奥帕尔在等价级数理论上的贡献
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摘要
基于原始文献,利用历史分析和比较的方法,首次研究了奥帕尔在等价级数理论方面的工作及影响.分析了奥帕尔的工作背景,指出图兰的工作是其直接基础;研究了他在泰勒级数及法布尔级数的等价级数理论,深入分析了其数学思想和方法.研究结果表明,奥帕尔对等价级数理论的发展做出了重要贡献,其工作对其他数学家有重要影响.
Based on the original literature,L.Alpár's Work and influence on theory of equivalent series were studied and evaluated by historical analysis and comparative method.Analyzing the background of his work from a series of papers,we found Alpár made further important contributions to the theory of equivalent series.He found several generalizations of the theorem of Turán concerning power series and later extended them to Faber series too.After profound analysis of his mathematical ideas and methods,we confirmed that Alpár made an important contribution to the development of the theory of equivalent series and also had some significant influence on other mathematicians.
Based on the original literature,L.Alpár's Work and influence on theory of equivalent series were studied and evaluated by historical analysis and comparative method.Analyzing the background of his work from a series of papers,we found Alpár made further important contributions to the theory of equivalent series.He found several generalizations of the theorem of Turán concerning power series and later extended them to Faber series too.After profound analysis of his mathematical ideas and methods,we confirmed that Alpár made an important contribution to the development of the theory of equivalent series and also had some significant influence on other mathematicians.
引文
[1] Indlekofer K H.On Turán's equivalent power series [C]// P Erd?s. Studies in Pure Mathematics.Birkh?user,1983:357-379.
[2] Indlekofer K H. A survey of Turán's equivalent power series [C]// P Jáons. Number Theory,Analysis,and Combinatorics.Budapest,2011:127-143.
[3] Horváth J. A panorama of hungarian mathematics in the twentieth century I [M]. Berlin:Springer Verlag,2006:351-354.
[4] Hardy G H,Littlewood J E. Theorems concerning the summability of series by Borel's exponential method [J]. Rend del Circolo Mat di Palermo,1916,41(1):36-53.
[5] Carleman T. Some theorems concerning the convergence of power series on the circle of convergence [J]. Arkiv f?r Math Astr och Fye,1920,15(1):1-13.
[6] Clunie J,Vermes P. Regular Sonnenschein type summability methods [J]. Acad Roy Belg Bull Sci,1959,45(5): 930-954.
[7] Sledd W. Regularity conditions for Karamata Matrices [J]. London Math Soc,1963,38:105-107.
[8] Turán P. A remark concerning the behaviour of a power series on the periphery of its convergence circle [J]. Publ Inst Math Acad Serbe Sci,1958,12:19-26.
[9] Bajsanski B. Généralisation d'un théorème de Carleman [J]. Publ Ins Math,1958,12:101-108.
[10] Turán P. On some connections of theories of functions and series [J]. Math Rev,1956,17(6):598.
[11] Ishiguro K,Meyer-K?nig W,Zeller K. Die Turánsche Matrix als Produkt einer Euler-Knopp and einer Taylor-Matrix [J]. Kodai Math,1990,13(1):47-51.
[12] Halász G. On the behaviour of Taylor series under conformal mappings of the circle of convergence [J]. Studia Sci Math Hung,1966,1:389-401.
[13] Alpár L. Remarque sur la sommabilité des séries de Taylor sur leurs cercles de convergence I [J].Magyar Tud Akad Mat Kutató Int K?zl,1958,3:1-12.
[14] Alpár L. Remarque sur la sommabilité des séries de Taylor sur leurs cercles de convergence II [J]. Magyar Tud Akad Mat Kutató Int K?zl,1958,3:141-158.
[15] Alpár L. Remarque sur la sommabilité des séries de Taylor sur leurs cercles de convergence III [J]. Magyar Tud Akad Mat Kutató Int K?zl,1960,5:97-152.
[16] Turán P. On some examples in theory of power series [J]. Bull Amer Math Soc,1948,54:932-936.
[17] Alpár L. Egyes hatványsorok abszolút konvergenciája a konvergencia k?r kerületén [J]. Mat Lapok,1960,11:312-322.
[18] Halász G. On Taylor series absolutely convergent on the circumference of the circle of convergence I I [J]. Publ Math Debrecen,1968,15(1):23-31.
[19] Alpár L. Sur une classe particulière de séries de Fourier à certaines puissances absolument convergentes [J]. Studia Sci Math Hung,1968,3:279-284.
[20] Alpár L. Sur certaines transformées des séries de puissance absolument convergentes sur la frontière de leur cercle de convergence [J]. Publ Math Inst Hung Acad Sci,1962,7:287-315.
[21] Richards L. Karamata type operators on the space of uniformly convergent power series [J]. Acta Math Hung,1981,37(1-3):173-184.
[22] Alpár L. Sur une classe particuliere de séries de Fourier ayant de sommes partielles bornées [J]. Studia Sci Math Hung,1966,1:189-204.
[23] Alpár L. Sur certaines transformations des séries de Faber [J]. Magyar Tud Akad Mat Kutató Int K?zl,1964,9:283-296.
[24] Alpár L. Convergence et représentation conforme [J]. Magyar Tud Akad Mat Kutató Int K?zl,1964,9:503-514.
[25] Alpár L. Convergence absolue et représentation conforme [J]. Studia Sci Math Hung,1966,1:379-388.
[26] Alpár L. Convergence uniforme et représentation conforme [J]. Periodica Math Hung,1971,1(3):187-193.
[27] Alpár L. Sur certains changements de variable des series de Faber [J]. Studia Sei Math Hung,1978,13:173-180.
[28] Alpár L. Cesàro summability and conformal mapping [J]. Colloq Math Soc Janos Bolyai,1980,35:101-125.
[29] Alpár L. On the linear transformations of series summable in the sense of Cesàro [J]. Acta Math Hung,1982,39(1-3):233-243.
[30] Schwarz W. Bemerkungen zu einem Satz der Herren Turán und Clunie über das Verhalten von Potenzreihen auf dem Rande des Konvergenzkreises [J]. Publ Math Debrecen,1969,16:67-73.
[31] Schwarz W. Bemerkungen zu einem Satz der Herren Turán und Clunie über das Verhalten von Potenzreihen auf dem Rande des Konvergenzkreises II [J]. Publ Math Debrecen,1971,18:129-137.
[32] Indlekofer K H. Bemerkungenü ber?quivalente Potenzreihen von Funktionen mit einem gewissen Stetigkeitsmodul [J]. Monat Math,1972,76(2):124-129.
[33] Halász G. On Taylor series absolutely convergent on the circumference of the circle of convergence III [J]. Acta Math Hung,1974,25(1-2):81-87.
[34] Warlimont R. Euler-Summierbarkeit konform ?quivalenter Reihen [J]. Monat Math,1976,81(1):63-68.
[35] Indlekofer K H,Trautner R. Fortsetzbare ?quivalente Potenzreihen [J]. Publ Math Debrecen,1981,28:25-30.
[36] 徐传胜. 圣彼得堡数学学派研究 [M]. 北京:科学出版社,2017:1-3.
[2] Indlekofer K H. A survey of Turán's equivalent power series [C]// P Jáons. Number Theory,Analysis,and Combinatorics.Budapest,2011:127-143.
[3] Horváth J. A panorama of hungarian mathematics in the twentieth century I [M]. Berlin:Springer Verlag,2006:351-354.
[4] Hardy G H,Littlewood J E. Theorems concerning the summability of series by Borel's exponential method [J]. Rend del Circolo Mat di Palermo,1916,41(1):36-53.
[5] Carleman T. Some theorems concerning the convergence of power series on the circle of convergence [J]. Arkiv f?r Math Astr och Fye,1920,15(1):1-13.
[6] Clunie J,Vermes P. Regular Sonnenschein type summability methods [J]. Acad Roy Belg Bull Sci,1959,45(5): 930-954.
[7] Sledd W. Regularity conditions for Karamata Matrices [J]. London Math Soc,1963,38:105-107.
[8] Turán P. A remark concerning the behaviour of a power series on the periphery of its convergence circle [J]. Publ Inst Math Acad Serbe Sci,1958,12:19-26.
[9] Bajsanski B. Généralisation d'un théorème de Carleman [J]. Publ Ins Math,1958,12:101-108.
[10] Turán P. On some connections of theories of functions and series [J]. Math Rev,1956,17(6):598.
[11] Ishiguro K,Meyer-K?nig W,Zeller K. Die Turánsche Matrix als Produkt einer Euler-Knopp and einer Taylor-Matrix [J]. Kodai Math,1990,13(1):47-51.
[12] Halász G. On the behaviour of Taylor series under conformal mappings of the circle of convergence [J]. Studia Sci Math Hung,1966,1:389-401.
[13] Alpár L. Remarque sur la sommabilité des séries de Taylor sur leurs cercles de convergence I [J].Magyar Tud Akad Mat Kutató Int K?zl,1958,3:1-12.
[14] Alpár L. Remarque sur la sommabilité des séries de Taylor sur leurs cercles de convergence II [J]. Magyar Tud Akad Mat Kutató Int K?zl,1958,3:141-158.
[15] Alpár L. Remarque sur la sommabilité des séries de Taylor sur leurs cercles de convergence III [J]. Magyar Tud Akad Mat Kutató Int K?zl,1960,5:97-152.
[16] Turán P. On some examples in theory of power series [J]. Bull Amer Math Soc,1948,54:932-936.
[17] Alpár L. Egyes hatványsorok abszolút konvergenciája a konvergencia k?r kerületén [J]. Mat Lapok,1960,11:312-322.
[18] Halász G. On Taylor series absolutely convergent on the circumference of the circle of convergence I I [J]. Publ Math Debrecen,1968,15(1):23-31.
[19] Alpár L. Sur une classe particulière de séries de Fourier à certaines puissances absolument convergentes [J]. Studia Sci Math Hung,1968,3:279-284.
[20] Alpár L. Sur certaines transformées des séries de puissance absolument convergentes sur la frontière de leur cercle de convergence [J]. Publ Math Inst Hung Acad Sci,1962,7:287-315.
[21] Richards L. Karamata type operators on the space of uniformly convergent power series [J]. Acta Math Hung,1981,37(1-3):173-184.
[22] Alpár L. Sur une classe particuliere de séries de Fourier ayant de sommes partielles bornées [J]. Studia Sci Math Hung,1966,1:189-204.
[23] Alpár L. Sur certaines transformations des séries de Faber [J]. Magyar Tud Akad Mat Kutató Int K?zl,1964,9:283-296.
[24] Alpár L. Convergence et représentation conforme [J]. Magyar Tud Akad Mat Kutató Int K?zl,1964,9:503-514.
[25] Alpár L. Convergence absolue et représentation conforme [J]. Studia Sci Math Hung,1966,1:379-388.
[26] Alpár L. Convergence uniforme et représentation conforme [J]. Periodica Math Hung,1971,1(3):187-193.
[27] Alpár L. Sur certains changements de variable des series de Faber [J]. Studia Sei Math Hung,1978,13:173-180.
[28] Alpár L. Cesàro summability and conformal mapping [J]. Colloq Math Soc Janos Bolyai,1980,35:101-125.
[29] Alpár L. On the linear transformations of series summable in the sense of Cesàro [J]. Acta Math Hung,1982,39(1-3):233-243.
[30] Schwarz W. Bemerkungen zu einem Satz der Herren Turán und Clunie über das Verhalten von Potenzreihen auf dem Rande des Konvergenzkreises [J]. Publ Math Debrecen,1969,16:67-73.
[31] Schwarz W. Bemerkungen zu einem Satz der Herren Turán und Clunie über das Verhalten von Potenzreihen auf dem Rande des Konvergenzkreises II [J]. Publ Math Debrecen,1971,18:129-137.
[32] Indlekofer K H. Bemerkungenü ber?quivalente Potenzreihen von Funktionen mit einem gewissen Stetigkeitsmodul [J]. Monat Math,1972,76(2):124-129.
[33] Halász G. On Taylor series absolutely convergent on the circumference of the circle of convergence III [J]. Acta Math Hung,1974,25(1-2):81-87.
[34] Warlimont R. Euler-Summierbarkeit konform ?quivalenter Reihen [J]. Monat Math,1976,81(1):63-68.
[35] Indlekofer K H,Trautner R. Fortsetzbare ?quivalente Potenzreihen [J]. Publ Math Debrecen,1981,28:25-30.
[36] 徐传胜. 圣彼得堡数学学派研究 [M]. 北京:科学出版社,2017:1-3.