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"Fourier series."
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Function Spaces of Logarithmic Smoothness: Embeddings and Characterizations
In this paper we present a comprehensive treatment of function spaces with logarithmic smoothness (Besov, Sobolev, Triebel-Lizorkin).
We establish the following results:
The key tools behind our results
are limiting interpolation techniques and new characterizations of Besov and Sobolev norms in terms of the behavior of the Fourier
transforms for functions such that their Fourier transforms are of monotone type or lacunary series.
eBook
Adaptive event-triggered control for a class of nonlinear systems with periodic disturbances
This paper investigates the adaptive event-triggered control problem for a class of nonlinear systems subject to periodic disturbances. To reduce the communication burden, a reliable relative threshold strategy is proposed. Fourier series expansion and radial basis function neural network are combined into a function approximator to model suitable time-varying disturbed function of known periods in strict-feedback systems. By combining the Lyapunov stability theory and the backstepping technique, the proposed adaptive control approach ensures that all the signals in the closed-loop system are bounded, and the tracking error can be regulated to a compact set around zero in finite time. Finally, simulation results are presented to verify the effectiveness of the theoretical results.
Journal Article
A nonlinear vibration isolator supported on a flexible plate: analysis and experiment
To address low-frequency vibration isolation, an issue that engineers often face, this paper first studies the nonlinear energy transfer of a flexible plate, with arbitrary boundary, with the coupling of high-static-low-dynamic-stiffness (HSLDS) isolator. The nonlinear coupled dynamic equation was derived via the Lagrange method, and the improved Fourier series and Rayleigh–Ritz methods provide modal coefficients of the arbitrary boundary flexible plate with nonlinear vibration isolators. The Galerkin and harmonic balance methods approximate the frequency response functions of power flow for the coupled system. The numerical method, via direct integration of the dynamic equation, validates the analytical results of the frequency response functions. In addition, the finite element simulation, used here, validates the analytical results of the mode shapes for flexible plate. The experiment is carried out to validate the isolation performance of the nonlinear vibrator supported on a flexible plate. On these bases, increasing damping and controlling HSLDS can improve the low-frequency isolation efficiency, and nonlinear jumping-phenomena could disappear over a low-frequency range (either frequency overlap or frequency jump). Hence, a properly configured flexible plate could improve the bearing capacity and low-frequency isolation efficiency while avoiding frequency mistune. An explanation for these is offered in the article.
Journal Article
Degree of convergence of the functions of trigonometric series in Sobolev spaces and its applications
In this paper, we study the degree of convergence of the functions of Fourier series and conjugate Fourier series in Sobolev spaces using Riesz means. We also study some applications of our main results and observe that our results are much better than earlier results.
Journal Article
Cascade Object Detection and Remote Sensing Object Detection Method Based on Trainable Activation Function
Object detection is an important process in surveillance system to locate objects and it is considered as major application in computer vision. The Convolution Neural Network (CNN) based models have been developed by many researchers for object detection to achieve higher performance. However, existing models have some limitations such as overfitting problem and lower efficiency in small object detection. Object detection in remote sensing hasthe limitations of low efficiency in detecting small object and the existing methods have poor localization. Cascade Object Detection methods have been applied to increase the learning process of the detection model. In this research, the Additive Activation Function (AAF) is applied in a Faster Region based CNN (RCNN) for object detection. The proposed AAF-Faster RCNN method has the advantage of better convergence and clear bounding variance. The Fourier Series and Linear Combination of activation function are used to update the loss function. The Microsoft (MS) COCO datasets and Pascal VOC 2007/2012 are used to evaluate the performance of the AAF-Faster RCNN model. The proposed AAF-Faster RCNN is also analyzed for small object detection in the benchmark dataset. The analysis shows that the proposed AAF-Faster RCNN model has higher efficiency than state-of-art Pay Attention to Them (PAT) model in object detection. To evaluate the performance of AAF-Faster RCNN method of object detection in remote sensing, the NWPU VHR-10 remote sensing data set is used to test the proposed method. The AAF-Faster RCNN model has mean Average Precision (mAP) of 83.1% and existing PAT-SSD512 method has the 81.7%mAP in Pascal VOC 2007 dataset.
Journal Article
Convergence Almost Everywhere of Partial Sums and Féjer Means of Vilenkin–Fourier Series
We characterize subsequences$$\\{S_{n_k}\\}$$S n k of partial sums with respect to (bounded or unbounded) Vilenkin systems of$$fın L^1(G_m)$$f ∈ L 1 ( G m ) for which almost everywhere convergence holds. Moreover, we construct an explicit$$fın L^p(G_m), \\ 1\\le p<ınfty $$f ∈ L p ( G m ) , 1 ≤ p < ∞ whose partial sums (satisfying the same conditions which guarantee almost everywhere convergence) diverges on any set of measure zero. We also prove a similar divergence result for Vilenkin–Féjer means.
Journal Article
Rate of Convergence for Double Rational Fourier Series
We calculate the rate of convergence of the double rational Fourier series for regular, bounded, measurable, and two-variable functions. The rectangular oscillation of the two-variable function is used to quantify this rate. Additionally, we give an approximation of convergence rate of the double rational Fourier series for continuous functions with generalized bounded variation.
Journal Article
Fractional fourier series and its applications
In this article, we use a new multiplication to propose a definition of fractional Fourier series, regarding the Jumarie type of modified Riemann-Liouville fractional derivatives. Furthermore, several examples are provided to illustrate the applications of fractional Fourier series.
Journal Article
Dynamic behavior analysis of a spinning Timoshenko beam-rigid disk with nonlinear elastic boundaries under axial loading
In this study, an attempt is made to model and investigate the dynamic behavior of the spinning Timoshenko beam-disk with nonlinear elastic boundaries, in which an unbalanced concentrated mass and axial loads are considered. In order to satisfy the elastic boundary conditions of the spinning beam-disk containing translational and rotational stiffnesses as well as nonlinear stiffnesses, an improved version of Fourier series is employed for the admission function construction. Nonlinear dynamic behavior of the spinning beam-disk and its boundary supporting system are described based on energy principle, and the system governing equations of spinning beam-disk are formulated by the
Lagrange
equation of the second type. The time-domain response is then obtained by solving the system dynamic equations through
Runge–Kutta
method, while the reliability of the current model is verified through the comparison with those predicted by harmonic balance method. Then, the effect of sweep direction and nonlinear elastic boundary parameters on system dynamic behavior of the spinning Timoshenko beam-disk is investigated and addressed. The results show that the dynamic responses of the spinning beam-disk with nonlinear elastic boundary are sensitive to the initial values of calculation, and the nonlinear elastic boundary parameters make the spinning beam-disk exhibit complex dynamic behavior. Analysis of Poincare points in the phase diagram can better determine the dynamic behavior of spinning beam-disk, and a set of suitable nonlinear elastic boundary parameters can suppress the complex dynamic response of the spinning beam-disk.
Journal Article