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Problems in Mathematical Alnalysis III Kaczkornowak 1
Copyright 2003 American Mathematical Society Duplication prohibited Please report unauth Thank You! http://dx.doi.org/10.1090/stml/021 Problems in Mathematical Analysis HI Integration Copyright 2003 American Mathematical Society Duplication prohibited Please report unauth Thank You! This page intentionally left blank Copyright 2003 American Mathematical Society Duplication prohibited Please report unauth Thank You! STUDENT MATHEMATICAL LIBRARY Volume 21 Problems in Mathematical Analysis III Integration W J Kaczor M.T Nowak #AMS AMERICAN MATHEMATICA L SOCIET Y Copyright 2003 American Mathematical Society Duplication prohibited Please report unauth Thank You! Editorial Boar d David Bressoud , Chai r Danie l L GorofT Car l Pomeranc e 2000 Mathematics Subject Classification Primar y 00A07 , 26A42 ; Secondary 26A45 , 26A46 , 26D1 , 28A1 For additiona l informatio n an d updates o n this book , visi t www.ams.org/bookpages/stml-21 Library o f Congres s Cataloging-in-Publicatio n D a t a Kaczor, W J (Wieslaw a J.) , 949 [Zadania z analizy matematycznej English ] Problems i n mathematica l analysis I Rea l numbers , sequence s an d serie s / W J Kaczor , M T Nowak p cm — (Studen t mathematica l library , ISS N 520-91 ; v 4) Includes bibliographica l references ISBN 0-821 8-2050- (softcove r ; alk paper ) Mathematica l analysis I Nowak , M T (Mari a T.) , 951 - II Title III Series QA300K32513 200 515'.076— dc2199-08703 Copying an d reprinting Individua l reader s o f thi s publication , an d nonprofi t libraries actin g fo r them , ar e permitte d t o mak e fai r us e of th e material , suc h a s to copy a chapte r fo r us e in teachin g o r research Permissio n i s grante d t o quot e brie f passages fro m thi s publicatio n i n reviews, provide d th e customary acknowledgmen t of the sourc e i s given Republication, systemati c copying , o r multiple reproductio n o f any materia l i n this publication i s permitted onl y unde r licens e fro m th e American Mathematica l Society Requests fo r suc h permissio n shoul d b e addresse d t o th e Acquisition s Department , American Mathematica l Society , 20 Charles Street , Providence , Rhod e Islan d 02904 2294, USA Requests ca n also be made b y e-mail t o reprint-permission@ams.org © 200 b y the American Mathematica l Society Al l rights reserved The America n Mathematica l Societ y retain s al l right s except thos e grante d t o the United State s Government Printed i n the United State s o f America @ Th e paper use d i n this boo k i s acid-free an d fall s withi n th e guideline s established t o ensure permanenc e an d durability Visit th e AMS home pag e a t http://www.ams.org / 10 8 07 06 05 04 Copyright 2003 American Mathematical Society Duplication prohibited Please report unauth Thank You! Contents Preface vi i Part Problem s Chapter Th e Riemann-Stieltje s Integra l §1.1 Propertie s o f th e Riemann-Stieltje s Integra l §1.2 Function s1 o f Bounde d Variatio n §1.3 Furthe r Propertie s o f th e Riemann-Stieltje s Integra 1l §1.4 Prope r Integral s §1.5 Imprope r Integral s §1.6 Integra l Inequalitie s §1.7 Jorda n Measur e Chapter Th e Lebesgu e Integra l §2.1 Lebesgu e Measur e o n th e Rea l Lin e §2.2 Lebesgu e Measurabl e Function s 6 §2.3 Lebesgu e Integratio n §2.4 Absolut e Continuity , Differentiatio n an d Integratio n Copyright 2003 American Mathematical Society Duplication prohibited Please report unauth Thank You! Contents VI §2.5 Fourie r Serie s Part Solution s Chapter Th e Riemann-Stieltje s Integra l §1.1 Propertie s o f th e Riemann-Stieltje s Integra l §1.2 Function s o1 f Bounde d Variatio n §1.3 Furthe r Propertie s o f the Riemann-Stieltje s Integra1l §1.4 Prope r Integral s §1.5 Imprope r Integral s §1.6 Integra l Inequalitie s 20 §1.7 Jorda n Measur e 22 Chapter Th e Lebesgu e Integra l 24 §2.1 Lebesgu e Measur e o n th e Rea l Lin e 24 §2.2 Lebesgu e Measurabl e Function s 26 §2.3 Lebesgu e Integratio n 28 §2.4 Absolut e Continuity , Differentiatio n an d Integratio n 29 §2.5 Fourie r Serie s Bibliography - Book s 35 Index 35 Copyright 2003 American Mathematical Society Duplication prohibited Please report unauth Thank You! Preface This i s a seque l t o ou r book s Problems in Mathematical Analysis I, II (Volume s an d i n th e Studen t Mathematica l Librar y series) The boo k deal s with th e Riemann-Stieltje s integra l an d th e Lebesgu e integral fo r rea l function s o f one rea l variable Th e boo k i s organize d in a wa y simila r t o tha t o f the firs t tw o volumes , tha t is , i t i s divide d into tw o parts : problem s an d thei r solutions Eac h sectio n start s with a numbe r o f problem s tha t ar e moderat e i n difficulty , bu t som e of th e problem s ar e actuall y theorems Thu s i t i s no t a typica l prob lem book , bu t rathe r a supplemen t t o undergraduat e an d graduat e textbooks i n mathematica l analysis W e hop e tha t thi s boo k wil l b e of interes t t o undergraduat e students , graduat e students , instructor s and researche s i n mathematical analysi s an d it s applications W e also hope tha t i t wil l b e suitabl e fo r independen t study The first chapte r of the book is devoted to Riemann an d Riemann Stieltjes integrals I n Sectio n w e conside r th e Riemann-Stieltje s integral wit h respec t t o monotoni c functions , an d i n Sectio n w e turn t o integratio n wit h respec t t o function s o f bounde d variation In Sectio n w e collec t famou s an d no t s o famou s integra l inequal ities Amon g others , on e ca n fin d OpiaP s inequalit y an d Steffensen' s inequality W e clos e th e chapte r wit h th e sectio n entitle d "Jorda n measure" Th e Jorda n measure , als o calle d conten t b y som e authors , Copyright 2003 American Mathematical Society Duplication prohibited Please report unauth Thank You! Vlll Preface is no t a measur e i n th e usua l sens e becaus e i t i s no t coun t ably addi tive However , i t i s closely connecte d wit h th e Rieman n integral , an d we hope that thi s section will give the student a deeper understandin g of the idea s underlyin g th e calculus Chapter deals with the Lebesgu e measur e an d integration Sec tion present s man y problem s connecte d wit h convergenc e theo rems tha t permi t th e interchang e o f limi t an d integral ; LP spaces o n finite interval s ar e als o considere d here I n th e nex t section , absolut e continuity an d the relation between differentiation an d integration ar e discussed W e present a proo f o f th e theore m o f Banac h an d Zareck i which state s tha t a function / i s absolutely continuou s o n a finite in terval [a , b] i f and onl y i f it i s continuous an d o f bounded variatio n o n [a, b], an d map s set s o f measur e zer o int o set s o f measur e zero Fur ther, th e concep t o f approximate continuit y i s introduced I t i s worth noting her e that ther e i s a certain analog y betwee n tw o relationships : the relationshi p betwee n Rieman n integrabilit y an d continuity , o n the on e hand , an d th e relationshi p betwee n approximat e continuit y and Lebesgu e integrability , o n th e othe r hand Namely , a bounde d function o n [a , b] i s Rieman n integrabl e i f an d onl y i f i t i s almost ev erywhere continuous ; an d similarly , a bounde d functio n o n [a , b] is measurable, an d s o Lebesgu e integrable , i f an d onl y i f i t i s almos t everywhere approximatel y continuous Th e las t sectio n i s devoted t o the Fourie r series Give n th e existenc e o f extensiv e literatur e o n th e subject, e.g , th e book s b y A Zygmun d "Trigonometri c Series" , b y N K Bar i " A Treatis e o n Trigonometri c Series" , an d b y R E Ed wards "Fourie r Series" , w e foun d i t difficul t t o decid e wha t materia l to includ e i n a boo k whic h i s primaril y addresse d t o undergraduat e students Consequently , w e hav e mainl y concentrate d o n Fourie r co efficients o f function s fro m variou s classe s an d o n basi c theorem s fo r convergence o f Fourie r series All the notatio n an d definition s use d i n this volume ar e standard One ca n find the m i n the textbook s [27 ] and [28] , which als o provid e the reade r wit h th e sufficien t theoretica l background However , t o avoid ambiguit y an d t o mak e the boo k self-containe d w e start almos t every sectio n wit h a n introductor y paragrap h containin g basi c defi nitions an d theorem s use d i n th e section Ou r referenc e convention s Copyright 2003 American Mathematical Society Duplication prohibited Please report unauth Thank You! Preface IX are bes t explaine d b y th e followin g examples : 2.1 or I , 2.1 o r II, 2.1 , whic h denot e th e numbe r o f th e proble m i n thi s volume , in Volum e I o r i n Volum e II , respectively W e als o us e notatio n an d terminology give n i n th e first tw o volumes Many problem s hav e bee n borrowe d freel y fro m proble m section s of journals lik e the America n Mathematica l Monthl y an d Mathemat ics Today (Russian) , an d fro m variou s textbooks an d proble m books ; of thos e onl y book s ar e liste d i n th e bibliography W e woul d lik e t o add tha t man y problem s i n Sectio n com e fro m th e boo k o f Ficht enholz [1 ] an d Sectio n i s influence d b y th e boo k o f Rogosinsk i [26] Regrettably , i t wa s beyon d ou r scop e t o trac e al l th e origina l sources, an d w e offer ou r sincer e apologies if we have overlooked som e contributions Finally, w e woul d lik e t o than k severa l peopl e fro m th e Depart ment o f Mathematics o f Maria Curie-Sklodowsk a Universit y t o who m we are indebted Specia l mentio n shoul d b e mad e o f Tadeusz Kuczu mow an d Witol d Rzymowsk i fo r suggestion s o f severa l problem s an d solutions, an d o f Stanisla w Pru s fo r hi s counselin g an d Te X support Words o f gratitud e g o t o Richar d J Libera , Universit y o f Delaware , for hi s generou s hel p wit h Englis h an d th e presentatio n o f th e ma terial W e ar e ver y gratefu l t o Jadwig a Zygmun t fro m th e Catholi c University o f Lublin , wh o s draw n al l th e figure s an d helpe d u s with incorporatin g the m int o th e text W e than k ou r student s wh o helped u s i n th e lon g an d tediou s proces s o f proofreading Specia l thanks g o t o Pawe l Sobolewsk i an d Przemysla w Widelski , wh o hav e read th e manuscrip t wit h muc h car e and thought , an d provide d man y useful suggestions Withou t thei r assistanc e som e errors , no t onl y ty pographical, coul d have passed unnoticed However , we accept ful l responsibility fo r an y mistake s o r blunder s tha t remain W e woul d like t o tak e thi s opportunit y t o than k th e staf f a t th e AM S fo r thei r long-lasting cooperation , patienc e an d encouragement W J Kaczor , M T Nowa k Copyright 2003 American Mathematical Society Duplication prohibited Please report unauth Thank You! 344 Solutions : Th e Lebesgu e Integra l Put b 2k) sin ^ an Pn =n^2(4 + dT = ~Yl k^ a n l+ b l / c = l fc=l If li m P n = , the n n—>oo n k-K Pn>n YJA n 2 bl) sin — > - £ k + (a2k + b\) > T , where th e las t inequalit y follow s fro m th e Cauch y inequality Thi s shows tha t li m T n = Conversely , i f li m T n = , the n w e hav e n—>oo n—*o o Nn * o o, b 2k)sm2^+n ^ Pn = n^2(4 + + b\) sin ^ {a\ fc = l fc = iVn+ l iVn , 22 ° ° - V " kb k > -b [n/2]([n/2] + l ) [n / ] " fc=l which implie s tha t [n/2]6[ n/2] tend s t o zer o a s n —* oo OO 2.5.58 I f ^ n sinnx i s th e Fourie r serie s o f a bounde d function , n=l oo then th e sequenc e {cr n(:r)} o f th e Cesar o mean s o f ^2 b ns'mnx i s ra=l bounded (se e 2.5.41 ) I n particular , {cr n(7r/n)} i s bounded , an d on e can procee d a s i n th e solutio n o f th e previou s proble m t o sho w tha t nbn = 0(1 ) No w assum e tha t nb n = 0(1 ) ; tha t is , ther e i s C > such tha t nb n < C , fo r n G N Ou r m i s t o sho w tha t ther e i s a constant M suc h tha t fo r al l x an d n , \Sn{x)\ = y ^ 6/ e sin kx < M fc=i Without los s o f generalit y w e ca n assum e tha t < x < 7r The n fo r ]v^X < x < jj w e hav e iV i(^)| < / bk sin/ex + fc=i Y^ 6f c sin fcx fc=JV+l (an empt y su m bein g counte d a s zero ) an d NN 2^ bk sin kx < x Y^ kbk < KC fc=i fe=i Moreover, summatio n b y part s give s N^ bk sin /ex fc=iV+l n-l ] T (6 fc - 6fc+i)(J3 fc(x) - D N(x)) + n (D n (x) - L>TV(X) ) fc=JV+l where |£>fc(s) ^sinj^