A note on the time series which is the product of two stationary time series

The time series […,x_-1y_-1,x₀y₀,x₁y₁,…]> which is the product of two stationary time series x_t and y_t is studied. Such sequences arise in the study of nonlinear time series, censored time series, amplitude modulated time series, time series with random parameters, and time series with missing observations. The mean and autocovariance function of the product sequence are derived.     

BMO- andL p -conditions for power series and Dirichlet series with positive coefficients

It is known that theL _p -norms of the sums of power series can be estimated from below and above by means of their coefficients, provided these coefficients are nonnegative. In the paper we prove analogous estimates for theL _p -norms of the sums of Dirichlet series. Our main result gives exact lower and upper estimates for the BMO-norm of the sums of power series and Dirichlet series, respectively, by means of their coefficients.     

Hadamard grade of power series

The Hadamard product of two power series ∑anzn and ∑bnzn is the power series ∑anbnzn. We define the (Hadamard) grade of a power series A to be the least number (finite or infinite) of algebraic power series, the Hadamard product of which equals A. We study and discuss this notion.     

Eisenstein series and Ramanujan-type series for 1/π

Using certain representations for Eisenstein series, we uniformly derive several Ramanujan-type series for 1/π.     

On p-adic quaternionic Eisenstein series

We show that certain p-adic Eisenstein series for quaternionic modular groups of degree 2 become “real” modular forms of level p in almost all cases. To prove this, we introduce a U(p) type operator. We also show that there exists a p-adic Eisenstein series of the above type that has transcendental coefficients. Former examples of p-adic Eisenstein series for Siegel and Hermitian modular groups are both rational (i.e., algebraic).     

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.     

Transformation of the triple series of Gelfand, Graev, and Retakh into a series of the same type and related problems

A transformation of the triple series T related to the GrassmanianG _2,4 into a series of the same structure type is obtained. This transformation generalizes the reduction formula of Gelfand, Graev, and Retakh taking the series T to the Gauss function under two additional conditions and two more general reduction formulas taking the series T to the Appell function F ₁ and to the Horn function G ₂ under one of the additional conditions. The approach used to analyze the series T is based on the representation of the initial series T in terms of series with convenient computational properties.     

Chirp transforms and Chirp series

Motivated by the recent work on the non-harmonic Fourier atoms initiated by T. Qian and the non-harmonic Fourier series which originated from the celebrated work of Paley and Wiener, we introduce an integral version of the non-harmonic Fourier series, called Chirp transform. As an integral transform with kernel ei?(t)θ(ω), the Chirp transform is an unitary isometry from L²(R,d?) onto L²(R,dθ) and it can be explicitly defined in terms of generalized Hermite polynomials. The corresponding Chirp series take einθ(t) as a basis which in some sense is dual to the theory of non-harmonic Fourier series which take eiλnt as a basis. The Chirp version of the Shannon sampling theorem and the Poisson summation formula are also considered by dealing with sampling points which may non-equally distributed. Since the Chirp transform interchanges weighted derivatives into multiplications, it plays a role in solving certain differential equations with variable coefficients. In addition, we extend T. Qian's theorem on the characterization of a measure to be a linear combination of a number of harmonic measures on the unit disc with positive integer coefficients to that with positive rational coefficients.     

A class of trigonometric series

Trigonometric series with coefficientsa _k → 0 under the condition $$(\exists p \in R,p > 1):\left( {\sum\nolimits_{n = 1}^\infty {\left\{ {\sum\nolimits_{k = n}^\infty {|\Delta a_k |p_{/n} } } \right\}^{1/p}< \infty } } \right)$$ are considered. It is shown that, under these conditions, the cosine series is a Fourier series for which the conditiona _n In n → 0 is the criterion for convergence in the metric of L. For the sine series, this is true under the further assumption that ∑ _n=1 ^∞ |a _n |/n<∞.     

Principal series and generalized principal series Whittaker functions with peripheral K-types on the real symplectic group of rank 2

First, we give some explicit formulas of principal series Whittaker functions on the real symplectic group of rank 2 with arbitrary one-dimensional K-types. These formulas are extension of Ishii??s formulas for Whittaker functions with minimal K-types. Secondly, we compute explicit formulas of the holonomic system for the radial part of Whittaker functions with peripheral K-types belonging to the generalized principal series representations induced from the Siegel maximal parabolic subgroup (i.e., P _S-series). Thirdly, we derive eight power series solutions for our holonomic system utilizing the embedding of the P _S-series into various principal series, from the power series Whittaker functions belonging to the principal series.     

Convergence and growth of multiple Dirichlet series

This article investigates the convergence and growth of multiple Dirichlet series. The Valiron formula of Dirichlet series is extended to n-tuple Dirichlet series and an equivalence relation between the order of n-tuple Dirichlet series and its coefficients and exponents is obtained.     

Dirichlet series in function fields

We study certain functions from the p-adic integers to a locally compact field of characteristic p: finite linear combinations of exponentials and their uniform limits (which we call Dirichlet series). Our main result is that such a Dirichlet series is determined by its restriction to an arbitrarily small open subset of the p-adic integers.     

Matrices related to Dirichlet series

We attach a certain n×n matrix An to the Dirichlet series . We study the determinant, characteristic polynomial, eigenvalues, and eigenvectors of these matrices. The determinant of An can be understood as a weighted sum of the first n coefficients of the Dirichlet series L(s)⁻¹. We give an interpretation of the partial sum of a Dirichlet series as a product of eigenvalues. In a special case, the determinant of An is the sum of the Möbius function. We disprove a conjecture of Barrett and Jarvis regarding the eigenvalues of An.     

Increasing the convergence rate of series

This paper deals with the question of obtaining from the sequence {s_n} of partial sums of a convergent series s a new sequence {t_n} which converges to the same limit s as s_n, but more rapidly. When the general term u_n of the series s possesses certain types of expansion involving inverse powers of n, it is shown how t_n is obtained by adding a fixed number of terms to s_n. When the series s is convergent, these terms tend to zero as n tends to infinity, but they are such as to make t_n much more rapidly convergent to s—in fact we can make the convergence rate as great as we wish. Explicit general formulas are obtained for a wide range of important series.     

Hilbert series of subspace arrangements

The vanishing ideal I of a subspace arrangement V₁∪V₂∪?∪V_m⊆V is an intersection I₁∩I₂∩?∩I_m of linear ideals. We give a formula for the Hilbert polynomial of I if the subspaces meet transversally. We also give a formula for the Hilbert series of the product ideal J=I₁I₂?I_m without any assumptions about the subspace arrangement. It turns out that the Hilbert series of J is a combinatorial invariant of the subspace arrangement: it only depends on the intersection lattice and the dimension function. The graded Betti numbers of J are determined by the Hilbert series, so they are combinatorial invariants as well. We will also apply our results to generalized principal component analysis (GPCA), a tool that is useful for computer vision and image processing.     

Transformation and summation formulas for double hypergeometric series

A number of new transformation formulas for double hypergeometric series are presented. The series appearing here are the so-called Kampé de Fériet functions of type F_1:1;2^0:3;4(1,1) and F_0:2;2^1:2;2(1,1). The transformation formulas relate such double series to a single hypergeometric series of ₄F₃(1) type. By specializing certain parameters, a list of new summation formulas for F_0:2;2^1:2;2(1,1) series is obtained. The origin of the results comes from studying symmetries of the 9-j coefficient appearing in quantum theory of angular momentum.     

Abstract theory of universal series and an application to Dirichlet series

We present an abstract theory of universal series; in particular, we give a necessary and sufficient condition for the existence of universal series of a certain type. Most of the known results can be proved or strengthened by using this condition. We also obtain new results, for example, related to universal Dirichlet series. To cite this article: V. Nestoridis, C. Papadimitropoulos, C. R. Acad. Sci. Paris, Ser. I 341 (2005).     

A recursive formula for even order harmonic series

A useful recursive formula for obtaining the infinite sums of even order harmonic series Σ^∞_n=1 (1/n^2k), k = 1, 2, …, is derived by an application of Fourier series expansion of some periodic functions. Since the formula does not contain the Bernoulli numbers, infinite sums of even order harmonic series may be calculated by the formula without the Bernoulli numbers. Infinite sums of a few even order harmonic series, which are calculated using the recursive formula, are tabulated for easy reference.     

Automatic β-expansions of formal Laurent series over finite fields

We consider β-expansions of formal Laurent series over finite fields. If the base β is a Pisot or Salem series, we prove that the β-expansion of a Laurent series α is automatic if and only if α is algebraic.     

Principal series representations and harmonic spinors

Let G be a real reductive Lie group and G/H a reductive homogeneous space. We consider Kostant's cubic Dirac operator D on G/H twisted with a finite-dimensional representation of H. Under the assumption that G and H have the same complex rank, we construct a nonzero intertwining operator from principal series representations of G into the kernel of D. The Langlands parameters of these principal series are described explicitly. In particular, we obtain an explicit integral formula for certain solutions of the cubic Dirac equation D=0 on G/H.