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In this way, Iyengar's inequality and a number of other useful inequalities are generalized.",{"EN":129},"Some estimates of an integral in terms of the L p-norm of the (n+1)st derivative of its integrand",{"VOID":131},"[\"2881094497812987094\"]",{"EN":133},"",{"VOID":135},"10.1023\u002FA:1022894413541","PUBLICATION","VERIFIED","2024-05-02T12:56:48.295+00:00","Auto Verify",[141],"EN","https:\u002F\u002Flink.springer.com\u002Farticle\u002F10.1023\u002FA:1022894413541",[144,161],{"id":145,"sortIndex":23,"researcher":22,"roles":146,"affiliations":147,"properties":156,"displayName":158,"givenName":22,"familyName":22},"5a38c2a6-6217-47ca-b899-5e5449e71d03",[],[148],{"id":149,"sortIndex":23,"affiliation":150,"properties":22},"36eeb73f-f9e7-4d67-9ef3-174ac602bb4a",{"id":149,"createTime":22,"updateTime":22,"relativeEntities":151,"slug":22,"properties":152,"entityType":22,"verifyStatus":22,"verifyTime":22,"verifyNote":22,"languages":22,"translateLanguages":22,"viewCount":22,"url":22,"parentIds":155,"statistic":22},[],{"title":153},{"EN":154},"Department of Applied Mathematics and Informatics, Jiaozuo Institute of Technology, Jiaozuo City, Henan, People's Republic of China",[],{"title":157,"gsAuthor":159},{"EN":158},"Bai-Ni Guo",{"VOID":160},"[\"Cn71SKoAAAAJ\"]",{"id":162,"sortIndex":163,"researcher":22,"roles":164,"affiliations":165,"properties":172,"displayName":174,"givenName":22,"familyName":22},"9a21968a-d53e-4724-b22c-dfdedb39b45e",1,[],[166],{"id":149,"sortIndex":23,"affiliation":167,"properties":22},{"id":149,"createTime":22,"updateTime":22,"relativeEntities":168,"slug":22,"properties":169,"entityType":22,"verifyStatus":22,"verifyTime":22,"verifyNote":22,"languages":22,"translateLanguages":22,"viewCount":22,"url":22,"parentIds":171,"statistic":22},[],{"title":170},{"EN":154},[],{"title":173,"gsAuthor":175},{"EN":174},"Feng Qi",{"VOID":176},"[\"Q-wGSBYAAAAJ\"]","ARTICLE",{"url":22,"publisher":179,"properties":22},{"id":6,"createTime":7,"updateTime":8,"relativeEntities":180,"slug":10,"properties":181,"entityType":20,"verifyStatus":21,"verifyTime":22,"verifyNote":22,"languages":22,"translateLanguages":22,"viewCount":23,"subjectFields":185,"manageAffiliations":194,"indexDatabases":205,"url":22,"thumbnailPath":22,"statistic":220,"gsStatistic":22,"type":114,"analyzePriority":22},[],{"issn":182,"title":183,"eissn":184},{"VOID":15},{"EN":17},{"VOID":13},[186,190],{"id":26,"createTime":22,"updateTime":22,"relativeEntities":187,"label":188,"description":189,"parentId":22,"standard":22,"scholarHubFieldId":22},[],{"EN":29},{},{"id":32,"createTime":22,"updateTime":22,"relativeEntities":191,"label":192,"description":193,"parentId":22,"standard":22,"scholarHubFieldId":22},[],{"EN":35},{},[195,200],{"id":39,"createTime":22,"updateTime":22,"relativeEntities":196,"slug":22,"properties":197,"entityType":22,"verifyStatus":22,"verifyTime":22,"verifyNote":22,"languages":22,"translateLanguages":22,"viewCount":22,"url":22,"parentIds":199,"statistic":22},[],{"title":198},{"EN":43},[45],{"id":47,"createTime":22,"updateTime":22,"relativeEntities":201,"slug":22,"properties":202,"entityType":22,"verifyStatus":22,"verifyTime":22,"verifyNote":22,"languages":22,"translateLanguages":22,"viewCount":22,"url":22,"parentIds":204,"statistic":22},[],{"title":203},{"EN":51},[],[206,213],{"id":55,"indexDatabase":207,"url":68,"indexYears":22,"academicFieldIds":212,"indexDatabaseRanking":22},{"id":57,"createTime":22,"updateTime":22,"relativeEntities":208,"label":209,"description":210,"key":64,"publicationTags":211,"standard":22},[],{"EN":60,"VI":60},{"EN":62,"VI":63},[66,67],[70],{"id":72,"indexDatabase":214,"url":83,"indexYears":84,"academicFieldIds":219,"indexDatabaseRanking":88},{"id":74,"createTime":22,"updateTime":22,"relativeEntities":215,"label":216,"description":217,"key":80,"publicationTags":218,"standard":22},[],{"EN":77,"VI":77},{"EN":77,"VI":79},[82],[86,87],{"impactFactor":23,"impactFactorByYear":221,"i10Index":23,"i10IndexLast5Year":23,"totalPublication":91,"totalPublicationByYear":222,"totalCitation":23,"totalCitationByYear":223,"totalCitationPerPublication":23,"totalCitationPerPublicationByYear":224,"hindexLast5Year":23,"hindex":23},{},{"1975":93,"1976":94,"1977":95,"1978":96,"1979":93,"1980":97,"1981":93,"1982":98,"1983":99,"1984":93,"1985":93,"1986":96,"1987":96,"1988":94,"1989":100,"1990":101,"1991":93,"1992":95,"1993":102,"1994":103,"1995":95,"1996":104,"1997":99,"1998":96,"1999":96,"2000":96,"2001":103,"2002":103,"2003":94,"2004":104,"2005":104,"2006":105,"2007":103,"2008":93,"2009":97,"2010":104,"2011":97,"2012":97,"2013":99,"2014":104,"2015":99,"2016":96,"2017":106,"2018":107,"2019":95,"2020":108,"2021":109,"2022":110,"2023":108,"2024":111},{},{},{"total":103,"publishYear":226,"statisticByYear":227},2003,{},"2003-03-01","ERROR_IN_ANALYZE_CITATION","2026-08-14T02:01:03.968+00:00",[88,66],[233,235,237,239,241,243,245,247,249,251,253,255,257,259,261,263,265,267],{"id":22,"text":234,"url":22,"identifiers":22},"R. P. Agarwal and S. S. Dragomir, An application of Hayashi's inequality for differentiable functions, Comput. Math. Appl., 32(1996), 95-99.",{"id":22,"text":236,"url":22,"identifiers":22},"P. Cerone and S. S. Dragomir, Lobatto type quadrature rules for functions with bounded derivative, Math. Ineq. Appl., 3(2000), 197-209; RGMIA Res. Rep. Coll., 2(1999), 133–146; http:\u002F\u002Frgmia.vu.edu.au\u002Fv2n2.html.",{"id":22,"text":238,"url":22,"identifiers":22},"P. Cerone and S. S. Dragomir, On a weighted generalization of Iyengar type inequalities involving the bounded first derivative, Math. Ineq. Appl., 3(2000), 35-44.; RGMIA Res. Rep. Coll., 2(1999), 147–157; http:\u002F\u002Frgmia.vu.edu.au\u002Fv2n2.html.",{"id":22,"text":240,"url":22,"identifiers":22},"L.-H. Cui and B.-N. Guo, On proofs of an integral inequality and its generalizations, J. Zhengzhou Grain College, 17(1996), Supplement, 152-154, 158 (Chinese).",{"id":22,"text":242,"url":22,"identifiers":22},"B.-N. Guo and F. Qi, Proofs of an integral inequality, Mathematics and Informatics Quarterly, 7(1997), 182-184.",{"id":22,"text":244,"url":22,"identifiers":22},"K. S. K. Iyengar, Note on an inequality, Math. Student, 6(1938), 75-76.",{"id":22,"text":246,"url":22,"identifiers":22},"J.-Ch. Kuang, Applied inequalities, Hunan Education Press (Changsha, China, 1993) (Chinese).",{"id":22,"text":248,"url":22,"identifiers":22},"G. V. MilovanoviĆ and J. E. PeČariĆ, Some considerations on Iyengar's inequality and some related applications, Univ. Beograd Publ. Elektrotehn. Fak. Ser. Mat., 544–576(1976), 166-170.",{"id":22,"text":250,"url":22,"identifiers":22},"D. S. MitrinoviĆ, Analytic inequalities, Springer (Berlin-Heidelberg, 1970).",{"id":22,"text":252,"url":22,"identifiers":22},"D. S. MitrinoviĆ, J. E. PeČariĆ, and A. M. Fink, Inequalities involving functions and their integrals and derivatives, Kluwer (Dordrecht, 1991).",{"id":22,"text":254,"url":22,"identifiers":22},"F. Qi, Inequalities for an integral, Math. Gaz., 80(1996), 376-377.",{"id":22,"text":256,"url":22,"identifiers":22},"F. Qi, Further generalizations of inequalities for an integral, Univ. Beograd Publ. Elektrotehn. Fak. Ser. Mat., 8(1997), 79-83.",{"id":22,"text":258,"url":22,"identifiers":22},"F. Qi, Inequalities for a multiple integral, Acta Math. Hungar., 84(1999), 19-26.",{"id":22,"text":260,"url":22,"identifiers":22},"F. Qi, P. Cerone and S. S. Dragomir, Some new Iyengar type inequalities, RGMIA Res. Rep. Coll., 5(2002); http:\u002F\u002Frgmia.vu.edu.au\u002Fv5n2.html.",{"id":22,"text":262,"url":22,"identifiers":22},"F. Qi and Y.-J. Zhang, Inequalities for a weighted integral, Adv. Stud. Contemp. Math., 4(2002), 93-101; RGMIA Res. Rep. Coll., 2(1999), 967–975; http:\u002F\u002Frgmia.vu.edu.au\u002Fv2n7.html.",{"id":22,"text":264,"url":22,"identifiers":22},"F. Qi, Inequalities for a weighted multiple integral, J. Math. Anal. Appl., 253(2001), 381-388; RGMIA Res. Rep. Coll., 2(1999), 991–997; http:\u002F\u002Frgmia.vu.edu.au\u002Fv7n2.html.",{"id":22,"text":266,"url":22,"identifiers":22},"M. E. Taylor, Partial differential equations. I — Basic theory, Springer (Berlin-Heidelberg-New York, 1996).",{"id":22,"text":268,"url":22,"identifiers":22},"VasiĆ and G. V. MilonanoviĆ, On an inequality of Iyengar, Univ. Beograd Publ. Elektrotehn. Fak. Ser. Mat., 544-576(1976), 18-24.",false,{"id":271,"createTime":272,"updateTime":273,"relativeEntities":274,"slug":275,"properties":276,"entityType":136,"verifyStatus":137,"verifyTime":287,"verifyNote":139,"languages":22,"translateLanguages":22,"viewCount":23,"primaryUrl":288,"fullTextUrl":22,"authors":289,"publicationType":177,"publisherRelationship":306,"citationCount":94,"citationInfo":358,"publishDate":363,"publishYear":359,"citationAnalyzeStatus":364,"lastCitationAnalyze":365,"indexDatabases":366,"openAccess":22,"references":22,"isForceReanalyzing":269},"b25a597b-61cf-438e-80e4-8834ad3d7dee","2023-11-29T03:56:13.255+00:00","2026-07-25T13:14:47.434+00:00",[],"On-the-modulus-of-continuity-in-connection-with-a-problem-of-J-Szabados-concerning-strong-approximation",{"abstract":277,"title":279,"gsPaper":281,"references":283,"doi":285},{"EN":278},"Пустьf(x) — интегрируемая 2π-периодическая функция, aω(f,δ) иs\nn(x)=sn(f, x). соответственно, модуль непрерывности иn-ая сумма Фурье этой функции. В настоящей работе, продолжающей исследования Г. Фрейда, Л. Лейндлера—E. M. Никищина, И. Сабадоша и К. И. Осколкова, доказывается следующая теорема.Если Ω(u) — выпуклая или вогнутая непрерывная функция и если (1)\n                  1\n                  \n                    \n                  \n                  \n$$\\left\\| {\\left. {\\sum\\limits_{k = 1}^\\infty  \\Omega  (|S_k (x) - f(x)|)} \\right\\|_C } \\right.$$\n\n                \nто\n\n                  1\n                  \n                    \n                  \n                  \n$$\\omega (f;\\delta ) = O\\left( {\\delta \\int\\limits_\\delta ^1 {\\frac{{\\bar \\Omega (v)}}{{v^2 }}dv} } \\right),$$\n\n                \n где ¯Ω(v) —функция, обратная к Ω(и). При этом существует функция f0(х), удовлетворяющая условию (1), для которой\n                  \n                    \n                  \n                  \n$$\\omega (f;\\delta ) = c\\delta \\int\\limits_\\delta ^1 {\\frac{{\\bar \\Omega (v)}}{{v^2 }}dv}    (c > 0).$$\n\n                 ЕслиΩ(u)— вогнутая функция, то интеграл\n                  \n                    \n                  \n                  \n$$\\int\\limits_\\delta ^1 {\\frac{{\\bar \\Omega (v)}}{{v^2 }}dv} $$\n\n                 можно заменить на\n                  \n                    \n                  \n                  \n$$\\int\\limits_{\\bar \\Omega (\\delta )}^1 {\\frac{{du}}{{\\Omega (u)}}.} $$\n\n                . Отсюда вытекает, что еслиΩ(u) — функция типа модуля непрерывности, то для того, чтобы (1) всегда влекло принадлежность f(x) классу Lip 1, необходимо и достаточно условие\n                  \n                    \n                  \n                  \n$$\\int\\limits_0^1 {\\frac{{du}}{{\\Omega (u)}}}\u003C \\infty .$$\n\n                \n",{"EN":280},"On the modulus of continuity in connection with a problem of J. Szabados concerning strong approximation",{"VOID":282},"[\"3513505216442715588\"]",{"VOID":284},"А. В. Ефимов, Линейные методы приближения непрерывных периодических функций,Матем. сб.,54 (1961), 51–90.\nG. Freud, Über die Sättigungsklasse der starken Approximation durch Teilsummen der Fourierschen Reihe,Acta Math. Acad. Sci. Hungar.,20 (1969), 275–279.\nL.Leindler, On strong approximation of Fourier series,Approximation Theory (Proc. Conf. Poznan, 1972); 129–140 (Warszawa, 1975).\nL. Leindler andE. M. Nikišin, Note on strong approximation by Fourier series,Acta Math. Acad. Sci. Hungar.,24 (1973), 223–227.\nK. I. Oskolkov, On strong summability of Fourier series and differentiability of functions,Analysis Math.,2 (1976), 41–47.\nJ. Szabados, On a problem of L. Leindler concerning strong approximation by Fourier series,Analysis Math.,2 (1976), 155–161.",{"VOID":286},"10.1007\u002FBF02116979","2024-05-16T15:11:20.596+00:00","http:\u002F\u002Flink.springer.com\u002F10.1007\u002FBF02116979",[290],{"id":291,"sortIndex":23,"researcher":22,"roles":292,"affiliations":294,"properties":303,"displayName":305,"givenName":22,"familyName":22},"5b60aba0-e0db-4f76-8a46-5be5cd4f87ae",[293],"AUTHOR",[295],{"id":296,"sortIndex":23,"affiliation":297,"properties":22},"7f93f5a9-ac24-4a4a-94f9-b03740c1c9d4",{"id":296,"createTime":22,"updateTime":22,"relativeEntities":298,"slug":22,"properties":299,"entityType":22,"verifyStatus":22,"verifyTime":22,"verifyNote":22,"languages":22,"translateLanguages":22,"viewCount":22,"url":22,"parentIds":302,"statistic":22},[],{"title":300},{"VI":301},"Bolyai Institute, Szeged, Hungary",[],{"title":304},{"VI":305},"V. 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In particular, it is shown that they are completely characterized in terms of an algebraic decomposition with a Fredholm linear relation and a bounded nilpotent operator. The behaviour of a polynomial in them is also investigated.",{"EN":377},"B-Fredholm linear relations in Hilbert spaces",{"VOID":379},"[\"6649844396692654313\"]",{"VOID":381},"10.1007\u002Fs10476-020-0030-1","2024-05-01T10:23:59.130+00:00","https:\u002F\u002Flink.springer.com\u002Farticle\u002F10.1007\u002Fs10476-020-0030-1",[385,400],{"id":386,"sortIndex":23,"researcher":22,"roles":387,"affiliations":388,"properties":397,"displayName":399,"givenName":22,"familyName":22},"b76581e7-3260-44a8-b69f-8dd710c81a69",[293],[389],{"id":390,"sortIndex":23,"affiliation":391,"properties":22},"d21b9058-59ac-4e33-b014-20ed1346c0fe",{"id":390,"createTime":22,"updateTime":22,"relativeEntities":392,"slug":22,"properties":393,"entityType":22,"verifyStatus":22,"verifyTime":22,"verifyNote":22,"languages":22,"translateLanguages":22,"viewCount":22,"url":22,"parentIds":396,"statistic":22},[],{"title":394},{"VI":395},"Département de Mathématiques, Université de Sfax, Faculté des Sciences de Sfax, Sfax, Tunisie",[],{"title":398},{"VI":399},"A. Ghorbel",{"id":401,"sortIndex":163,"researcher":22,"roles":402,"affiliations":403,"properties":410,"displayName":412,"givenName":22,"familyName":22},"72c88742-2a57-4624-a1ba-a156b9cc51ed",[293],[404],{"id":390,"sortIndex":23,"affiliation":405,"properties":22},{"id":390,"createTime":22,"updateTime":22,"relativeEntities":406,"slug":22,"properties":407,"entityType":22,"verifyStatus":22,"verifyTime":22,"verifyNote":22,"languages":22,"translateLanguages":22,"viewCount":22,"url":22,"parentIds":409,"statistic":22},[],{"title":408},{"VI":395},[],{"title":411,"gsAuthor":413},{"VI":412},"M. 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Alvarez, Quasi-Fredholm and Semi-B-Fredholm linear relations, Mediterr. J. Math., 14 (2017), Paper 22, 26 pp.","https:\u002F\u002Flink.springer.com\u002Farticle\u002F10.1007\u002Fs00009-016-0828-z",{"doi":477},"10.1007\u002Fs00009-016-0828-z",{"id":479,"text":480,"url":481,"identifiers":482},"4c68646b-0035-4279-8000-0006b275d4fa","T. Alvarez, Y. Chamkha and M. Mnif, Quasi-Fredholm linear relations in Hilbert spaces, Filomat, 31 (2017), 2575–2585.","https:\u002F\u002Flink.springer.com\u002F10.1007\u002Fs10440-022-00541-7",{"doi":483},"10.1007\u002Fs10440-022-00541-7",{"id":479,"text":485,"url":481,"identifiers":486},"M. Berkani, On a class of quasi-Fredholm operators, Integral Equations Operator Theory, 34 (1999), 244–249.",{"doi":483},{"id":22,"text":488,"url":22,"identifiers":489},"M. Berkani and N. Castro-Gonzalez, Unbounded B-Fredholm operators on Hilbert spaces, Proc. Edinb. Math. Soc., 51 (2008), 285–296.",{},{"id":479,"text":491,"url":481,"identifiers":492},"M. Berkani and A. Ouahab, Théor`eme de l’application spectrale pour le spectre essentiel quasi-Fredholm, Proc. Amer. Math. Soc., 125 (1997), 763–774.",{"doi":483},{"id":479,"text":494,"url":481,"identifiers":495},"E. Chafai and T. Alvarez, Ascent, essential ascent, descent and essential descent for a linear relation in a linear space, Filomat, 31 (2017), 709–721.",{"doi":483},{"id":479,"text":497,"url":481,"identifiers":498},"E. Chafai and M. Mnif, Spectral mapping theorem for ascent, essential ascent, descent and essential descent spectrum of linear relations, Acta Math. Sci., 34 (2014), 1212–1224.",{"doi":483},{"id":479,"text":500,"url":481,"identifiers":501},"Y. Chamkha and M. Mnif, The class of B-Fredholm linear relations, Complex Anal. Oper. Theory, 9 (2015), 1681–1699.",{"doi":483},{"id":479,"text":503,"url":481,"identifiers":504},"R. W. Cross, Multivalued Linear Operators, Pure and Applied Mathematics, Marcel Dekker (1998).",{"doi":483},{"id":479,"text":506,"url":481,"identifiers":507},"F. Fakhfakh and M. Mnif, Perturbation theory of lower semi-Browder multivalued linear operators, Publ. Math. Debrecen., 78 (2011), 595–606.",{"doi":483},{"id":509,"text":510,"url":511,"identifiers":512},"78f85935-8c33-4fae-9490-b9f87e3f8c56","A. Ghorbel and M. Mnif, Drazin inverse of multivalued operators and its applications, Monatsh. Math., 189 (2019), 273–293.","https:\u002F\u002Flink.springer.com\u002Farticle\u002F10.1007\u002Fs00605-018-1257-9",{"doi":513},"10.1007\u002Fs00605-018-1257-9",{"id":479,"text":515,"url":481,"identifiers":516},"J. Ph. Labrousse, Les opérateurs quasi-Fredholm: Une généralisation des opérateurs semi-Fredholm, Rend. Circ. Mat. Palermo (2), 29 (1980), 161–258.",{"doi":483},{"id":479,"text":518,"url":481,"identifiers":519},"J. Ph. Labrousse, A. Sandovici, H. S. V. de Snoo and H. Winkler, The Kato decomposition of quasi-Fredholm relations, Oper. Matrices, 4 (2010), 1–51.",{"doi":483},{"id":521,"text":522,"url":523,"identifiers":524},"2f21986b-a1a7-44e7-91e7-1d8fbf442125","M. Mbekhta, Ascente, descente et spectre essentiel quasi-Fredholm, Rend. Cir. Mat. Palermo (2), 46 (1997), 175–196.","https:\u002F\u002Flink.springer.com\u002Farticle\u002F10.1007\u002FBF02977027",{"doi":525},"10.1007\u002FBF02977027",{"id":479,"text":527,"url":481,"identifiers":528},"A. Sandovici, Some basic properties of polynomials in a linear relation in linear spaces, in: Operator Theory in Inner Product Spaces, Oper. Theory Adv. Appl., vol. 175, Birkhauser (Basel, 2007), pp. 231–240.",{"doi":483},{"id":530,"text":531,"url":532,"identifiers":533},"21020faf-b4a4-4c1e-8ee4-0533ee089ee6","A. Sandovici and H. Snoo, An index formula for the product of linear relations, Linear Algebra Appl., 431 (2009), 2160–2171.","https:\u002F\u002Fwww.sciencedirect.com\u002Fscience\u002Farticle\u002Fpii\u002FS0024379509003632",{"doi":534},"10.1016\u002Fj.laa.2009.07.011",{"id":479,"text":536,"url":481,"identifiers":537},"A. Sandovici, H. Snoo and H. Winkler, Ascent, descent, nullity, defect and related notions for linear relations in linear spaces, Linear Algebra Appl., 423 (2007), 456–497.",{"doi":483},{"id":539,"text":540,"url":541,"identifiers":542},"8d5eeab4-693f-475e-aa72-a3b2a98400fe","D. Wilcox, Essential spectra of linear relations, Linear Algebra Appl., 462 (2014), 110–125.","https:\u002F\u002Fwww.sciencedirect.com\u002Fscience\u002Farticle\u002Fpii\u002FS0024379514004558",{"doi":543},"10.1016\u002Fj.laa.2014.07.006",{"id":22,"text":545,"url":22,"identifiers":546},"D. Wilcox, Multivalued semi-Fredholm operators in normed linear spaces, Thesis, University of Cape Town (2002).",{},{"id":548,"createTime":549,"updateTime":550,"relativeEntities":551,"slug":552,"properties":553,"entityType":136,"verifyStatus":137,"verifyTime":564,"verifyNote":139,"languages":22,"translateLanguages":22,"viewCount":23,"primaryUrl":565,"fullTextUrl":22,"authors":566,"publicationType":177,"publisherRelationship":579,"citationCount":163,"citationInfo":631,"publishDate":634,"publishYear":632,"citationAnalyzeStatus":364,"lastCitationAnalyze":550,"indexDatabases":635,"openAccess":22,"references":22,"isForceReanalyzing":269},"88f9e5ca-20ae-4432-8160-a0e9b8503e08","2024-01-27T11:37:33.296+00:00","2026-07-20T01:08:13.562+00:00",[],"On-the-strong-summation-of-Fourier-series-with-variable-exponents",{"abstract":554,"title":556,"gsPaper":558,"references":560,"doi":562},{"EN":555},"Пустьf 2π-периодическ ая суммируемая функц ия, as\n\n                  k\n                \n(x) еë сумма Фурье порядк аk. В связи с известным ре зультатом Зигмунда о сильной суммируемости мы уст анавливаем, что если λn→∞, то сущес твует такая функцияf, что почти всюду\n                  \n                    \n                  \n                  \n$$\\mathop {\\lim \\sup }\\limits_{n \\to \\infty } \\left\\{ {\\frac{1}{n}\\mathop \\sum \\limits_{k = n + 1}^{2n} |s_k (x) - f(x)|^{\\lambda _{2n} } } \\right\\}^{1\u002F\\lambda _{2n} }  = \\infty .$$\n\n                 Отсюда, в частности, вы текает, что если λn↿∞, т о существует такая фун кцияf, что почти всюду\n                  \n                    \n                  \n                  \n$$\\mathop {\\lim \\sup }\\limits_{n \\to \\infty } \\left\\{ {\\frac{1}{n}\\mathop \\sum \\limits_{k = 0}^n |s_k (x) - f(x)|^{\\lambda _k } } \\right\\}^{1\u002F\\lambda _n }  = \\infty .$$\n\n                 Пусть, далее, ω-модуль н епрерывности и\n                  \n                    \n                  \n                  \n$$H^\\omega   = \\{ f:\\parallel f(x + h) - f(x)\\parallel _c  \\leqq K_f \\omega (h)\\} .$$\n\n                . Мы доказываем, что есл и λ\n                  n\n                ↿∞, то необходимым и достаточным условие м для того, чтобы для всехf∈H\nω выполнялос ь соотношение\n                  \n                    \n                  \n                  \n$$\\mathop {\\lim }\\limits_{n \\to \\infty } \\left\\{ {\\frac{1}{n}\\mathop \\sum \\limits_{k = n + 1}^{2n} |s_k (x) - f(x)|^{\\lambda _n } } \\right\\}^{1\u002F\\lambda _n }  = 0(x \\in [0;2\\pi ])$$\n\n                 является условие\n                  \n                    \n                  \n                  \n$$\\omega \\left( {\\frac{1}{n}} \\right) = o\\left( {\\frac{1}{{\\log n}} + \\frac{1}{{\\lambda _n }}} \\right).$$\n\n                 Это же условие необхо димо и достаточно для того, чтобы выполнялось соотнош ение\n                  \n                    \n                  \n                  \n$$\\mathop {\\lim }\\limits_{n \\to \\infty } \\frac{1}{{n + 1}}\\mathop \\sum \\limits_{k = 0}^n |s_k (x) - f(x)|^{\\lambda _k }  = 0(f \\in H^\\omega  ,x \\in [0;2\\pi ]).$$\n\n                \n",{"EN":557},"On the strong summation of Fourier series with variable exponents",{"VOID":559},"[\"6294983449796350969\"]",{"VOID":561},"G. Alexits, Problem 3,On Approximation Theory (Proc. Conf. Oberwolfach, 1963), Birkhäuser (Basel, 1964), 179–190.\nD. Králik, Über ein Problem der starken Summierbarkeit von Fourierreihen,Acta Math. Acad. Sci. Hungar.,17 (1966), 303–312.\nL. Leindler, On a problem of strong summability of Fourier series,Acta Math. Acad. Sci. Hungar.,19 (1968), 87–94.\nK. Tandori, Bemerkung zur starken Summation der Fourierreihen,Acta Math. Acad. Sci. Hungar.,19 (1968), 271–285.\nV.Totik, On the strong approximation of Fourier series,Acta Math. Acad. Sci. Hungar. (to appear).\nP. Turán, On the strong summability of Fourier series,J. Indian Math. Soc.,12 (1948), 8–12.\nA. Zygmund, On the convergence and summability of power series on the circle of convergence,Proc. London Math. Soc.,47 (1941), 326–350.\nA.Zygmund,Trigonometric series.I (Cambridge, 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Nelson, Polyharmonic cardinal splines: a minimization property,J. Approx. Theory,63(1990), 303–320.",{"doi":483},{"id":479,"text":906,"url":481,"identifiers":907},"S. M. Nikol'skii,Approximation of functions of several variables and imbedding theorems, Springer (New York, 1975).",{"doi":483},{"id":22,"text":909,"url":22,"identifiers":910},"Ю. H. Субботин, Экстр емальные задачи теор ии приближений функц ий при неполной инфор мации,Труды МИАН,145(1980), 152–168.",{},{"id":22,"text":912,"url":22,"identifiers":913},"Sun Yongsheng,The theory of approximation (I), Beijing Normal Univ. Press (Beijing, 1989).",{},{"id":22,"text":915,"url":22,"identifiers":916},"Сунь Юншен иЛи Чу нь, Наилучшее прибли жение некоторых клас сов гладких функций н а действительной оси сплайнами высшего по рядка,Матем. заметк и,48(1990), 100–109.",{},{"id":22,"text":918,"url":22,"identifiers":919},"Sun Yongsheng andLi Chun, Optimal recovery for W r2 (R) in (L)2(R),Acta Math. Sinica, N. 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Трофимов, Прибл ижение усеченными ср едними арифметическ ими частных сумм ряда Фурье на некоторых кл ассах, определяемых п олигармоническим оп ератором,Исследова ния по современным пр облемам конструктив ной теории функций (М осква, 1961), 251–253.",{},{"id":930,"createTime":931,"updateTime":932,"relativeEntities":933,"slug":934,"properties":935,"entityType":136,"verifyStatus":137,"verifyTime":944,"verifyNote":139,"languages":22,"translateLanguages":22,"viewCount":23,"primaryUrl":945,"fullTextUrl":22,"authors":946,"publicationType":177,"publisherRelationship":962,"citationCount":22,"citationInfo":22,"publishDate":1014,"publishYear":1015,"citationAnalyzeStatus":21,"lastCitationAnalyze":1016,"indexDatabases":1017,"openAccess":22,"references":1018,"isForceReanalyzing":269},"60ba0cf7-2c2f-4aa8-8032-fbb3eb729d94","2024-01-04T07:49:34.740+00:00","2026-04-21T10:37:42.226+00:00",[],"Remarks-to-a-new-result-of-P-L-Ul-yanov",{"abstract":936,"title":938,"gsPaper":940,"doi":942},{"EN":937},"P. L. Ul'yanov has recently proved a new type of equivalence theorems in connection with the classical equivalence theorem by A. Plessner. Making use of certain results proved by us more than thirty years ago, we extend Ul'yanov's results into more general equivalence theorems.",{"EN":939},"Remarks to a new result of P. L. Ul'yanov",{"VOID":941},"[\"7624629456959722686\"]",{"VOID":943},"10.1007\u002FBF02771072","2024-05-02T17:42:48.314+00:00","https:\u002F\u002Flink.springer.com\u002Farticle\u002F10.1007\u002FBF02771072",[947],{"id":948,"sortIndex":23,"researcher":22,"roles":949,"affiliations":950,"properties":959,"displayName":961,"givenName":22,"familyName":22},"3dc09377-9225-4335-92ee-74b0df0b1bcd",[293],[951],{"id":952,"sortIndex":23,"affiliation":953,"properties":22},"5452b603-f5df-4d85-83ef-fe55e0600641",{"id":952,"createTime":22,"updateTime":22,"relativeEntities":954,"slug":22,"properties":955,"entityType":22,"verifyStatus":22,"verifyTime":22,"verifyNote":22,"languages":22,"translateLanguages":22,"viewCount":22,"url":22,"parentIds":958,"statistic":22},[],{"title":956},{"VI":957},"Bolyai Institute, University of Szeged, Szeged, Hungary",[],{"title":960},{"VI":961},"L. Leindler",{"url":945,"publisher":963,"properties":1009},{"id":6,"createTime":7,"updateTime":8,"relativeEntities":964,"slug":10,"properties":965,"entityType":20,"verifyStatus":21,"verifyTime":22,"verifyNote":22,"languages":22,"translateLanguages":22,"viewCount":23,"subjectFields":969,"manageAffiliations":978,"indexDatabases":989,"url":22,"thumbnailPath":22,"statistic":1004,"gsStatistic":22,"type":114,"analyzePriority":22},[],{"issn":966,"title":967,"eissn":968},{"VOID":15},{"EN":17},{"VOID":13},[970,974],{"id":26,"createTime":22,"updateTime":22,"relativeEntities":971,"label":972,"description":973,"parentId":22,"standard":22,"scholarHubFieldId":22},[],{"EN":29},{},{"id":32,"createTime":22,"updateTime":22,"relativeEntities":975,"label":976,"description":977,"parentId":22,"standard":22,"scholarHubFieldId":22},[],{"EN":35},{},[979,984],{"id":39,"createTime":22,"updateTime":22,"relativeEntities":980,"slug":22,"properties":981,"entityType":22,"verifyStatus":22,"verifyTime":22,"verifyNote":22,"languages":22,"translateLanguages":22,"viewCount":22,"url":22,"parentIds":983,"statistic":22},[],{"title":982},{"EN":43},[45],{"id":47,"createTime":22,"updateTime":22,"relativeEntities":985,"slug":22,"properties":986,"entityType":22,"verifyStatus":22,"verifyTime":22,"verifyNote":22,"languages":22,"translateLanguages":22,"viewCount":22,"url":22,"parentIds":988,"statistic":22},[],{"title":987},{"EN":51},[],[990,997],{"id":55,"indexDatabase":991,"url":68,"indexYears":22,"academicFieldIds":996,"indexDatabaseRanking":22},{"id":57,"createTime":22,"updateTime":22,"relativeEntities":992,"label":993,"description":994,"key":64,"publicationTags":995,"standard":22},[],{"EN":60,"VI":60},{"EN":62,"VI":63},[66,67],[70],{"id":72,"indexDatabase":998,"url":83,"indexYears":84,"academicFieldIds":1003,"indexDatabaseRanking":88},{"id":74,"createTime":22,"updateTime":22,"relativeEntities":999,"label":1000,"description":1001,"key":80,"publicationTags":1002,"standard":22},[],{"EN":77,"VI":77},{"EN":77,"VI":79},[82],[86,87],{"impactFactor":23,"impactFactorByYear":1005,"i10Index":23,"i10IndexLast5Year":23,"totalPublication":91,"totalPublicationByYear":1006,"totalCitation":23,"totalCitationByYear":1007,"totalCitationPerPublication":23,"totalCitationPerPublicationByYear":1008,"hindexLast5Year":23,"hindex":23},{},{"1975":93,"1976":94,"1977":95,"1978":96,"1979":93,"1980":97,"1981":93,"1982":98,"1983":99,"1984":93,"1985":93,"1986":96,"1987":96,"1988":94,"1989":100,"1990":101,"1991":93,"1992":95,"1993":102,"1994":103,"1995":95,"1996":104,"1997":99,"1998":96,"1999":96,"2000":96,"2001":103,"2002":103,"2003":94,"2004":104,"2005":104,"2006":105,"2007":103,"2008":93,"2009":97,"2010":104,"2011":97,"2012":97,"2013":99,"2014":104,"2015":99,"2016":96,"2017":106,"2018":107,"2019":95,"2020":108,"2021":109,"2022":110,"2023":108,"2024":111},{},{},{"pages":1010,"volume":1012},{"VOID":1011},"31-39",{"VOID":1013},"24","1998-12-01",1998,"2026-04-21T10:37:42.222+00:00",[88,66],[1019,1022,1025,1028,1031,1034,1037,1040,1043,1046,1049,1052],{"id":479,"text":1020,"url":481,"identifiers":1021},"G. Alexits, Über den Einfluß der Struktur einer Funktion auf die Konvergenz fast überall ihrer Fourierreihe,Acta Math. Acad. Sci. Hungar.,4(1953), 95–101.",{"doi":483},{"id":22,"text":1023,"url":22,"identifiers":1024},"G. Alexits undD. Králik, Über die Bedeutung der strukturellen Eigenschaften einer Funktion für die Konvergenz ihrer Orthogonalentwicklungen,Acta Sci. Math. (Szeged),18(1957), 131–139.",{},{"id":22,"text":1026,"url":22,"identifiers":1027},"L. Leindler, Über verschiedene Konvergenzarten trigonometrischer Reihen,Acta Sci. Math. (Szeged),25(1964), 233–249.",{},{"id":479,"text":1029,"url":481,"identifiers":1030},"L. Leindler, Über Strukturbedingungen für Fourierreihen,Math. Zeitschr.,88(1965), 418–431.",{"doi":483},{"id":22,"text":1032,"url":22,"identifiers":1033},"L. Leindler, Über verschiedene Konvergenzarten trigonometrischer Reihen. III,Acta Sci. Math. (Szeged),27(1966), 205–215.",{},{"id":22,"text":1035,"url":22,"identifiers":1036},"L. Leindler, On the converses of inequalities of Hardy and Littlewood,Acta Sci. Math. (Szeged),58(1993), 187–192.",{},{"id":22,"text":1038,"url":22,"identifiers":1039},"J. Marcinkiewicz, Sur une nouvelle condition pour la convergence presque partout des séries de Fourier,Ann. Scuola Norm. Sup. Pisa,8(1939), 239–240.",{},{"id":479,"text":1041,"url":481,"identifiers":1042},"A. Plessner, Über Konvergenz von trigonometrischen Reihen,J. Reine Angew. Math.,155(1926), 15–25.",{"doi":483},{"id":22,"text":1044,"url":22,"identifiers":1045},"M. K. Potapov, On the equivalence of convergence criteria for Fourier series (in Russian),Mat. Sb.,68(1965), 111–127.",{},{"id":22,"text":1047,"url":22,"identifiers":1048},"S. B. Stechkin, On the theorem of Kolmogorov-Seliverstov (in Russian),Izv. Akad. Nauk SSSR,17(1953), 499–512.",{},{"id":22,"text":1050,"url":22,"identifiers":1051},"P. L. Ul'yanov, On some equivalent conditions of convergence of series and integrals (in Russian),Uspekhi Mat. Nauk,8(1953), 133–141.",{},{"id":22,"text":1053,"url":22,"identifiers":1054},"P. L. Ul'yanov, On moduli of continuity and Foruier coefficients (in Russian),Vestnik Moskov. Univ. Ser. I Mat. Mekh.,1(1995), 37–52.",{},{"id":1056,"createTime":1057,"updateTime":1058,"relativeEntities":1059,"slug":1060,"properties":1061,"entityType":136,"verifyStatus":137,"verifyTime":1070,"verifyNote":139,"languages":22,"translateLanguages":22,"viewCount":23,"primaryUrl":1071,"fullTextUrl":22,"authors":1072,"publicationType":177,"publisherRelationship":1088,"citationCount":1139,"citationInfo":1140,"publishDate":1143,"publishYear":1141,"citationAnalyzeStatus":21,"lastCitationAnalyze":1144,"indexDatabases":1145,"openAccess":22,"references":1146,"isForceReanalyzing":269},"456c9496-4faa-4bdd-805f-bd95bb63eb54","2024-01-05T04:09:11.201+00:00","2026-03-31T16:01:47.910+00:00",[],"Pointwise-convergence-of-expansions-with-respect-to-certain-product-systems",{"abstract":1062,"title":1064,"gsPaper":1066,"doi":1068},{"EN":1063},"Изучается μ-почти всю ду сходимость рядов п о системе из произведений функци й системы\n                  \n                    \n                  \n                 такой, что для некото рой последовательно сти сг-алгебр\n                  \n                    \n                  \n                 функция ϕn является An-измеримой,\n                  \n                    \n                  \n                . Доказан о, что система произвед ений по такой системе Ф является системой сходимости. В качестве специального случая получается, что если э лементы Ф являются независимы ми функциями с равным нулю средним значением и ¦ϕn¦=1 (n=1,2, …), то система произведени й функций из Ф является системой сходимости. Эти результаты влекут, чт о системами сходимос ти являются системы Уол ша—Пэли и Уолша—Качм ажа, а также перестановка Качмаж а тригонометрической системы.",{"EN":1065},"Pointwise convergence of expansions with respect to certain product systems",{"VOID":1067},"[\"10385247174802507435\",\"753081458079035643\"]",{"VOID":1069},"10.1007\u002FBF02079908","2024-05-06T08:09:51.036+00:00","https:\u002F\u002Flink.springer.com\u002Farticle\u002F10.1007\u002FBF02079908",[1073],{"id":1074,"sortIndex":23,"researcher":22,"roles":1075,"affiliations":1076,"properties":1085,"displayName":1087,"givenName":22,"familyName":22},"2527d0ac-b959-45b8-a37d-66b75017bc97",[293],[1077],{"id":1078,"sortIndex":23,"affiliation":1079,"properties":22},"cda75da1-1027-4e3e-8249-18f4f0d77e88",{"id":1078,"createTime":22,"updateTime":22,"relativeEntities":1080,"slug":22,"properties":1081,"entityType":22,"verifyStatus":22,"verifyTime":22,"verifyNote":22,"languages":22,"translateLanguages":22,"viewCount":22,"url":22,"parentIds":1084,"statistic":22},[],{"title":1082},{"VI":1083},"Department of Mathematics, Eötvös Loránd University, Budapest, Hungary",[],{"title":1086},{"VI":1087},"F. 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Carleson, On convergence and growth of partial sums of Fourier series,Acta Math.,116 (1966), 135–157.",{"doi":483},{"id":479,"text":1151,"url":481,"identifiers":1152},"P. Billard, Sur la convergence presque partout des séries de Fourier-Walsh des fonctions de l'espaceL 2 (0,1),Studia Math.,28 (1967), 363–388.",{"doi":483},{"id":479,"text":1154,"url":481,"identifiers":1155},"D. Waterman, W-systems are the Walsh functions,Bull. Amer. Math. Soc.,75 (1969), 139–142.",{"doi":483},{"id":22,"text":1157,"url":22,"identifiers":1158},"F. Schipp, Über die Konvergenz von Reihen nach Produktsystemen,Acta Sci. Math. (Szeged),35 (1973), 13–16.",{},{"id":1160,"text":1161,"url":1162,"identifiers":1163},"5f18f28f-95e7-4da3-b602-ec76376f534f","F. Schipp, OnL p-norm convergence of series with respect to product systems,Anal. Math.,2 (1976), 49–64.","https:\u002F\u002Flink.springer.com\u002Farticle\u002F10.1007\u002FBF02079907",{"doi":1164},"10.1007\u002FBF02079907",{"id":479,"text":1166,"url":481,"identifiers":1167},"G. Alexits,Convergence problems of orthogonal series, Pergamon Press (New York-Oxford-Paris, 1961).",{"doi":483},{"id":22,"text":1169,"url":22,"identifiers":1170},"J. L. Bretagnolle, S. D. Chatterji etP. A. Meyer,Ecole d'Eté de Probabilités: Processus Stochastiques, Springer (New York-Heidelberg-Berlin, 1973).",{},{"id":22,"text":1172,"url":22,"identifiers":1173},"F. Schipp, On a generalization of the concept of orthogonality,Acta Sci. Math. (Szeged),37 (1975), 279–285.",{},{"id":479,"text":1175,"url":481,"identifiers":1176},"J. Neveu,Martingales à temps discret, Masson et Cie (Paris, 1972).",{"doi":483},{"id":22,"text":1178,"url":22,"identifiers":1179},"Р. Ш. Липцер иА. Н. Ши ряев,Статистика слу чайных процессов, На ука (Москва, 1974).",{},{"id":479,"text":1181,"url":481,"identifiers":1182},"L. A. Balasov andA. I. Rubinstein, Series with respect to the Walsh system and generalization,J. Soviet Math.,1 (1973), 727–763.",{"doi":483},{"id":22,"text":1184,"url":22,"identifiers":1185},"Ф. Шипп, О некоторы х перестановках рядо в по системе Уолша,Ма тем. заметки,18 (1975), 193–201.",{},{"id":479,"text":1187,"url":481,"identifiers":1188},"Wo Sang Young, on rearrangements of Walsh-Fourier series and Hardy-Littlewood type maximal inequalities,Bull. Amer. Math. Soc.,80 (1974), 490–494.",{"doi":483},{"id":479,"text":1190,"url":481,"identifiers":1191},"J. Gosselin, Almost everywhere convergence of Vilenkin-Fourier series,Trans. Amer. Math. Soc.,185 (1973), 345–370.",{"doi":483},{"id":1193,"createTime":1194,"updateTime":1195,"relativeEntities":1196,"slug":1197,"properties":1198,"entityType":136,"verifyStatus":137,"verifyTime":1207,"verifyNote":139,"languages":1208,"translateLanguages":22,"viewCount":23,"primaryUrl":1209,"fullTextUrl":22,"authors":1210,"publicationType":177,"publisherRelationship":1249,"citationCount":362,"citationInfo":1302,"publishDate":1304,"publishYear":729,"citationAnalyzeStatus":1305,"lastCitationAnalyze":1306,"indexDatabases":1307,"openAccess":22,"references":1308,"isForceReanalyzing":269},"6001bd4a-dde6-4fce-80d9-f7e35ee8132e","2024-04-17T17:13:15.827+00:00","2026-03-03T00:04:25.213+00:00",[],"Composition-Operators-in-Grand-Lebesgue-Spaces",{"openalex":1199,"title":1201,"gsPaper":1203,"doi":1205},{"VOID":1200},"W4319458974",{"EN":1202},"Composition Operators in Grand Lebesgue 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Appell and P. P. Zabreiko, Nonlinear Superposition Operators, Cambridge University Press (Cambridge, 1990).",{"doi":1312},"10.1017\u002FCBO9780511897450",{"id":22,"text":1314,"url":22,"identifiers":1315},"G. Bourdaud, Personal communication (2019).",{},{"id":22,"text":1317,"url":22,"identifiers":1318},"G. Bourdaud, An introduction to composition operators in Sobolev spaces, Eurasian Math. J., 14 (2023), to appear.",{"doi":1319},"10.32523\u002F2077-9879-2023-14-1-39-54",{"id":22,"text":1321,"url":22,"identifiers":1322},"C. Capone and A. Fiorenza, On small Lebesgue spaces, J. Funct. Spaces Appl., 3 (2005), 73–89.",{"doi":1323},"10.1155\u002F2005\u002F192538",{"id":22,"text":1325,"url":22,"identifiers":1326},"R. M. Corless, H. H. Gonnet, D. E. Hare, D. J. Jeffrey and D. E. Knuth, On the Lambert W function, Adv. Comput. Math., 5 (1996), 329–359.",{"doi":1327},"10.1007\u002FBF02124750",{"id":22,"text":1329,"url":22,"identifiers":1330},"N. G. de Bruijn, Asymptotic Methods in Analysis, North-Holland (1961).",{},{"id":22,"text":1332,"url":22,"identifiers":1333},"G. Di Fratta and A. Fiorenza, A direct approach to the duality of grand and small Lebesgue spaces, Nonlinear Anal., 70 (2009), 2582–2592.",{"doi":1334},"10.1016\u002Fj.na.2008.03.044",{"id":22,"text":1336,"url":22,"identifiers":1337},"R. M. Dudley and R. Norvaiša, Concrete Functional Calculus, Springer Monographs in Mathematics, Springer (New York, 2011).",{"doi":1338},"10.1007\u002F978-1-4419-6950-7",{"id":22,"text":1340,"url":22,"identifiers":1341},"A. Fiorenza, Duality and reflexivity in grand Lebesgue spaces, Collect. Math., 51 (2000), 131–148.",{},{"id":22,"text":1343,"url":22,"identifiers":1344},"A. Fiorenza and G. E. Karadzhov, Grand and small Lebesgue spaces and their analogs, Z. Anal. Anwend., 23 (2004), 657–681.",{"doi":1345},"10.4171\u002FZAA\u002F1215",{"id":22,"text":1347,"url":22,"identifiers":1348},"L. Greco, T. Iwaniec and C. Sbordone, Inverting the p-harmonic operator, Manuscripta Math., 92 (1997), 249–258.",{"doi":1349},"10.1007\u002FBF02678192",{"id":22,"text":1351,"url":22,"identifiers":1352},"T. Iwaniec and C. Sbordone, On the integrability of the Jacobian under minimal hypotheses, Arch. Rational Mech. Anal., 119 (1992), 129–143.",{"doi":1353},"10.1007\u002FBF00375119",{"id":22,"text":1355,"url":22,"identifiers":1356},"A. Karapetyants and M. Lanza de Cristoforis, Composition operators in generalized Morrey spaces, Z. Anal. Anwend. (2023), to appear.",{"doi":1206},{"id":22,"text":1358,"url":22,"identifiers":1359},"A. N. Karapetyants and S. G. Samko, On grand and small Bergman spaces, Math. Notes, 104 (2018), 431–436; translated from Mat. Zametki, 104 (2018), 439–446 (in Russian).",{"doi":1360},"10.1134\u002FS0001434618090109",{"id":22,"text":1362,"url":22,"identifiers":1363},"Y. Katznelson, An Introduction to Harmonic Analysis, Dover (1976).",{},{"id":22,"text":1365,"url":22,"identifiers":1366},"T. Runst and W. Sickel, Sobolev Spaces of Fractional Order, Nemytskij Operators and Nonlinear Partial Differential Equations, De Gruyter (Berlin, 1996).",{"doi":1367},"10.1515\u002F9783110812411",{"id":22,"text":1369,"url":22,"identifiers":1370},"S. M. Umarkhadzhiev, Generalization of the notion of grand Lebesgue space, Russian Math. (Iz. VUZ), 58 (2014), 35–43; translated from Izv. Vyssh. Uchebn. Zaved. Mat. (2014), no. 4, 42–51 (in Russian).",{"doi":1371},"10.3103\u002FS1066369X14040057",{"id":1373,"createTime":1374,"updateTime":1375,"relativeEntities":1376,"slug":1377,"properties":1378,"entityType":136,"verifyStatus":137,"verifyTime":1388,"verifyNote":139,"languages":22,"translateLanguages":22,"viewCount":23,"primaryUrl":1389,"fullTextUrl":22,"authors":1390,"publicationType":177,"publisherRelationship":1404,"citationCount":22,"citationInfo":22,"publishDate":1456,"publishYear":1457,"citationAnalyzeStatus":1305,"lastCitationAnalyze":1458,"indexDatabases":1459,"openAccess":22,"references":22,"isForceReanalyzing":269},"16b6b17b-06ba-4299-80d9-7359bdfa1f95","2023-12-22T15:38:29.865+00:00","2026-02-28T08:49:28.282+00:00",[],"On-the-regular-summability-of-multiple-function-series-and-Lebesgue-functions",{"abstract":1379,"title":1381,"gsPaper":1383,"references":1384,"doi":1386},{"EN":1380},"Основной целью работ ы является обобщение одного результата Кратца и Т раутнера [4], известного для одном ерных функциональны х рядов, на кратные ряды. Этот рез ультат касается суммируемо сти функционального ряда почти всюду при слабых пред положениях. В частности, он примен им к суммируемости по Чезаро и по Риссу. Мы рассматриваемd-кр атный ряд\n                  \n                    \n                  \n                  \n$$\\mathop \\sum \\limits_{k_1  = 0}^\\infty   \\cdots \\mathop \\sum \\limits_{k_d  = 0}^\\infty  c_{k_1 ,...,k_d } f_{k_1 ,...,k_d } (x),  \\mathop \\sum \\limits_{k_1  = 0}^\\infty   \\cdots \\mathop \\sum \\limits_{k_d  = 0}^\\infty  c_{k_1 ,...,k_d }^2\u003C \\infty $$\n\n                \n и предполагается, что функции\n                  \n                    \n                  \n                  \n$$f_{k_1 ,...,k_d } (x)$$\n\n                 интегрируе мы по пространству с полож ительной мерой и имеют почти вс юду ограниченные фун кции Лебега для метода суммирова ния Т. Метод Т определяетсяd-мерной матрицей\n                  \n                    \n                  \n                  \n$$T = \\{ a_{m_1 ,...,m_d ;k_1 ,...,k_d } \\} $$\n\n                 сл едующим образом:\n                  \n                    \n                  \n                  \n$$t_{m_1 ,...,m_d } (x) = \\mathop \\sum \\limits_{k_1  = 0}^\\infty   \\cdots \\mathop \\sum \\limits_{k_d  = 0}^\\infty  a_{m_1 ,...,m_d ;k_1 ,...,k_d } c_{k_1 ,...,k_d } f_{k_1 ,...,k_d } (x).$$\n\n                \n Эти средние существу ют, поскольку мы предп олагаем, что\n                  \n                    \n                  \n                  \n$$a_{m_1 ,...,m_d ;k_1 ,...,k_d }  = 0$$\n\n                ,если max(k\n1,...,k\nd) достаточно вели к (в зависимости, конеч но, отm\n1,...,m\nd). При некоторых дополнительных усло виях на матрицуТ (см. (7)– (9) в разделе 3) устанавлива ется почти всюду регулярная схо димость средних\n                  \n                    \n                  \n                  \n$$t_{m_1 ,...,m_d } (x) \\user2{} \\user2{(}m_1 \\user2{,}...\\user2{,}m_d \\user2{)} \\to \\infty $$\n\n                . Как вспомогательный результат, в работе об общается теорема Алексича [1] о сх одимости почти всюду некоторы х подпоследовательн остей частных сумм функцио нального ряда.",{"EN":1382},"On the regular summability of multiple function series and Lebesgue functions",{"VOID":1204},{"VOID":1385},"G. Alexits, On the convergence of function series,Acta Sci. Math. (Szeged),34 (1973), 1–9.\nG. Alexits andA. Sharma, The influence of Lebesgue functions on the convergence and summability of function series,Acta Sci. Math. (Szeged),33 (1972), 1–10.\nG. H. Hardy, On the convergence of certain multiple series,Proc. Cambridge Philosoph. Soc.,19 (1916–19), 86–95.\nW. Kratz andR. Trautner, On the summability of function series,Acta Math. Acad. Sci. Hungar.,33 (1979), 101–104.\nL. Leindler, Über die sehr starke Riesz-Summierbarkeit der Orthogonalreihen und Konvergenz lückenhafter Orthogonalreihen,Acta Math. Acad. Sci. Hungar.,13 (1962), 401–404.\nF. Móricz, On the convergence in a restricted sense of multiple series,Analysis Math.,5 (1979), 135–147.\nF. Móricz, Lebesgue functions and multiple function series. II,Acta Math. Acad. Sci.Hungar.,39 (1982), 95–105.\nG. Sunouchi, On the Riesz-summability of Fourier series,Tôhoku Math. J.,11 (1959), 319–326.\nK. Tandori, Weitere Bemerkungen über die Konvergenz und Summierbarkeit der Funktionenreihen,Acta Math. Acad. Sci. 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