In the author’s recent publications, a parametric system biorthogonal to the corresponding segment of the exponential Fourier system was unusually effective. On its basis, it was discovered that knowledge of a finite...In the author’s recent publications, a parametric system biorthogonal to the corresponding segment of the exponential Fourier system was unusually effective. On its basis, it was discovered that knowledge of a finite number of Fourier coefficients of function f from an infinite-dimensional set of elementary functions allows f to be accurately restored (the phenomenon of over-convergence). Below, parametric biorthogonal systems are constructed for classical trigonometric Fourier series, and the corresponding phenomena of over-convergence are discovered. The decisive role here was played by representing the space L2 as an orthogonal sum of two corresponding subspaces. As a result, fast parallel algorithms for reconstructing a function from its truncated trigonometric Fourier series are proposed. The presented numerical experiments confirm the high efficiency of these convergence accelerations for smooth functions. In conclusion, the main results of the work are summarized, and some prospects for the development and generalization of the proposed approaches are discussed.展开更多
The current paper considers the problem of recovering a function using a limited number of its Fourier coefficients. Specifically, a method based on Bernoulli-like polynomials suggested and developed by Krylov, Lanczo...The current paper considers the problem of recovering a function using a limited number of its Fourier coefficients. Specifically, a method based on Bernoulli-like polynomials suggested and developed by Krylov, Lanczos, Gottlieb and Eckhoff is examined. Asymptotic behavior of approximate calculation of the so-called ＂jumps＂ is studied and asymptotic L2 constants of the rate of convergence of the method are computed.展开更多
The key objective of this paper is to improve the approximation of a sufficiently smooth nonperiodic function defined on a compact interval by proposing alternative forms of Fourier series expansions. Unlike in classi...The key objective of this paper is to improve the approximation of a sufficiently smooth nonperiodic function defined on a compact interval by proposing alternative forms of Fourier series expansions. Unlike in classical Fourier series, the expansion coefficients herein are explicitly dependent not only on the function itself, but also on its derivatives at the ends of the interval. Each of these series expansions can be made to converge faster at a desired polynomial rate. These results have useful implications to Fourier or harmonic analysis, solutions to differential equations and boundary value problems, data compression, and so on.展开更多
Accurate acceleration acquisition is a critical issue in the robotic exoskeleton system,but it is difficult to directly obtain the acceleration via the existing sensing systems.The existing algorithm-based acceleratio...Accurate acceleration acquisition is a critical issue in the robotic exoskeleton system,but it is difficult to directly obtain the acceleration via the existing sensing systems.The existing algorithm-based acceleration acquisition methods put more attention on finite-time convergence and disturbance suppression but ignore the error constraint and initial state irrelevant techniques.To this end,a novel radical bias function neural network(RBFNN)based fixed-time reconstruction scheme with error constraints is designed to realize high-performance acceleration estimation.In this scheme,a novel exponential-type barrier Lyapunov function is proposed to handle the error constraints.It also provides a unified and concise Lyapunov stability-proof template for constrained and non-constrained systems.Moreover,a fractional power sliding mode control law is designed to realize fixed-time convergence,where the convergence time is irrelevant to initial states or external disturbance,and depends only on the chosen parameters.To further enhance observer robustness,an RBFNN with the adaptive weight matrix is proposed to approximate and attenuate the completely unknown disturbances.Numerical simulation and human sub ject experimental results validate the unique properties and practical robustness.展开更多
该文通过研究无穷序列加速收敛方法 ,在 L evin t-变换的基础上 ,考虑了L evin t-变换的迭代过程 ,提出了 L evin t-变换迭代法 ,指出了这种方法能加快序列的收敛速度 ,给出了理论证明 ,并且通过具体实例给予了证实。同时 ,此法形成了...该文通过研究无穷序列加速收敛方法 ,在 L evin t-变换的基础上 ,考虑了L evin t-变换的迭代过程 ,提出了 L evin t-变换迭代法 ,指出了这种方法能加快序列的收敛速度 ,给出了理论证明 ,并且通过具体实例给予了证实。同时 ,此法形成了循环加速的过程 ,适合于在计算机上进行计算 ,从而在实际应用中具有明显的优越性。对于交错级数部分和序列的加速收敛 。展开更多
The paper considers the Krylov-Lanczos and the Eckhoff approximations for recovering a bivariate function using limited number of its Fourier coefficients. These approximations are based on certain corrections associa...The paper considers the Krylov-Lanczos and the Eckhoff approximations for recovering a bivariate function using limited number of its Fourier coefficients. These approximations are based on certain corrections associated with jumps in the partial derivatives of the approximated function. Approximation of the exact jumps is accomplished by solution of systems of linear equations along the idea of Eckhoff. Asymptotic behaviors of the approximate jumps and the Eckhoff approximation are studied. Exact constants of the asymptotic errors are computed. Numerical experiments validate theoretical investigations.展开更多
Convergence acceleration of the classical trigonometric interpolation by the Eckhoff method is considered, where the exact values of the ＂jumps＂ are approximated by solution of a system of linear equations. The accu...Convergence acceleration of the classical trigonometric interpolation by the Eckhoff method is considered, where the exact values of the ＂jumps＂ are approximated by solution of a system of linear equations. The accuracy of the ＂jump＂ approximation is explored and the corresponding asymptotic error of interpolation is derived. Numerical results validate theoretical estimates.展开更多
In this paper, we investigate not only the acceleration problem of the q-Bernstein polynomials Bn（f, q; x） to B∞ （f, q; x） but also the convergence of their iterated Boolean sum. Using the methods of exact estima...In this paper, we investigate not only the acceleration problem of the q-Bernstein polynomials Bn（f, q; x） to B∞ （f, q; x） but also the convergence of their iterated Boolean sum. Using the methods of exact estimate and theories of modulus of smoothness, we get the respective estimates of the convergence rate, which suggest that q-Bernstein polynomials have the similar answer with the classical Bernstein polynomials to these two problems.展开更多
文摘In the author’s recent publications, a parametric system biorthogonal to the corresponding segment of the exponential Fourier system was unusually effective. On its basis, it was discovered that knowledge of a finite number of Fourier coefficients of function f from an infinite-dimensional set of elementary functions allows f to be accurately restored (the phenomenon of over-convergence). Below, parametric biorthogonal systems are constructed for classical trigonometric Fourier series, and the corresponding phenomena of over-convergence are discovered. The decisive role here was played by representing the space L2 as an orthogonal sum of two corresponding subspaces. As a result, fast parallel algorithms for reconstructing a function from its truncated trigonometric Fourier series are proposed. The presented numerical experiments confirm the high efficiency of these convergence accelerations for smooth functions. In conclusion, the main results of the work are summarized, and some prospects for the development and generalization of the proposed approaches are discussed.
文摘The current paper considers the problem of recovering a function using a limited number of its Fourier coefficients. Specifically, a method based on Bernoulli-like polynomials suggested and developed by Krylov, Lanczos, Gottlieb and Eckhoff is examined. Asymptotic behavior of approximate calculation of the so-called ＂jumps＂ is studied and asymptotic L2 constants of the rate of convergence of the method are computed.
文摘The key objective of this paper is to improve the approximation of a sufficiently smooth nonperiodic function defined on a compact interval by proposing alternative forms of Fourier series expansions. Unlike in classical Fourier series, the expansion coefficients herein are explicitly dependent not only on the function itself, but also on its derivatives at the ends of the interval. Each of these series expansions can be made to converge faster at a desired polynomial rate. These results have useful implications to Fourier or harmonic analysis, solutions to differential equations and boundary value problems, data compression, and so on.
基金Project supported by the Move Robotics Technology Co.,Ltd.the National Natural Science Foundation of China(No.51705163)。
文摘Accurate acceleration acquisition is a critical issue in the robotic exoskeleton system,but it is difficult to directly obtain the acceleration via the existing sensing systems.The existing algorithm-based acceleration acquisition methods put more attention on finite-time convergence and disturbance suppression but ignore the error constraint and initial state irrelevant techniques.To this end,a novel radical bias function neural network(RBFNN)based fixed-time reconstruction scheme with error constraints is designed to realize high-performance acceleration estimation.In this scheme,a novel exponential-type barrier Lyapunov function is proposed to handle the error constraints.It also provides a unified and concise Lyapunov stability-proof template for constrained and non-constrained systems.Moreover,a fractional power sliding mode control law is designed to realize fixed-time convergence,where the convergence time is irrelevant to initial states or external disturbance,and depends only on the chosen parameters.To further enhance observer robustness,an RBFNN with the adaptive weight matrix is proposed to approximate and attenuate the completely unknown disturbances.Numerical simulation and human sub ject experimental results validate the unique properties and practical robustness.
文摘该文通过研究无穷序列加速收敛方法 ,在 L evin t-变换的基础上 ,考虑了L evin t-变换的迭代过程 ,提出了 L evin t-变换迭代法 ,指出了这种方法能加快序列的收敛速度 ,给出了理论证明 ,并且通过具体实例给予了证实。同时 ,此法形成了循环加速的过程 ,适合于在计算机上进行计算 ,从而在实际应用中具有明显的优越性。对于交错级数部分和序列的加速收敛 。
文摘The paper considers the Krylov-Lanczos and the Eckhoff approximations for recovering a bivariate function using limited number of its Fourier coefficients. These approximations are based on certain corrections associated with jumps in the partial derivatives of the approximated function. Approximation of the exact jumps is accomplished by solution of systems of linear equations along the idea of Eckhoff. Asymptotic behaviors of the approximate jumps and the Eckhoff approximation are studied. Exact constants of the asymptotic errors are computed. Numerical experiments validate theoretical investigations.
基金Supported in part by grant PS 1867 from the Armenian National Science and Education Fund (ANSEF) based in New York, USA
文摘Convergence acceleration of the classical trigonometric interpolation by the Eckhoff method is considered, where the exact values of the ＂jumps＂ are approximated by solution of a system of linear equations. The accuracy of the ＂jump＂ approximation is explored and the corresponding asymptotic error of interpolation is derived. Numerical results validate theoretical estimates.
文摘In this paper, we investigate not only the acceleration problem of the q-Bernstein polynomials Bn（f, q; x） to B∞ （f, q; x） but also the convergence of their iterated Boolean sum. Using the methods of exact estimate and theories of modulus of smoothness, we get the respective estimates of the convergence rate, which suggest that q-Bernstein polynomials have the similar answer with the classical Bernstein polynomials to these two problems.
