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Article

Keywords:
Dirichlet series; Banach algebra
Summary:
Consider the space $L$ of entire functions represented by multiple Dirichlet series that becomes a non uniformly convex Banach space which is also proved to be dense, countable and separable. Continuing further, for the given space $L$ the characterization of bounded linear transformations in terms of matrix and characterization of linear functional has been obtained.
References:
[1] Akanksha, A., Srivastava, G. S.: Spaces of vector-valued Dirichlet series in a half plane. Front. Math. China 9 (2014), 1239-1252. DOI 10.1007/s11464-014-0396-0 | MR 3260996 | Zbl 1308.30004
[2] Azpeitia, A. G.: A remark on the Ritt order of entire functions defined by Dirichlet series. Proc. Am. Math. Soc. 12 (1961), 722-723. DOI 10.2307/2034864 | MR 0126534 | Zbl 0126.29101
[3] Azpeitia, A. G.: On the maximum modulus and the maximum term of an entire Dirichlet series. Proc. Am. Math. Soc. 12 (1961), 717-721. DOI 10.2307/2034863 | MR 0126533 | Zbl 0126.29003
[4] Azpeitia, A. G.: On the Ritt order of entire Dirichlet series. Q. J. Math., Oxf. II. Ser. 15 (1964), 275-277. DOI 10.1093/qmath/15.1.275 | MR 0168736 | Zbl 0129.29003
[5] Behnam, H. S., Srivastava, G. S.: Spaces of analytic functions represented by Dirichlet series of two complex variables. Approximation Theory Appl. 18 (2002), 1-14. DOI 10.1007/BF02837109 | MR 1942346 | Zbl 1017.32007
[6] Cahen, E.: Sur la fonction $\zeta(s)$ de Riemann et sur des fonctions analogues. Ann. Sci. Éc. Norm. Supér. (3) 11 (1894), 75-164 French \99999JFM99999 25.0702.01. DOI 10.24033/asens.401 | MR 1508903
[7] Dagene, E.: The central index of Dirichlet series. Litov. Mat. Sb. 8 (1968), 503-520 Russian. MR 0245763 | Zbl 0183.06902
[8] Daoud, S.: On the class of entire Dirichlet functions of several complex variables having finite order point. Port. Math. 43 (1986), 417-427. MR 0911448 | Zbl 0636.46019
[9] Estermann, T.: On certain functions represented by Dirichlet series. Proc. Lond. Math. Soc. (2) 27 (1928), 435-448 \99999JFM99999 54.0366.03. DOI 10.1112/plms/s2-27.1.435 | MR 1575403
[10] Janusauskas, A. I.: Elementary theorems on convergence of double Dirichlet series. Sov. Math., Dokl. 18 (1977), 610-614 translation from Dokl. Akad. Nauk. SSSR 234 1977 34-37. MR 0442265 | Zbl 0388.40003
[11] Kong, Y.: On some $q$-orders and $q$-types of Dirichlet-Hadamard product function. Acta Math. Sin., Chin. Ser. Chinese 52 (2009), 1165-1172. MR 2640624 | Zbl 1212.30004
[12] Kong, Y., Gan, H.: On orders and types of Dirichlet series of slow growth. Turk. J. Math. 34 (2010), 1-11. MR 2654411 | Zbl 1189.30004
[13] Kumar, N., Chutani, L., Manocha, G.: Certain results on a class of entire Dirichlet series in two variables. Sci. Magna 11 (2016), 33-40. MR 3817299
[14] Kumar, N., Manocha, G.: A class of entire Dirichlet series as an FK-space and a Fréchet space. Acta Math. Sci., Ser. B, Engl. Ed. 33 (2013), 1571-1578. DOI 10.1016/S0252-9602(13)60105-8 | MR 3116603 | Zbl 1313.30007
[15] Kumar, N., Manocha, G.: On a class of entire functions represented by Dirichlet series. J. Egypt. Math. Soc. 21 (2013), 21-24. DOI 10.1016/j.joems.2012.10.008 | MR 3040754 | Zbl 1277.30004
[16] Kumar, N., Manocha, G.: A study of various results for a class of entire Dirichlet series with complex frequencies. Math. Bohem. 143 (2018), 1-9. DOI 10.21136/MB.2017.0066-16 | MR 3778046 | Zbl 06861588
[17] Larsen, R.: Banach Algebras: An Introduction. Pure and Applied Mathematics 24. Marcel Dekker, New York (1973). MR 0487369 | Zbl 0264.46042
[18] Larsen, R.: Functional Analysis: An Introduction. Pure and Applied Mathematics 15. Marcel Dekker, New York (1973). MR 0461069 | Zbl 0261.46001
[19] Leonard, I. E.: Banach sequence spaces. J. Math. Anal. Appl. 54 (1976), 245-265. DOI 10.1016/0022-247X(76)90248-1 | MR 0420216 | Zbl 0343.46010
[20] Liang, M., Gao, Z.: Convergence and growth of multiple Dirichlet series. Acta Math. Sci., Ser. B, Engl. Ed. 30 (2010), 1640-1648. DOI 10.1016/S0252-9602(10)60157-9 | MR 2778633 | Zbl 1240.30016
[21] Rahman, Q. I.: A note on entire functions defined by Dirichlet series. Math. Student 24 (1956), 203-207. MR 0086125 | Zbl 0082.05801
[22] Rahman, Q. I.: On means of entire functions. Q. J. Math., Oxf. II. Ser. 7 (1956), 192-195. DOI 10.1093/qmath/7.1.192 | MR 0095941 | Zbl 0073.06601
[23] Rahman, Q. I.: On the lower order of entire functions defined by Dirichlet series. Q. J. Math., Oxf. II. Ser. 7 (1956), 96-99. DOI 10.1093/qmath/7.1.96 | MR 0098830 | Zbl 0074.29901
[24] Rahman, Q. I.: On the maximum modulus and the coefficients of an entire Dirichlet series. Tôhoku Math. J., II. Ser. 8 (1956), 108-113. DOI 10.2748/tmj/1178245013 | MR 0080153 | Zbl 0074.29803
[25] Rahman, Q. I.: On entire functions defined by a Dirichlet series. Proc. Am. Math. Soc. 10 (1959), 213-215. DOI 10.1090/S0002-9939-1959-0122968-X | MR 0122968 | Zbl 0095.27803
[26] Rahman, Q. I.: On entire functions defined by a Dirichlet series: Correction. Proc. Am. Math. Soc. 11 (1960), 624-625. DOI 10.1090/S0002-9939-1960-0122969-X | MR 0122969 | Zbl 0107.28202
[27] Sarkar, P. K.: On the Gol'dberg order and Gol'dberg type of an entire function of several complex variables represented by multiple Dirichlet series. Indian J. Pure Appl. Math. 13 (1982), 1221-1229. MR 0679065 | Zbl 0513.32005
[28] Tanaka, C.: Note on Dirichlet series. V: On the integral functions defined by Dirichlet series. I. Tôhoku Math. J., II. Ser. 5 (1953), 67-78. DOI 10.2748/tmj/1178245352 | MR 0057320 | Zbl 0053.37502
[29] Tanaka, C.: Note on Dirichlet series. VI: On the integral functions defined by Dirichlet series. II. Mem. Sch. Sci. Eng., Waseda Univ. 17 (1953), 85-94. MR 0060013
[30] Tanaka, C.: Note on Dirichlet series. X: Remark on S. Mandelbrojt's theorem. Proc. Japan Acad. 29 (1953), 423-426. DOI 10.3792/pja/1195570554 | MR 0062253 | Zbl 0052.30002
[31] Vaish, S. K.: On the coefficients of entire multiple Dirichlet series of several complex variables. Bull. Math. Soc. Sci. Roum., Nouv. Sér. 46 (2003), 195-202. MR 2094187 | Zbl 1084.30511
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