicklungskoeffizienten der krummlinigen Distributionen hergeleitet, und im zweiten Teil wird die Differentiation der trigonometrischen Reihe der Funktion von zwei Veränderlichen mit dem krummlinigen Sprung untersucht. Zur Illustration der Anwendung der hergeleiteten Formeln werden einige numerische Beispiele angeführt.

     

A Second-Order Synchrosqueezing Transform with a Simple Form of Phase Transformation

To model a non-stationary signal as a superposition of amplitude and frequency-modulated Fourier-like oscillatory modes is important to extract information, such as the underlying dynamics, hidden in the signal. Recently, the synchrosqueezed wavelet transform (SST) and its variants have been developed to estimate instantaneous frequencies and separate the components of non-stationary multicomponent signals. The short-time Fourier transform-based SST (FSST for short) reassigns the frequency variable to sharpen the time-frequency representation and to separate the components of a multicomponent non-stationary signal. However, FSST works well only with multicomponent signals having slowly changing frequencies. To deal with multicomponent signals having fast-changing frequencies, the second-order FSST (FSST2 for short) was proposed. The key point for FSST2 is to construct a phase transformation of a signal which is the instantaneous frequency when the signal is a linear chirp. In this paper we consider a phase transformation for FSST2 which has a simpler expression than that used in the literature. In the study the theoretical analysis of FSST2 with this phase transformation, we observe that the proof for the error bounds for the instantaneous frequency estimation and component recovery is simpler than that with the conventional phase transformation. We also provide some experimental results which show that this FSST2 performs well in non-stationary multicomponent signal separation.     

On inverse methods for the resolution of the Gibbs phenomenon

When Fourier expansions, or more generally spectral methods, are used for the representation of nonsmooth functions, then one has to face the so-called Gibbs phenomenon. Considerable progresses have been made these last years to overcome the Gibbs phenomenon, using direct or inverse approaches, both in the discrete or continuous framework. A discrete inverse method for the global or local reconstruction of a non-smooth function starting from its oscillating (trigonometric) polynomial interpolant is introduced and both its capabilities and limits are emphasized.     

From Local Cosine Bases to Global Harmonics

In this paper, the reproduction of trigonometric polynomials with two-overlapping local cosine bases is investigated. This study is motivated by the need to represent most effectively a Fourier series in the form of a localized cosine series for the purpose of local analysis, thus providing a vehicle for the transition from classical harmonic analysis to analysis by Wilson-type wavelets. It is shown that there is one and only one class, which is a one-parameter family, of window functions that allows pointwise reproduction of all global harmonics, where the parameter is the order of smoothness of the window functions. It turns out that this class of window functions is also optimal in the sense that all global harmonics are reproduced by using a minimal number of the local trigonometric basis functions.     

On the convergence rate of Fourier series

The rate of convergence of a Fourier‐series representation of a given function depends on the nature of the function and of its derivatives. This is shown by using the graphical outputs of a desk computer for different cases. For full‐range series, the effects of continuity and discontinuity of the function and its first derivatives are shown first. The advantage of half‐range formulae due to the free choice of function in the second half‐range are demonstrated next, along with the importance of choice of sine and cosine series according to the function being represented. Finally, a method is given of modifying the given function in a simple manner whereby a dramatic increase in the rate of convergence of the Fourier series is obtained. Examples of the Gibbs overshoot and its elimination are included.     

A formulation to obtain semi-analytical planetary theories using true anomalies as temporal variables

The main methods used to obtain analytical theories of perturbed motion in celestial mechanics are based on the expansion of the disturbing function in trigonometric series of the mean anomalies (or longitudes). In this paper a new method based on the double Fourier series expansion using the true anomalies (or longitudes) is developed. The method involves a semi-analytical technique to allow the expansion of the inverse of the distance with great accuracy, and a new integration technique using a linear combination of the true anomalies based on an iterative method to integrate each term of the expansion of the Lagrange planetary equations.     

一般随机缺项三角级数表示断片的Bouligand维数

该文对［１］中的Ｂｏｕｌｉｇａｎｄ维数计算公式进行了改进，用对称原理和简化原理，得到了一般随机缺项三角级数所表示断片的Ｂｏｕｌｉｇａｎｄ维数的一些计算公式．     

Adaptive Fourier decomposition‐based Dirac type time‐frequency distribution

The Dirac‐type time‐frequency distribution (TFD), regarded as ideal TFD, has long been desired. It, until the present time, cannot be implemented, due to the fact that there has been no appropriate representation of signals leading to such TFD. Instead, people have been developing other types of TFD, including the Wigner and the windowed Fourier transform types. This paper promotes a practical passage leading to a Dirac‐type TFD. Based on the proposed function decomposition method, viz., adaptive Fourier decomposition, we establish a rigorous and practical Dirac‐type TFD theory. We do follow the route of analytic signal representation of signals founded and developed by Garbo, Ville, Cohen, Boashash, Picinbono, and others. The difference, however, is that our treatment is theoretically throughout and rigorous. To well illustrate the new theory and the related TFD, we include several examples and experiments of which some are in comparison with the most commonly used TFDs. Copyright © 2016 John Wiley & Sons, Ltd.     

Hyperbolic Wavelets and Multiresolution in H^{2}(\mathbb{T})

In signal processing and system identification for H²(\BbbT)H^{2}(\Bbb{T}) and H²(\BbbD)H^{2}(\Bbb{D}) the traditional trigonometric bases and trigonometric Fourier transform are replaced by the more efficient rational orthogonal bases like the discrete Laguerre, Kautz and Malmquist-Takenaka systems and the associated transforms. These bases are constructed from rational Blaschke functions, which form a group with respect to function composition that is isomorphic to the Blaschke group, respectively to the hyperbolic matrix group. Consequently, the background theory uses tools from non-commutative harmonic analysis over groups and the generalization of Fourier transform uses concepts from the theory of the voice transform. The successful application of rational orthogonal bases needs a priori knowledge of the poles of the transfer function that may cause a drawback of the method. In this paper we give a set of poles and using them we will generate a multiresolution in H²(\BbbT)H^{2}(\Bbb{T}) and H²(\BbbD)H^{2}(\Bbb{D}). The construction is an analogy with the discrete affine wavelets, and in fact is the discretization of the continuous voice transform generated by a representation of the Blaschke group over the space H²(\BbbT)H^{2}(\Bbb{T}). The constructed discretization scheme gives opportunity of practical realization of hyperbolic wavelet representation of signals belonging to H²(\BbbT)H^{2}(\Bbb{T}) and H²(\BbbD)H^{2}(\Bbb{D}) if we can measure their values on a given set of points inside the unit circle or on the unit circle. Convergence properties of the hyperbolic wavelet representation will be studied.     

The central limit theorem for lacunary series

In this paper, the central limit theorem for lacunary trigonometric series is proved. Two gap conditions by Erdos and Takahashi are extended and unified. The criterion for the Fourier character of lacunary series is also given.

     

Decomposition theorems and approximation by a ``floating" system of exponentials

The main problem considered in this paper is the approximation of a trigonometric polynomial by a trigonometric polynomial with a prescribed number of harmonics. The method proposed here gives an opportunity to consider approximation in different spaces, among them the space of continuous functions, the space of functions with uniformly convergent Fourier series, and the space of continuous analytic functions. Applications are given to approximation of the Sobolev classes by trigonometric polynomials with prescribed number of harmonics, and to the widths of the Sobolev classes. This work supplements investigations by Maiorov, Makovoz and the author where similar results were given in the integral metric.