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We analyze the asymptotic behavior of the Apostol-Bernoulli polynomials Bn(χ; λ) in detail. The starting point is their Fourier series on [0, 1] which, it is shown, remains valid as an asymptotic expansion over compact subsets of the complex plane. This is used to determine explicit estimates on the constants in the approximation, and also to analyze oscillatory phenomena which arise in certain cases. These results are transferred to the Apostol-Euler polynomials En(χ; λ) via a simple relation linking them to the Apostol-Bernoulli polynomials. [ABSTRACT FROM AUTHOR]

In order to reduce the Gibbs phenomenon exhibited by the partial Fourier sums of a periodic function $ f$ $ [-\pi,\pi]$ Driscoll and Fornberg considered so-called singular Fourier-Padé approximants constructed from the Hermite-Padé approximants of the system of functions $ (1,g_{1} (z),g_{2} (z))$ $ g_{1} (z)=\log (1-z)$ is analytic, such that $ \operatorname{Re}(g_{2} (e^{it}))=f (t)$. [ABSTRACT FROM AUTHOR]

Modified Fourier series have recently been introduced as an adjustment of classical Fourier series for the approximation of nonperiodic functions defined on $ d$ In this paper we extend Eckhoff's method to the convergence acceleration of multivariate modified Fourier series. By suitable augmentation of the approximation basis we demonstrate how to increase the convergence rate to an arbitrary algebraic order. Moreover, we illustrate how numerical stability of the method can be improved by utilising appropriate auxiliary functions. In the univariate setting it is known that Eckhoff's method exhibits an auto-correction phenomenon. We extend this result to the multivariate case. Finally, we demonstrate how a significant reduction in the number of approximation coefficients can be achieved by using a hyperbolic cross index set. [ABSTRACT FROM AUTHOR]