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

Sepulcre, Juan Matías; Vidal, Tomás

doi:10.1007/s00009-014-0444-8

Cited by 3 publications

(7 citation statements)

References 6 publications

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“…Concurrently with the development above, another purpose in this paper is to show that any solution of a functional equation of type (1.4), where S ⊂ C is a certain set that contains 0, cannot be analytic at the origin, except for the solutions given by polynomials (see Proposition 3.1). This also extends [12,Proposition 5], which showed that, under the condition of positive coefficients b j > 0, the only solution of such a functional equation that is analytic at 0 is the trivial solution f ≡ 0. Moreover, under the hypothesis of analyticity on an annulus S centered at 0, we shall obtain the form of other solutions of such functional equations which are analytic on C \ {0} (see Proposition 3.4).…”

Section: Introductionsupporting

confidence: 69%

“…We next prove that the functions f w0 (s) = e w0 Log s , with w 0 belonging to the set of zeros of g(s), are linearly independent in the vector space of functions analytic on Ω. The proof is similar to that of [12,Proposition 6]. Proof.…”

Section: The Practical Correspondence Between Almost Periodic Functions and Their Underlying Functional Equationsmentioning

confidence: 92%

“…where n ∈ N, n ≥ 2, a j are complex numbers, γ j are pair-wise distinct positive numbers and S is a certain set in C. Hence the member on the left is certainly summable, and the expressions of the form (3.1) are included in this type of functional equations, which means that we will handle a particular case of the functional equations of type (1.3) which was already considered in [12]. In fact, we will show more solutions of the functional equation (3.1) (extended to the whole C or with other domains) which are not of the form f w0 (s) indicated above.…”

Section: Solutions Of the Functional Equations Associated With Exponential Polynomialsmentioning

confidence: 99%

“…Note that the proposition above is related to [12,Proposition 5], which shows that, under the condition of positive coefficients a j > 0, the only solution of the functional equation (3.1) that is analytic at 0 is the trivial solution f ≡ 0. As P (0) = 0, the constant solution f 0 (s) = 1 is a particular case, but P (s) has infinitely many zeros (all of them lie in a vertical strip, see for example [9]) which provide other solutions f w0 (s) which are not polynomials.…”

Section: Solutions Of the Functional Equations Associated With Exponential Polynomialsmentioning

confidence: 99%

“…which is of type (1.2). Indeed, it is clear that the functions of the form f n,j (s) = e αn,j Log s , where α n,j is a zero of P n (s) and Log s denotes the principal branch of the logarithm of s = 0, are solutions analytic on C \ (−∞, 0] of the functional equation (1.4) (see also [12]). It is worth noting that this connection was previously shown for the special cases…”

Section: Introductionmentioning

confidence: 99%

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A class of functional equations associated with almost periodic functions

Sepulcre

Vidal

2020

Aequat. Math.

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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.

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Section: Introductionsupporting

confidence: 69%

Section: The Practical Correspondence Between Almost Periodic Functions and Their Underlying Functional Equationsmentioning

confidence: 92%

Section: Solutions Of the Functional Equations Associated With Exponential Polynomialsmentioning

confidence: 99%

Section: Solutions Of the Functional Equations Associated With Exponential Polynomialsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

A class of functional equations associated with almost periodic functions

Sepulcre

Vidal

2020

Aequat. Math.

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The Analytic Solutions of the Functional Equations $$a_{1} f (\gamma _{1} s) + a_{2} f (\gamma _{2} s) + \cdots + a_{n} f (\gamma _{n} s) = h(s)$$

Shen

2023

Mediterr. J. Math.

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Source and genesis of Ca-Cl type brines in Qaidam Basin, Qinghai-Tibetan Plateau: evidence from hydrochemistry as well as B and Li isotopes

Li,

Pan,

et al. 2024

Front. Environ. Sci.

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Ca-Cl brine is a highly mineralized special water body occurring in oceanic ridges or basins. The deep part of the western Qaidam Basin is rich in Ca-Cl brine, which is essential for the formation of (MgSO4-deficient) potash deposits. Previous studies on this type of brine in the Qaidam Basin focused on its source tracing, but different carriers led to different conclusions. Moreover, the genesis of Ca-Cl brines in inland basins still remains unclear. In essence, a solid understanding of the relatively abundant Ca sources for the formation of Ca-Cl brines in inland basins has not been reached. In addition, a metallogenic model for this type of brine has not yet been established. Through hydrochemical analysis as well as Band Li isotope tracing, this study clarified that the main elements (Na and Cl) in these Ca-Cl brines mainly originated from the dissolution of existing evaporites in the inner basin. Further, trace elements in Ca-Cl brines (such as Band Li) mainly originated from lixiviation; atmospheric rainfall infiltrated and interacted with ore-rich mother rocks along the basin-margin fracture system under suitable temperature-pressure conditions; alternatively, they may have originated from mixing with magmatic hydrothermal fluids. This study showed that Ca-Cl brines in the western part of the Qaidam Basin have a typical “binary structure” in terms of their sources. In addition, the sources of Ca in Ca-Cl brines were further discussed based on B and Li isotope data, thus exploring a new genesis of Ca-Cl brines in inland basins. Finally, by integrating the above findings, an ore-forming model of this type of water body in the western part of the Qaidam Basin is proposed. This study is valuable for understanding the genesis mechanism of Ca-Cl brines in inland basins, and also has practical significance for the prediction, exploration, and evaluation of this type of brine resource.

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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

Cited by 3 publications

References 6 publications

A class of functional equations associated with almost periodic functions

A class of functional equations associated with almost periodic functions

The Analytic Solutions of the Functional Equations $$a_{1} f (\gamma _{1} s) + a_{2} f (\gamma _{2} s) + \cdots + a_{n} f (\gamma _{n} s) = h(s)$$

Source and genesis of Ca-Cl type brines in Qaidam Basin, Qinghai-Tibetan Plateau: evidence from hydrochemistry as well as B and Li isotopes

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