On the stability of entire multiple Dirichlet series

M. M. Dolynyuk; O. B. Skaskiv

doi:10.15330/ms.43.2.171-179

Let

$D^p(\lambda)$ be the class of entire multiple Dirichlet series of the form

$F(z)=\sum\nolimits_{\|n\|=0}^{+\infty}{a_{n}\textrm{e}^{(z,\lambda_{n})}}$ ,

$z\in\mathbb{C}^{p},$ \

$p\geq 1$ , with exponents satisfying the conditions

$\lambda_n=(\lambda^{(1)}_{n_1},\ldots,\lambda^{(p)}_{n_p})$ ,

$n=(n_1,\ldots,n_p)\in\mathbb{Z}^{p}_{+}$ ,

$0\le\lambda^{(j)}_{k}\leq\lambda^{(j)}_{k+1}\to +\infty$ \

$(0\leq k\to +\infty)$ ;\

$w\colon [0,+\infty)\to [0,+\infty)$ {a} nondecreasing function, and

$\nu_1(t)=\sum_{\|\lambda_n\|\leq t}e^{w(\|\lambda_n\|)}$ ,

$\|a\|=a_1+\ldots+a_p$ ,

$(a,b)=a_1b_1+\ldots+a_pb_p$ , for

$a=(a_1,\ldots, a_p),\ b=(b_1,\ldots, b_p)\in\mathbb{C}^p$ .\ If

$\int_{1}^{+\infty}t^{-1}d\ln\nu_1(t)$ is finite and

$F_w\in D^p(\lambda)$ ,\

$F_w(z)=\sum\nolimits_{\|n\|=0}^{+\infty}{a_{n}\textrm{e}^{w(\|\lambda_n\|)+(z,\lambda_{n})}}$ .\ {Then}

$\ln\max\{|a_n|e^{w(\|\lambda_n\|)+(\sigma,\lambda_n)}\colon n\in\mathbb{Z}^{p}_{+}\}\sim\ln\max\{|a_n|e^{(\sigma,\lambda_n)}\colon n\in\mathbb{Z}^{p}_{+}\}$ as

$|\sigma| \to +\infty$ \

$(\sigma\in K\setminus E)$ , for {an} arbitrary cone

$K$ in

$\mathbb{R}^p_+$ with vertex at the point

$O$ such that

$\overline{K}\backslash\{O\}\subset\mathbb{R}^p_+,$ and {a} measurable set

$E\subset \mathbb{R}_{+}^{p}$ such that

$\tau_{p}(E\cap K)=\int\nolimits_{E\cap K}\frac{d\sigma_1\ldots d\sigma_p}{|\sigma|^{p-1}}$ is finite.

1. Gaisin A.M. The estimate of a Dirichlet series with Fejer gaps// Dokl. RAN. – 2000. – V.370, №6. – P. 735–737.

2. Skaskiv O.B., Trakalo O.M. On the stability of the maximum term of the entire Dirichlet series// Ukr. Mat. Zh. – 2005. – V.57, №4. – P. 571–576. (in Ukrainian); English transl. in Ukr. Math. J. – 2005. – V.57, №4. – P. 686–693.

3. Leont’ev A.F. Exponential series. – Мoscow: Nauka, 1976. – 536 p.

4. Mulyava O.M. On the abscissa of the convergence of the Dirichlet series// Mat. Stud. – 1998. – V.8, №2. – P. 171–176. (in Ukrainian)

5. Skaskiv O.B. The stability of the maximum of a sequence of linear functions// International conference dedicated to 125-th anniversary of Hans Hahn: Book of abstracts. – International conf. (Chernivtsi, June 27–July 3, 2004). – P. 100–101. (in Ukrainian)

6. Skaskiv O.B. The stability of the maximum of a sequence of linear functions// Mat. Visn. NTSh. – 2004. – V.1. – P. 120–129. (in Ukrainian)

7. Dolynyuk M., Skaskiv O. The stability of the maximal term of entire multiple Dirichlet series// International V.Ya. Skorobohatko mathematical conference: Book of abstracts. – International conf. (Drohobych, September 24–28, 2007). – Lviv, 2007. – P. 94. (in Ukrainian)

8. Skaskiv O.B., Trakalo O.M. Asymptotic estimates for Laplace integrals// Mat. Stud. – 2002. – V.18, №2. - P. 125–146. (in Ukrainian)

9. Skaskiv O.B., Zikrach D.Yu. On the best possible description of exceptional set in asymptotic estimates for Laplace–Stieltjes integrals// Mat. Stud. – 2011. – V.35, №2. – P. 131–141.

10. Skaskiv O.B. On the behaviour of the maximum term of a Dirichlet series defining an entire function// Mat. Zametki. – 1985. – V.37, №1. – P. 41–47.; English transl. in Math. Notes, 1985, V.37, №1, 24–28.

	On the stability of entire multiple Dirichlet series
Author	M. M. Dolynyuk, O. B. Skaskiv mira0201@rambler.ru, olskask@gmail.com Ivan Franko National University of Lviv
Abstract	Let $D^p(\lambda)$ be the class of entire multiple Dirichlet series of the form $F(z)=\sum\nolimits_{\\|n\\|=0}^{+\infty}{a_{n}\textrm{e}^{(z,\lambda_{n})}}$ , $z\in\mathbb{C}^{p},$ \ $p\geq 1$ , with exponents satisfying the conditions $\lambda_n=(\lambda^{(1)}_{n_1},\ldots,\lambda^{(p)}_{n_p})$ , $n=(n_1,\ldots,n_p)\in\mathbb{Z}^{p}_{+}$ , $0\le\lambda^{(j)}_{k}\leq\lambda^{(j)}_{k+1}\to +\infty$ \ $(0\leq k\to +\infty)$ ;\ $w\colon [0,+\infty)\to [0,+\infty)$ {a} nondecreasing function, and $\nu_1(t)=\sum_{\\|\lambda_n\\|\leq t}e^{w(\\|\lambda_n\\|)}$ , $\\|a\\|=a_1+\ldots+a_p$ , $(a,b)=a_1b_1+\ldots+a_pb_p$ , for $a=(a_1,\ldots, a_p),\ b=(b_1,\ldots, b_p)\in\mathbb{C}^p$ .\ If $\int_{1}^{+\infty}t^{-1}d\ln\nu_1(t)$ is finite and $F_w\in D^p(\lambda)$ ,\ $F_w(z)=\sum\nolimits_{\\|n\\|=0}^{+\infty}{a_{n}\textrm{e}^{w(\\|\lambda_n\\|)+(z,\lambda_{n})}}$ .\ {Then} $\ln\max\{\|a_n\|e^{w(\\|\lambda_n\\|)+(\sigma,\lambda_n)}\colon n\in\mathbb{Z}^{p}_{+}\}\sim\ln\max\{\|a_n\|e^{(\sigma,\lambda_n)}\colon n\in\mathbb{Z}^{p}_{+}\}$ as $\|\sigma\| \to +\infty$ \ $(\sigma\in K\setminus E)$ , for {an} arbitrary cone $K$ in $\mathbb{R}^p_+$ with vertex at the point $O$ such that $\overline{K}\backslash\{O\}\subset\mathbb{R}^p_+,$ and {a} measurable set $E\subset \mathbb{R}_{+}^{p}$ such that $\tau_{p}(E\cap K)=\int\nolimits_{E\cap K}\frac{d\sigma_1\ldots d\sigma_p}{\|\sigma\|^{p-1}}$ is finite.
Keywords	Dirichlet series; maximal term; stability; exceptional set; asymptotic estimate; Laplace-Stieltjes integral
DOI	doi:10.15330/ms.43.2.171-179
Reference	1. Gaisin A.M. The estimate of a Dirichlet series with Fejer gaps// Dokl. RAN. – 2000. – V.370, №6. – P. 735–737. 2. Skaskiv O.B., Trakalo O.M. On the stability of the maximum term of the entire Dirichlet series// Ukr. Mat. Zh. – 2005. – V.57, №4. – P. 571–576. (in Ukrainian); English transl. in Ukr. Math. J. – 2005. – V.57, №4. – P. 686–693. 3. Leont’ev A.F. Exponential series. – Мoscow: Nauka, 1976. – 536 p. 4. Mulyava O.M. On the abscissa of the convergence of the Dirichlet series// Mat. Stud. – 1998. – V.8, №2. – P. 171–176. (in Ukrainian) 5. Skaskiv O.B. The stability of the maximum of a sequence of linear functions// International conference dedicated to 125-th anniversary of Hans Hahn: Book of abstracts. – International conf. (Chernivtsi, June 27–July 3, 2004). – P. 100–101. (in Ukrainian) 6. Skaskiv O.B. The stability of the maximum of a sequence of linear functions// Mat. Visn. NTSh. – 2004. – V.1. – P. 120–129. (in Ukrainian) 7. Dolynyuk M., Skaskiv O. The stability of the maximal term of entire multiple Dirichlet series// International V.Ya. Skorobohatko mathematical conference: Book of abstracts. – International conf. (Drohobych, September 24–28, 2007). – Lviv, 2007. – P. 94. (in Ukrainian) 8. Skaskiv O.B., Trakalo O.M. Asymptotic estimates for Laplace integrals// Mat. Stud. – 2002. – V.18, №2. - P. 125–146. (in Ukrainian) 9. Skaskiv O.B., Zikrach D.Yu. On the best possible description of exceptional set in asymptotic estimates for Laplace–Stieltjes integrals// Mat. Stud. – 2011. – V.35, №2. – P. 131–141. 10. Skaskiv O.B. On the behaviour of the maximum term of a Dirichlet series defining an entire function// Mat. Zametki. – 1985. – V.37, №1. – P. 41–47.; English transl. in Math. Notes, 1985, V.37, №1, 24–28.
Pages	171-179
Volume	43
Issue	1
Year	2015
Journal	Matematychni Studii
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