Sets of values of equivalent almost periodic functions

Sepulcre, Juan Matías; Vidal, Tomás

doi:10.1007/s11139-020-00344-0

Cited by 6 publications

(10 citation statements)

References 8 publications

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“…In this way, by analogy with the recent developments which we made for the space AP(R, C), the Besicovitch almost periodic functions, and other spaces of generalized almost periodic functions (see [16,17,19], respectively), this equivalence relation leads to refine the Bochnertype property or normality in the sense that the condition of almost periodicity in AP(R, X) implies that every sequence of translates has a subsequence that converges, with respect to the topology of AP(R, X), to an equivalent function (see Theorem 3.7 and Corollary 3.8). This extends our previous work on the case of numerical almost periodic functions (see [16,17], but also [18,20]). Moreover, we point out that the proof given here of the main result, and specifically that of Lemma 3.2, is different from those of previous papers.…”

Section: J S E P U L C R E a N D T V I D A Lsupporting

confidence: 85%

A note on spaces of almost periodic functions with values in Banach spaces

Sepulcre¹,

Vidal²

2022

Can. Math. Bull.

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In this paper, we consider an equivalence relation on the space $AP(\mathbb {R},X)$ of almost periodic functions with values in a prefixed Banach space X. In this context, it is known that the normality or Bochner-type property, which characterizes these functions, is based on the relative compactness of the family of translates. Now, we prove that every equivalence class is sequentially compact and the family of translates of a function belonging to this subspace is dense in its own class, i.e., the condition of almost periodicity of a function $f\in AP(\mathbb {R},X)$ yields that every sequence of translates of f has a subsequence that converges to a function equivalent to f. This extends previous work by the same authors on the case of numerical almost periodic functions.

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Section: J S E P U L C R E a N D T V I D A Lsupporting

confidence: 85%

A note on spaces of almost periodic functions with values in Banach spaces

Sepulcre¹,

Vidal²

2022

Can. Math. Bull.

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“…This new equivalence relation, which is widely used in this paper, coincides with that of Bohr for the particular case of general Dirichlet series whose sets of exponents have an integral basis (see [22,Proposition 1]). Other important results derived from this equivalence relation can be seen in [19,20,21,22,23].…”

Section: Introductionmentioning

confidence: 73%

“…In the context of the complex functions which can be represented by a Dirichlet-like series (in particular those almost periodic functions in AP (U, C)), we established in 2018 a new equivalence relation among them, say the * -equivalence, which led to refining Bochner's result that characterizes the almost periodicity (see [19,Theorem 5]) and a thorough extension of Bohr's equivalence theorem (see [22,Theorem 1]). This new equivalence relation, which is widely used in this paper, coincides with that of Bohr for the particular case of general Dirichlet series whose sets of exponents have an integral basis (see [22,Proposition 1]). Other important results derived from this equivalence relation can be seen in [19,20,21,22,23].…”

Section: Introductionmentioning

confidence: 99%

"On the real projections of zeros of analytic almost periodic functions"

Sepulcre¹,

Vidal²

2022

CJM

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"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."

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“…Based on the Bohr's equivalence relation, which was considered in [1, p. 173] for general Dirichlet series, we defined in [7][8][9][10] new equivalence relations in the more general context of the classes S Λ of exponential sums of type (1).…”

Section: The Class Of Functions Equivalent To the Riemann Zeta Functionmentioning

confidence: 99%

“…Throughout this work, we will use a generalization of Bohr's equivalence relation, defined in Sect. 2, which was used in [9] (see also [10]) to get a result like Bohr's equivalence theorem extended to certain classes of almost periodic functions in vertical strips {s ∈ C : α < Re s < β}.…”

Section: Introductionmentioning

confidence: 99%

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

Sepulcre

Vidal

2021

Results Math

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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 $$\{s\in {\mathbb {C}}:\mathrm{Re}\, s>1\}$$ { s ∈ C : Re s > 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 $$\mathrm{Re}\, s=1$$ Re s = 1 . In particular, regarding the Riemann zeta function $$\zeta (s)$$ ζ ( s ) , for every $$\sigma _0>1$$ σ 0 > 1 we can assure the existence of a relatively dense set of real numbers $$\{t_m\}_{m\ge 1}$$ { t m } m ≥ 1 such that the parametrized curve traced by the points $$(\mathrm{Re} (\zeta (\sigma +it_m)),\mathrm{Im}(\zeta (\sigma +it_m)))$$ ( Re ( ζ ( σ + i t m ) ) , Im ( ζ ( σ + i t m ) ) ) , with $$\sigma \in (1,\sigma _0)$$ σ ∈ ( 1 , σ 0 ) , makes a prefixed finite number of turns around the origin.

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Sets of values of equivalent almost periodic functions

Cited by 6 publications

References 8 publications

A note on spaces of almost periodic functions with values in Banach spaces

A note on spaces of almost periodic functions with values in Banach spaces

"On the real projections of zeros of analytic almost periodic functions"

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

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