Composition series of generalized principal series; the case of strongly positive discrete series

In this paper we determine the composition series of the generalized principal seriesδ⋊σ assuming thatσ is strongly positive discrete series.     

Multiplier convergent series and uniform convergence of mapping series

In this paper, we introduce the frame property of complex sequence sets and study the uniform convergence of nonlinear mapping series in β-dual of spaces consisting of multiplier convergent series.     

Overconvergent series of rational functions and universal Laurent series

In this paper, series of rational functions with fixed poles, which have restricted growth near the poles are considered. If they converge with a geometric rate on a continuum, a phenomenon of overconvergence takes place, in the sense that the convergence extends to a certain maximal domain. From this result, some properties of universal Laurent series are derived.     

Differentiability of arithmetic Fourier series arising from Eisenstein series

Let \(k\in \mathbb {N}^*\) be even. We consider two trigonometric series \( F_k(x)= \sum _{n=1}^\infty \frac{\sigma _{k-1}(n)}{n^{k+1}} \sin (2\pi n x)\) and \(G_k(x)= \sum _{n=1}^\infty \frac{\sigma _{k-1}(n)}{n^{k+1}} \cos (2\pi n x),\) where \(\sigma _{k-1}\) is the divisor function. They converge on \(\mathbb {R}\) to continuous functions. In this paper, we examine the differentiability of \(F_k\) and \(G_k\). These functions are related to Eisenstein series, and their (quasi-)modular properties allow us to apply the method proposed by Itatsu in 1981 in the study of the Riemann series. We focus on the case \(k=2\) and we show that the sine series exhibits a different behaviour with respect to differentiability than the cosine series. We prove that the differentiability of \(F_2\) at an irrational x is related to the continued fraction expansion of x. We estimate the modulus of continuity of \(F_2\). We formulate a conjecture concerning differentiability of \(F_k\) and \(G_k\) for any k even.     

Behaviour of an associated conjugate series of a Fourier series

In this paper we obtain a generalisation of a theorem of Mohanty and Ray     

Summation of series

Three approaches are considered to the summation of some ‘standard’ series: an elementary analysis, an approach based on residues, and a technique employing the digamma function and partial fractions. This last procedure is the most convenient to use, but all three techniques have their limitations.     

On the series of absolute values of blocks of trigonometric series

Conditions of integrability with a power weight are obtained for the sums of series of absolute values of blocks of sine series and cosine series under certain special conditions imposed on the coefficients of these series.     

Non-Gaussian series and series with non-zero means: Practical implications for time series analysis

This paper presents a discussion of the relationship between the distribution of the input and the distribution of the output of linear filters. The purpose of the discussion is to focus attention on several fundamental aspects of developing ARIMA models which seem to be overlooked routinely in practice. The authors discuss the implications of non-normally distributed time series realizations and examine the effect of failure to estimate the constant parameter in developing ARIMA models.     

Ramanujan's Eisenstein series and new hypergeometric-like series for

Using hypergeometric identities and certain representations for Eisenstein series, we uniformly derive several new series representations for 1/π².     

Functional series representable as a sum of two universal series

Series with respect to systems Φ{φ_n(x)}_n=1^∞ of measurable and almost everywhere finite functions are discussed. A necessary and sufficient condition for representing any series with respect to a system Φ as a sum of two universal series is formulated. A consequence of the condition is that any series with respect to an arbitrary complete and orthonormal system Φ is a sum of two universal series.