Multiple Fourier Series Research Articles

Unconditional convergence of eigenfunction expansions for abstract and elliptic operators

We study the most general class of eigenfunction expansions for abstract normal operators with pure point spectrum in a complex Hilbert space. We find sufficient conditions for such expansions to be unconditionally convergent in spaces with two norms and also estimate the degree of this convergence. Our result essentially generalizes and complements the known theorems of Krein and of Krasnosel'skiĭ and Pustyl'nik. We apply it to normal elliptic pseudodifferential operators on compact boundaryless $C^{\infty }$ -manifolds. We find generic conditions for eigenfunction expansions induced by such operators to converge unconditionally in the Sobolev spaces $W^{\ell }_{p}$ with $p>2$ or in the spaces $C^{\ell }$ (specifically, for the $p$ -th mean or uniform convergence on the manifold). These conditions are sufficient and necessary for the indicated convergence on Sobolev or Hörmander function classes and are given in terms of parameters characterizing these classes. We also find estimates for the degree of the convergence on such function classes. These results are new even for differential operators on the circle and for multiple Fourier series.

Generalized Localization and Summability Almost Everywhere of Multiple Fourier Series and Integrals

Generalized Localization and Summability Almost Everywhere of Multiple Fourier Series and Integrals

On the generalized β-absolute convergence of single and multiple Fourier series

In this paper, we provide sufficient conditions for the generalized β-absolute convergence of multiple Fourier series of a function of p-(Λ1, ... ,ΛN)-bounded variation.

On Pringsheim Convergence of a Subsequence of Partial Sums of a Multiple Trigonometric Fourier Series

On Pringsheim Convergence of a Subsequence of Partial Sums of a Multiple Trigonometric Fourier Series

Mean-square Approximation of Iterated Ito and Stratonovich Stochastic Integrals: Method of Generalized Multiple Fourier Series. Application to Numerical Integration of Ito Sdes and Semilinear Spdes (third Edition)

This is the third edition of the monograph (first edition 2020, second edition 2021) devoted to the problem of mean-square approximation of iterated Ito and Stratonovich stochastic integrals with respect to components of the multidimensional Wiener process. The mentioned problem is considered in the book as applied to the numerical integration of non-commutative Ito stochastic differential equations and semilinear stochastic partial differential equations with nonlinear non-commutative trace class noise. The book opens up a new direction in researching of iterated stochastic integrals. For the first time we use the generalized multiple Fourier series converging in the sense of norm in Hilbert space for the expansion of iterated Ito stochastic integrals of arbitrary multiplicity k with respect to components of the multidimensional Wiener process (Chapter 1). Sections 1.11-1.13 (Chapter 1) are new and generalize the results of Chapter 1 obtained earlier by the author and are also closely related to the multiple Wiener stochastic integral introduced by Ito in 1951. The convergence with probability 1 as well as the convergence in the sense of n-th (n=2, 3,...) moment for the expansion of iterated Ito stochastic integrals have been proved (Chapter 1). Moreover, the rate of both types of convergence has been established. The main difference between the third and second editions of the book is that the third edition includes original material (Chapter 2, Sections 2.10-2.19) on a new approach to the series expansion of iterated Stratonovich stochastic integrals of arbitrary multiplicity k with respect to components of the multidimensional Wiener process. The above approach allowed us to generalize some of the author's earlier results and also to make significant progress in solving the problem of series expansion of iterated Stratonovich stochastic integrals. In particular, for iterated Stratonovich stochastic integrals of the fifth and sixth multiplicity, series expansions based on multiple Fourier-Legendre series and multiple trigonometric Fourier series are obtained. In addition, expansions of iterated Stratonovich stochastic integrals of multiplicities 2 to 4 were generalized. These results (Chapter 2) adapt the results of Chapter 1 for iterated Stratonovich stochastic integrals. Two theorems on expansion of iterated Stratonovich stochastic integrals of arbitrary multiplicity k based on generalized iterated Fourier series with pointwise convergence are formulated and proved (Chapter 2). The results of Chapters 1 and 2 can be considered from the point of view of the Wong-Zakai approximation for the case of a multidimensional Wiener process and the Wiener process approximation based on its series expansion using Legendre polynomials and trigonometric functions. The integration order replacement technique for iterated Ito stochastic integrals has been introduced (Chapter 3). Exact expressions are obtained for the mean-square approximation error of iterated Ito stochastic integrals of arbitrary multiplicity k (Chapter 1) and iterated Stratonovich stochastic integrals of multiplicities 1 to 4 (Chapter 5). Furthermore, we provided a significant practical material (Chapter 5) devoted to the expansions and mean-square approximations of specific iterated Ito and Stratonovich stochastic integrals of multiplicities 1 to 6 from the unified Taylor-Ito and Taylor-Stratonovich expansions (Chapter 4). These approximations were obtained using Legendre polynomials and trigonometric functions. The methods constructed in the book have been compared with some existing methods (Chapter 6). The results of Chapter 1 were applied (Chapter 7) to the approximation of iterated stochastic integrals with respect to the finite-dimensional approximation of the Q-Wiener process (for integrals of multiplicity k) and with respect to the infinite-dimensional Q-Wiener process (for integrals of multiplicities 1 to 3).

A New Proof of the Expansion of Iterated Ito Stochastic Integrals with Respect to the Components of a Multidimensional Wiener Process Based on Generalized Multiple Fourier Series and Hermite Polynomials

The article is devoted to a new proof of the expansion for iterated Ito stochastic integrals with respect to the components of a multidimensional Wiener process. The above expansion is based on Hermite polynomials and generalized multiple Fourier series in arbitrary complete orthonormal systems of functions in a Hilbert space. In 2006, the author obtained a similar expansion, but with a lesser degree of generality, namely, for the case of continuous or piecewise continuous complete orthonornal systems of functions in a Hilbert space. In this article, the author generalizes the expansion of iterated Ito stochastic integrals obtained by him in 2006 to the case of an arbitrary complete orthonormal systems of functions in a Hilbert space using a new approach based on the Ito formula. The obtained expansion of iterated Ito stochastic integrals is useful for constructing of high-order strong numerical methods for systems of Ito stochastic differential equations with multidimensional non-commutative noise.

Features of the Expansion of Multiple Stochastic Stratonovich Integrals Using Walsh and Haar Functions

The problem of the root-mean-square convergence for approximations of multiple stochastic Stratonovich integrals based on the generalized multiple Fourier series method using Walsh and Haar functions is considered. It is shown that when they are chosen to expand multiple stochastic integrals, the proof of the root-mean-square convergence of a subsequence of series partial sums, which is formed in a way that is quite natural for these functions, does not require the explicit fulfillment of any additional conditions, except for the condition of the existence of the multiple stochastic Stratonovich integral.

Besov spaces with generalized smoothness and summability of multiple Fourier series

The paper deals with summation methods for multiple Fourier series in the framework of Besov spaces with generalized smoothness. The main focus lies on approximation with respect to the (quasi-) metric in Lp, where 0<p<1 as well as to refined smoothness near zero.

Rod-packing arrangements of invariant tori in solenoidal vector fields with cubic symmetries

The arrangements of invariant tori that resemble rod packings with cubic symmetries are considered in three-dimensional solenoidal vector fields. To find them systematically, vector fields whose components are represented in the form of multiple Fourier series with finite terms are classified using magnetic groups. The maximal magnetic group compatible with each arrangement is specified on the assumption that the cores of the nested invariant tori are straight and located on the lines corresponding to the central axes of the rods packed. Desired rod-packing arrangements are demonstrated by selecting vector fields whose magnetic groups are the maximal ones and by drawing their integral curves that twine around invariant tori. In the demonstration of chiral arrangements, Beltrami flows (or force-free fields in plasma physics), which have the strongest chirality of all solenoidal vector fields satisfying the same vector Helmholtz equation, are used. As by-products, several chain-like arrangements of closed invariant tori were found. One of the chains consists of knotted invariant tori. In all vector fields (chiral or achiral) selected for the demonstration, the volume percentages of ordered regions formed by invariant tori in a unit cell were roughly measured with the aid of a supervised machine learning technique.

On the convergence of the Abel-Poisson means of multiple Fourier series

Let $ A_\varepsilon (x,f)$ be the Abel-Poisson means of an integrable function $f(x)$ on $n$-dimensional torus $ \mathbf{T}^n, \; \;\; i= 1,\ldots,n \; (n\geq 2) $ in the Euclidean $n$-space. The famous Bochner's theorem asserts that for any function $ f\in L^1(\mathbf{T}^n)$ the Abel-Poisson means $A_\varepsilon (x,f)$ are pointwise converge to $f(x)$ a.e., that is, $$ \underset{\varepsilon \rightarrow0^+}{\lim}\, A_\varepsilon (x,f)= f(x), \;\; a.e.\; x\in \mathbf{T}^n. $$ In this paper we investigate the rate of convergence of Abel-Poisson means at the so-called $\mu$-smoothness point of $f$ .

Generalized Localization and Summability Almost Everywhere of Multiple Fourier Series and Integrals

It is well known that Luzins conjecture has a positive solution for one-dimensional trigonometric Fourier series, but in the multidimensional case it has not yet found its confirmation for spherical partial sums of multiple Fourier series. Historically, progress in solving Luzins hypothesis has been achieved by considering simpler problems. In this paper, we consider three of these problems for spherical partial sums: the principle of generalized localization, summability almost everywhere, and convergence almost everywhere of multiple Fourier series of smooth functions. A brief overview of the work in these areas is given and unsolved problems are mentioned and new problems are formulated. Moreover, at the end of the work, a new result on the convergence of spherical sums for functions from Sobolev classes is proved.

A note on multiple rational Fourier series

A note on multiple rational Fourier series

Multiple Fourier series and lattice point problems

For the multiple Fourier series of the periodization of some radial functions on Rd, we investigate the behavior of the spherical partial sum. We show the Gibbs-Wilbraham phenomenon, the Pinsky phenomenon and the third phenomenon for the multiple Fourier series, involving the convergence properties of them. The third phenomenon is closely related to the lattice point problems, which is a classical theme of the analytic number theory. We also prove that, for the case of two or three dimensions, the convergence problem on the Fourier series is equivalent to the lattice point problems in a sense. In particular, the convergence problem at the origin in two dimensions is equivalent to Hardy's conjecture on Gauss's circle problem.

Partial best approximations and the absolute Cesaro summability of multiple Fourier series

The article is devoted to the problem of absolute Cesaro summability of multiple trigonometric Fourier series. Taking a central place in the theory of Fourier series this problem was developed quite widely in the one-dimensional case and the fundamental results of this theory are set forth in the famous monographs by N.K. Bari, A. Zigmund, R. Edwards, B.S. Kashin and A.A. Saakyan [1–4]. In the case of multiple series, the corresponding theory is not so well developed. The multidimensional case has own specifics and the analogy with the one-dimensional case does not always be unambiguous and obvious. In this article, we obtain sufficient conditions for the absolute summability of multiple Fourier series of the function f ∈ Lq(Is) in terms of partial best approximations of this function. Four theorems are proved and four different sufficient conditions for the |C; β¯|λ-summability of the Fourier series of the function f are obtained. In the first theorem, a sufficient condition for the absolute |C; β¯|λ- summability of the Fourier series of the function f is obtained in terms of the partial best approximation of this function which consists of s conditions, in the case when β1 = ... = βs = 1/q'. Other sufficient conditions are obtained for double Fourier series. Sufficient conditions for the |C; β1; β2|λ-summability of the Fourier series of the function f ∈ Lq(I2) are obtained in the cases β1 = 1/q', −1 &lt; β2 &lt; 1/q'(in the second theorem), 1/q'&lt; β1 &lt; +∞, β2 = 1/q', (in the third theorem), −1 &lt; β1 &lt; 1/q', 1/q' &lt; β2 &lt; +∞ (in the fourth theorem).

Almost Everywhere Convergence of Multiple Trigonometric Fourier Series of Functions from Sobolev Classes

Almost Everywhere Convergence of Multiple Trigonometric Fourier Series of Functions from Sobolev Classes

Pick’s Theorem and Convergence of Multiple Fourier Series

We add another brick to the large building comprising proofs of Pick’s theorem. Although our proof is not the most elementary, it is short and reveals a connection between Pick’s theorem and the pointwise convergence of multiple Fourier series of piecewise smooth functions.

On the almost everywhere convergence of multiple Fourier series of square summable functions

On the almost everywhere convergence of multiple Fourier series of square summable functions

Rectangular Summation of Multiple Fourier Series and Multi-parametric Capacity

We consider the class of multiple Fourier series associated with functions in the Dirichlet space of the polydisc. We prove that every such series is summable with respect to unrestricted rectangular partial sums, everywhere except for a set of zero multi-parametric logarithmic capacity. Conversely, given a compact set in the torus of zero capacity, we construct a Fourier series in the class which diverges on this set, in the sense of Pringsheim. We also prove that the multi-parametric logarithmic capacity characterizes the exceptional sets for the radial variation and radial limits of Dirichlet space functions. As a by-product of the methods of proof, the results also hold in the vector-valued setting.

Model-Based Analysis and Quantification of Bearing Faults in Induction Machines

The detection of rolling-element bearing fault can be accomplished by monitoring and interpreting a variety of signals, including the vibration, the acoustic noise, and the stator current. The existence of a bearing fault as well as its specific fault type can be readily determined by performing frequency spectral analysis on the monitored signals with various signal processing techniques. However, this traditional approach, despite being simple and intuitive, is not able to identify the severity of a bearing fault in a quantitative manner. Moreover, it is often times tedious and time-consuming to apply this approach to electric machines with different power ratings, as the bearing fault threshold values need to be manually calibrated for each motor running at every possible speed and carrying any possible load. This article, thus, proposes a quantitative approach to estimate the bearing fault severity based on the airgap displacement profile, which is reconstructed from the mutual inductance variation profile estimated from a quantitative electrical model that takes the stator current as input. In addition, the accuracy of the developed electrical model and the estimated bearing fault severity are validated by the simulation and experimental results, and the explicit airgap variation profile is reconstructed with the superposition of multiple Fourier series terms estimated from the stator current via the proposed scheme. The proposed method offers a quantitative and universal bearing fault indicator for induction machines with any power ratings and operating under any speed and load conditions.

Explicit One-Step Numerical Method with the Strong Convergence Order of 2.5 for Ito Stochastic Differential Equations with a Multi-Dimensional Nonadditive Noise Based on the Taylor–Stratonovich Expansion

The article is devoted to construction of effective procedures of the mean-square approximation for iterated Stratonovich stochastic integrals of multiplicities 1 to 5. We apply the method of generalized multiple Fourier series for approximation of iterated stochastic integrals. More precisely, we use multiple Fourier-Legendre series converging in the sense of norm in Hilbert space $L_2([t,T]^k),$ $k\in\mathbb{N}.$ Considered iterated Stratonovich stochastic integrals are part of the Taylor-Stratonovich expansion. That is why the results of the article can be applied to implementation of numerical methods with the orders 1.0, 1.5, 2.0 and 2.5 of strong convergence for Ito stochastic differential equations with multidimensional non-commutative noise.