Chaundy and Jolliffe proved that if {a n } is a non-increasing (monotonic) real sequence with lim n →∞ a n = 0, then a necessary and sufficient condition for the uniform convergence of the series ∑∞ n=1 a n sin nx...Chaundy and Jolliffe proved that if {a n } is a non-increasing (monotonic) real sequence with lim n →∞ a n = 0, then a necessary and sufficient condition for the uniform convergence of the series ∑∞ n=1 a n sin nx is lim n →∞ na n = 0. We generalize (or weaken) the monotonic condition on the coefficient sequence {a n } in this classical result to the so-called mean value bounded variation condition and prove that the generalized condition cannot be weakened further. We also establish an analogue to the generalized Chaundy-Jolliffe theorem in the complex space.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
In this paper the transcendence of values of certain trigonometric series with algebraic coefficients and its derivatives at algebraic points is proved under a general condition dependent only on the coefficients. The...In this paper the transcendence of values of certain trigonometric series with algebraic coefficients and its derivatives at algebraic points is proved under a general condition dependent only on the coefficients. The proof of the theorem is based on a criterion for linear independence over a number field.展开更多
The regularly present paper proposes a new natural weak condition to replace the quasimonotone and Ovarying quasimonotone conditions, and shows the conclusion of the main result in still holds. Moreover, we prove that...The regularly present paper proposes a new natural weak condition to replace the quasimonotone and Ovarying quasimonotone conditions, and shows the conclusion of the main result in still holds. Moreover, we prove that any O-regularly varying quasimonotone condition implies this condition, thus the vague implication of quasimonotonieity can be avoided in practice展开更多
基金supported by National Sciences and Engineering Research Council of CanadaNational Natural Science Foundation of China (Grant No. 10471130)
文摘Chaundy and Jolliffe proved that if {a n } is a non-increasing (monotonic) real sequence with lim n →∞ a n = 0, then a necessary and sufficient condition for the uniform convergence of the series ∑∞ n=1 a n sin nx is lim n →∞ na n = 0. We generalize (or weaken) the monotonic condition on the coefficient sequence {a n } in this classical result to the so-called mean value bounded variation condition and prove that the generalized condition cannot be weakened further. We also establish an analogue to the generalized Chaundy-Jolliffe theorem in the complex space.
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Subject supposed by the National Natural Science Foundation of China (No. 10171097)
文摘In this paper the transcendence of values of certain trigonometric series with algebraic coefficients and its derivatives at algebraic points is proved under a general condition dependent only on the coefficients. The proof of the theorem is based on a criterion for linear independence over a number field.
基金Supported in paxt by Natural Science Foundation of China under the grant number 10471130.
文摘The regularly present paper proposes a new natural weak condition to replace the quasimonotone and Ovarying quasimonotone conditions, and shows the conclusion of the main result in still holds. Moreover, we prove that any O-regularly varying quasimonotone condition implies this condition, thus the vague implication of quasimonotonieity can be avoided in practice
