A Note on Derivative of Sine Series with Square Root

Chaundy and Jolliffe proved that if a n is a nonnegative, nonincreasing real sequence, then series ∑ a n sin n x converges uniformly if and only if n a n ⟶ 0 . The purpose of this paper is to show that if n a n is nonincreasing and n a n ⟶ 0 , then the series f x = ∑ a n sin n x can be differentiated term-by-term on c , d for c , d > 0 . However, f ′ 0 may not exist.

Asymptotics of the Sum of a Sine Series with a Convex Slowly Varying Sequence of Coefficients

We study the asymptotic behavior in a neighborhood of zero of the sum of a sine series g(b,x)=∑k=1∞bksinkx whose coefficients constitute a convex slowly varying sequence b. The main term of the asymptotics of the sum of such a series was obtained by Aljančić, Bojanić, and Tomić. To estimate the deviation of g(b,x) from the main term of its asymptotics bm(x)/x, m(x)=[π/x], Telyakovskiĭ used the piecewise-continuous function σ(b,x)=x∑k=1m(x)−1k2(bk−bk+1). He showed that the difference g(b,x)−bm(x)/x in some neighborhood of zero admits a two-sided estimate in terms of the function σ(b,x) with absolute constants independent of b. Earlier, the author found the sharp values of these constants. In the present paper, the asymptotics of the function g(b,x) on the class of convex slowly varying sequences in the regular case is obtained.

Effect of Pantograph’s Main Structure on the Contact Quality in High-Speed Railway

In general, the electrical energy is provided to the high-speed train through the pantograph-catenary sliding contact. The variation of the dynamic contact force is expected to be small enough to keep the good current collection quality and to extend the service life of the pantograph-catenary system. In this paper, the two tension wires of the catenary are discretized by the sine-series expansions, a multibody dynamics theory based on relative coordinates is adopted to describe the dynamic behavior of the pantograph, and the standard deviation (STD) of dynamic contact force is used as the indicator to evaluate the contact quality. The objective is to investigate how the variations of the pantograph’s main structure influence the contact quality, which may support the structural design and parameter optimization of the pantograph in high-speed railway.