In this paper, the following question is considered: what conditions on a strictly increasing sequence of positive integers superscriptsubscriptsubscriptð'>ð'-ð'-1 guarantee that the sum of the ...series superscriptsubscriptð'-1superscriptsubscriptð'~subscriptð'>ð'-subscriptð'>ð'-11subscriptð'ð'~ð'"subscriptð'¤ð'~ð'¥ where subscriptð'ð'~ð'" are the Walsh–Fourier coefficients of a function ð'", belongs to the space superscriptð¿ð'01, ð'1, for any function ð'" of bounded variation? For ð', it is proved that such a sequence does not exist. For finite ð'1, sufficient conditions are obtained for the sequence subscriptð'>ð'-; these conditions are similar to the ones obtained by the first author in the trigonometric case.
Given a closed subset E of Lebesgue measure zero on the unit circle T there is a function f on T with uniformly convergent symmetric Fourier series Sn(f,ζ)=∑k=−nnfˆ(k)ζk⇉Tf(ζ),such that for every ...continuous function g on E, there is a subsequence of partial power sums Sn+(f,ζ)=∑k=0nfˆ(k)ζkof f, which converges to g uniformly on E. Here fˆ(k)=∫Tζ̄kf(ζ)dm(ζ),and m is the normalized Lebesgue measure on T.
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.
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.
In this paper we discuss and prove an analogy of the Carleson–Hunt theorem with respect to Vilenkin systems. In particular, we use the theory of martingales and give a new and shorter proof of the ...almost everywhere convergence of Vilenkin–Fourier series of
f
∈
L
p
(
G
m
)
for
p
>
1
in case the Vilenkin system is bounded. Moreover, we also prove sharpness by stating an analogy of the Kolmogorov theorem for
p
=
1
and construct a function
f
∈
L
1
(
G
m
)
such that the partial sums with respect to Vilenkin systems diverge everywhere.
In recent earthquakes, tunnels in earthquake-prone zones have suffered significant damage. The cracks in these tunnels contain a mix of transverse, longitudinal and inclined cracks, suggesting a ...complex 3D tunnel response in these zones. In this article, we semi-analytically solve for the complete 3D response of a shallow lined tunnel embedded in a linear elastic homogeneous half-space under seismic waves that are incident on the tunnel from arbitrary directions. The waves scattered by the tunnel are represented using cylindrical waves, and the coefficients of these waves are estimated by enforcing the stress-free condition at the tunnel and ground surface, and the compatibility condition at the soil-liner interface. Accurate enforcement of the ground surface stress-free condition is accomplished by solving an improper contour integral that requires special treatment at branch points, removable singularities and poles. The convergence of the solution series is investigated, and the converged stresses and displacements at various locations in the soil and tunnel liner are compared with the results from past semi-analytical and numerical studies for 2D and 3D scenarios. Robustness and accuracy of the algorithm over a wide range of tunnel liner-to-soil stiffness ratios, tunnel depth-to-radius ratios, tunnel liner thickness-to-radius ratios, incident wave frequencies and directions are then tested to demonstrate the applicability of this algorithm as a fast and reliable tool for the initial design of a tunnel in an earthquake-prone zone.
In this paper, an adaptive tracking control approach is developed for full-state constrained switched nonlinear systems that have actuator saturation, periodic disturbances and unknown control ...direction. To deal with the full-state constraints, the Barrier Lyapunov functions are introduced to limit the state variables within the corresponding constraint conditions. Meanwhile, the Fourier series expansion technology is employed to deal with unknown periodic disturbances and unknown nonlinear dynamics jointly. Additionally, a Nussbaum-type function is used in the controller design to cope with the and unknown control gain and input saturation. On the basis of the Lyapunov stability theory, it is demonstrated rigorously that all signals of the closed-loop system are uniformly ultimately bounded, and the proposed controller ensures that the tracking error is kept within a compact set close to zero. In the end, the validity of the designed control protocol is verified by a simulation example.
In this paper, we obtain degree of convergence of a function of two‐dimensional variables in generalized Hölder spaces by matrix means of its conjugate Fourier series. We also obtain degree of ...convergence of a function of N‐dimensional variables in generalized Hölder spaces by matrix means of its conjugate Fourier series. We also deduce important corollaries from our main results.
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