Almost periodic functions in terms of Bohr’s equivalence relation

In this paper we introduce an equivalence relation on the classes of almost periodic functions of a real or complex variable which is used to refine Bochner’s result that characterizes these spaces of functions. In fact, with respect to the topology of uniform convergence, we prove that the limit points of the family of translates of an almost periodic function are precisely the functions which are equivalent to it, which leads us to a characterization of almost periodicity. In particular we show that any exponential sum which is equivalent to the Riemann zeta function, ζ(s), can be uniformly approximated in {s = σ +i t : σ > 1} by certain vertical translates of ζ(s).The first author’s research was partially supported by Generalitat Valenciana under Project GV/2015/035

Equivalence classes of exponential polynomials with the same set of zeros

Through several equivalence binary relations, in this paper we identify, on the one hand, groups of exponential polynomials with the same set of zeros, and on the other, groups of functional equations of the form a1f(1z) + a2f(2z) + : : : + anf(nz) = 0; z 2 C that lead to equivalent exponential polynomials with the same set of zeros

On the Analytic Solutions of the Functional Equations w 1 f(a 1 z) + w 2 f(a 2 z) + ... + w n f(a n z) = 0

In this paper, it is showed that, given an integer number n ≥ 2, each zero of an exponential polynomial of the form w1az1+w2az2+⋯+wnazn, with non-null complex numbers w 1,w 2,…,w n and a 1,a 2,…,a n , produces analytic solutions of the functional equation w 1 f(a 1 z) + w 2 f(a 2 z) + ... + w n f(a n z) = 0 on certain domains of C, which represents an extension of some existing results in the literature on this functional equation for the case of positive coefficients a j and w j

On the real projections of zeros of analytic almost periodic functions

This paper deals with the sets of real projections of zeros of analytic almost periodic functions defined in a vertical strip. By using our equivalence relation introduced in the context of the complex functions which can be represented by a Dirichlet-like series, this work provides practical results in order to determine whether a real number belongs to the closure of such a set. Its main result shows that, in the case that the Fourier exponents of an analytic almost periodic function are linearly independent over the rational numbers, such a set has no isolated points.The first author has been partially supported by MICIU of Spain under project number PGC2018-097960-B-C22

Bochner-Type Property on Spaces of Generalized Almost Periodic Functions

Our paper is focused on spaces of generalized almost periodic functions which, as in classical Fourier analysis, are associated with a Fourier series with real frequencies. In fact, based on a pertinent equivalence relation defined on the spaces of almost periodic functions in Bohr, Stepanov, Weyl and Besicovitch’s sense, we refine the Bochner-type property by showing that the condition of almost periodicity of a function in any of these generalized spaces can be interpreted in the way that, with respect to the topology of each space, the closure of its set of translates coincides with its corresponding equivalence class.The first author’s research was supported by PGC2018-097960-B-C22 (MCIU/AEI/ERDF, UE)

A class of functional equations associated with almost periodic functions

In this paper we will get a class of functional equations involving a countable set of terms, summed by the well known Bochner–Fejér summation procedure, which are closely associated with the set of almost periodic functions. We will show that the zeros of a prefixed almost periodic function determine analytic solutions of such a functional equation associated with it, and we will obtain other solutions which are analytic or meromorphic on a certain domain.J. M. Sepulcre was supported by PGC2018-097960-B-C22 (MCIU/AEI/ERDF, UE)

Equivalent almost periodic functions in terms of the new property of almost equality

In this paper we introduce the notion of almost equality (or, more specifically, almost equality by translations) of complex functions of an unrestricted real variable in terms of the new concept of ϵ-translation number of a function with respect to other one, which is inspired by Bohr’s notion of ϵ-translation number associated with an almost periodic function. We develop the main properties of this new class of functions and obtain a characterization through a very important equivalence relation which we introduced in previous papers in the context of the almost periodicity.The first author was supported by PGC2018-097960-B-C22 (MCIU/AEI/ERDF, UE)

Modelling the spinning dust emission from LDN 1780

We study the anomalous microwave emission (AME) in the Lynds Dark Nebula (LDN) 1780 on two angular scales. Using available ancillary data at an angular resolution of 1 degree, we construct an SED between 0.408 GHz to 2997 GHz. We show that there is a significant amount of AME at these angular scales and the excess is compatible with a physical spinning dust model. We find that LDN 1780 is one of the clearest examples of AME on 1 degree scales. We detected AME with a significance > 20$\sigma$. We also find at these angular scales that the location of the peak of the emission at frequencies between 23-70 GHz differs from the one on the 90-3000 GHz map. In order to investigate the origin of the AME in this cloud, we use data obtained with the Combined Array for Research in Millimeter-wave Astronomy (CARMA) that provides 2 arcmin resolution at 30 GHz. We study the connection between the radio and IR emissions using morphological correlations. The best correlation is found to be with MIPS 70$\mu$m, which traces warm dust (T$\sim$50K). Finally, we study the difference in radio emissivity between two locations within the cloud. We measured a factor $\approx 6$ of difference in 30 GHz emissivity. We show that this variation can be explained, using the spinning dust model, by a variation on the dust grain size distribution across the cloud, particularly changing the carbon fraction and hence the amount of PAHs.Comment: 14 pages, 11 figures, submitted to MNRA

The Vortex-like Behavior of the Riemann Zeta Function to the Right of the Critical Strip

Based on an equivalence relation that was established recently on exponential sums, in this paper we study the class of functions that are equivalent to the Riemann zeta function in the half-plane {s ∈ C : Res > 1}. In connection with this class of functions, we first determine the value of the maximum abscissa from which the images of any function in it cannot take a prefixed argument. The main result shows that each of these functions experiments a vortex-like behavior in the sense that the main argument of its images varies indefinitely near the vertical line Re s = 1. In particular, regarding the Riemann zeta function ζ(s), for every σ0 > 1 we can assure the existence of a relatively dense set of real numbers {tm}m≥1 such that the parametrized curve traced by the points (Re(ζ(σ+itm)), Im(ζ(σ+itm))), with σ ∈ (1, σ0), makes a prefixed finite number of turns around the origin.The first author was supported by PGC2018-097960-B-C22 (MCIU/AEI/ERDF, UE). Open Access funding provided thanks to the CRUE-CSIC agreement with Springer Nature