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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
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
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
Rate of Convergence of Double Rational Fourier Series of Functions of Generalized Bounded Variation
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In this paper, the rate of convergence of double rational Fourier series and in particular, double trigonometric Fourier series of functions of generalized bounded variation is estimated.
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
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
THE JAIN-MONRAD CRITERION FOR ROUGH PATHS AND APPLICATIONS TO RANDOM FOURIER SERIES AND NON-MARKOVIAN HÖRMANDER THEORY
We discuss stochastic calculus for large classes of Gaussian processes, based on rough path analysis. Our key condition is a covariance measure structure combined with a classical criterion due to Jain and Monrad [Ann. Probab. 11 (1983) 46-57]. This condition is verified in many examples, even in absence of explicit expressions for the covariance or Volterra kernels. Of special interest are random Fourier series, with covariance given as Fourier series itself, and we formulate conditions directly in terms of the Fourier coefficients. We also establish convergence and rates of convergence in rough path metrics of approximations to such random Fourier series. An application to SPDE is given. Our criterion also leads to an embedding result for Cameron-Martin paths and complementary Young regularity (CYR) of the Cameron-Martin space and Gaussian sample paths. CYR is known to imply Malliavin regularity and also Itô-like probabilistic estimates for stochastic integrals (resp., stochastic differential equations) despite their (rough) pathwise construction. At last, we give an application in the context of non-Markovian Hörmander theory.
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