Ultimate Generalization to Monotonicity for Uniform Convergence of Trigonometric Series

Chaundy and Jolliffe [4] proved that if $\{a_{n}\}$ is a non-increasing (monotonic) real sequence with $\lim\limits_{n\to \infty}a_{n}=0$, then a necessary and sufficient condition for the uniform convergence of the series $\sum_{n=1}^{\infty}a_{n}\sin nx$ is $\lim\limits_{n\to \infty}na_{n}=0$. 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 and Jolliffe theorem in the complex space.Comment: 21 page

On L1L^{1}-Convergence of Fourier Series Under MVBVMVBV Condition

Let $f\in L_{2\pi}$ be a real-valued even function with its Fourier series $\frac{a_{0}}{2}+\sum_{n=1}^{\infty}a_{n}\cos nx,$ and let $S_{n}(f,x), n\geq 1,$ be the $n$-th partial sum of the Fourier series. It is well-known that if the nonnegative sequence $\{a_{n}\}$ is decreasing and $\lim\limits_{n\to \infty}a_{n}=0$, then $$\lim\limits_{n\to \infty}\Vert f-S_{n}(f)\Vert_{L}=0 {if and only if} \lim\limits_{n\to \infty}a_{n}\log n=0.$$ We weaken the monotone condition in this classical result to the so-called mean value bounded variation ($MVBV$) condition. The generalization of the above classical result in real-valued function space is presented as a special case of the main result in this paper which gives the $L^{1}$% -convergence of a function $f\in L_{2\pi}$ in complex space. We also give results on $L^{1}$-approximation of a function $f\in L_{2\pi}$ under the $$ condition.Comment: 13 Pages, Accepted by Canad. Math. Bul

First-Principles calculation of atomic hydrogen adsorption on Be(10\={1}0) thin films

We present a first-principles study of the atomic hydrogen adsorption onto the Be(10\={1}0) thin film. There are two types of Be(10\={1}0) surfaces according to the interlayer spacing between the surface and its nearest-neighbor layer. We show that the H adsorption features on these two kinds of surfaces are remarkably different. The work function, averaged electrostatic potential, and the local charge density consistently show that the charge is transferred from H to Be for L-type (see the text below) surfaces, while the transfer process is inverted for S-type surfaces.Comment: 7 figure

An exact and explicit formula for pricing Asian options with regime switching

This paper studies the pricing of European-style Asian options when the price dynamics of the underlying risky asset are assumed to follow a Markov- modulated geometric Brownian motion; that is, the appreciation rate and the volatility of the underlying risky asset depend on unobservable states of the economy described by a continuous-time hidden Markov process. We derive the exact, explicit and closed-form solutions for European-style Asian options in a two-state regime switching model.Comment: arXiv admin note: substantial text overlap with arXiv:1407.486

A new condition for the uniform convergence of certain trigonometric series

The present paper proposes a new condition to replace both the ($O$-regularly varying) quasimonotone condition and a certain type of bounded variation condition, and shows the same conclusion for the uniform convergence of certain trigonometric series still holds.Comment: 10 page

A Generalization of Monotonicity Condition and Applications

In the present paper, we introduce a new class of sequences called $NBVS$ to generalize $GBVS$, essentially extending monotonicity from "one sided" to "two sided", while some important classical results keep true.Comment: 21 papges; Accepted by Acta Math. Hunga

A Remark on "Two-Sided" Monotonicity Condition: An Application to LpL^{p} Convergence

To verify the universal validity of the "two-sided" monotonicity condition introduced in [8], we will apply it to include more classical examples. The present paper selects the $L^{p}$ convergence case for this purpose. Furthermore, Theorem 3 shows that our improvements are not trivial.Comment: 10 page

Some Remarks on the Best Approximation Rate of Certain Trigonometric Series

The main object of the present paper is to give a complete result regarding the best approximation rate of certain trigonometric series in general complex valued continuous function space under a new condition which gives an essential generalization to $O$-regularly varying quasimonotonicity. An application is present in Section 3.Comment: 14 page

Lie-point symmetries of the Lagrangian system on time scales

This letter investigates the Lie point symmetries and conserved quantities of the Lagrangian systems on time scales, which unify the Lie symmetries of the two cases for the continuous and the discrete Lagrangian systems. By defining the infinitesimal transformations' generators and using the invariance of differential equations under infinitesimal transformations, the determining equations of the Lie symmetries on time scales are established. Then the structure equations and the form of conserved quantities with delta derivatives are obtained. The letter also gives brief discussion on the Lie symmetries for the discrete systems. Finally, several examples are designed to illustrate these results.Comment: 14 pages,0 figure

On L1 Convergence of Fourier Series of Complex Valued Functions

In the present paper, we give a brief review of $L^{1}$-convergence of trigonometric series. Previous known results in this direction are improved and generalized by establishing a new condition.Comment: 13 pages, Accepted by Studia Sci. Math. Hunga