A GEOMETRIC PROOF FOR SOflE SIMPLE CASES OF :iartit!gale BRAILEY SIMS
|
|
- Evelyn Richard
- 5 years ago
- Views:
Transcription
1 A GEOMETRC ROOF FOR SOflE SMLE CASES OF :iartt!gale CONVERGENCE BRALEY SMS The use of real valued martigales i probability theory has grov steadily sice their detailed itroductio by Doob i 195. Vector valued martigales vere first studied explicitly i 196 by Chatterji ad idepedetly Scalora, ad vere rapidly recogised as a pwerful tool for the study of certai Baach space structures (see, for example, Diestel ad Uhl, 1977). Besides itroducig the idea of a vector valued aartigale this article gives a "geometric" proof of the followig covergece theorem which makes o use of probability theory ad ivolves a miim~ of measure theory. 'HU)RM 1. Let (,, fl be a martigale i L (,u;x/ uhere 1 < p < - m.d X is a uifom!? muex Bmroch space. The :here eaiste fw c L (Z,g;X) 1~6th lf + if (ad a3 if) 1 i\/? 5 K for all d.ere X >. Our proof is particularly simple whe p = ad X is a Hilbert space. Sice this case vould usually be itroduced first, the proof for other cases has bee relegated to foototes ad remarks. The article oves much to the suggestios of my colleagues, i particular. Xr. L. Kavalieris, made durig a series of semiars at the Uiversity of New Eglad. 5 1 RELMNARES. Throughout the article, 1 deotes a o-algebra of subsete of Ci ad p a coutably additive probability measure o Z (that is p(el for all E c Z ad ~(') = 1). R is a Hilbert space over the real field, vith ier product (-,-, ad X is a real Baach space. We say X ie uifomly cavez if: every sequece (z-l for which Z + Z r 1 have a comro limit as,m +-, is a Cauchy sequece (ad hece llz 11 ad 1i coverget).() t is a easy cosequece of the parallelogram rule that B is uifordy - Z Z m +Z covex (11-1 = YZ~ + YZ~~ - ~ ~ ). () Bocher itegmtiu for vector valued fuctiue. The reader iterested oly i real valued martigales could pass over this material provided B (the real umbers) is everywhere substituted for 6' or X ad itegrals (1) This form of the defiitio is most suited for our purpose. The equivalece with Clarkso's origial defiitio beig a simple matter. () For details see either Duford ad Schvartq1958 or Diestel ad Uhl, 1977.
2 are Fterpreted i the sese of Lebesgue. A fuctio j: R -+ X is u-measurcbze Ff there exists a sequece of sfmple fuctios (S with S (wl - f(wlll * for u-almost all w E R. A simple fuctio is of the form e 1, which,without loss of geerality, may be N assumed to be pairvise disjoit. Here wd elsevhere, X is the cl-aracteristic fuctio E of rhe set E. The itegral of such a slmple fuctio f is defhed i the obvious vay; for some z1,xz,..., xn e X ad El,..., E A u-measurable fuctio is itegrable if there exists a sequece S) of s'sple fuctios with R - f d *. (Lebesgue itegral) this case the orm lfmit, as * m, of S d~ exists for each E E Z ad is ide- pedet of ay particular choice for S. By defiitio, f du is this commo limit. L 'C For 1 S p < we deote by L Z,u;X) the space of f: R * X for which 1 = J' ilf(u)li E~)B r R < o. t is a sigificat result that the elemets of L (1,u;X are precisely those 1 i.aciioe which are itegrable i the above sese. t follws from K61der1s iequality d the filteess of u that the elemets of L fz,p,x) are itegrable. Further, as i the scalar case, 1. is a Baach space orm for this space. 13) 1-3. LZZ,u;lil is a Hilbert space. Fim. = j(fw),gu)) dp defier a ier-prod,wt o LZ(Z.~;%) X.3. Day r19411 proves the followig. A 3. L Z,p;X) is AfomZy mvez if 1 < p < - - X is 9orifsmLy cvez. p.71 Strictly, th space of equivalece chses modulo almost everyvhere zero fuctios
3 Let Z be a proper sub -algebra of 1. the L (Z u;x) is a proper subspace oj of L Z,u;X). i Clearly for ay give f e L (E,u;X) there ca be at most oe f 6 L (1 p;x) r o' ( with je f & = b fo dy for all E e Lo. Whe such a f exists it will be referred to as the -it-kmal eqecidtia 3f f with respect to Z ad deoted by E(f(ZO). L W 4. fo r Mo = L (Z,u;H) i the wditioal erpectatia of f :Ath respect to Z, df cd mzy if (f- fo,g) = for all g e Mo.??,w?. t is sufficiet to cosider g = de where z e A ad E e ZO (Mo is the closed liear spa of such fuctios). Nw, >:a if - fg,z XS = if ad oly if (1 if - fo)iui du,z) = for all r r a. E - While it is ot ecessary for our subsequet work, oe kportat cosequece sf ixs lemma is the existece of E(f 1 ~ for ~ ) every f e L(ZJp;B) a& sub a-algebra ZO. T-is follws sice fo is the foot of the per~edicular frm f' to H if ad oly if fo is :he closest poit frw No to f, the existece of which is emured by the uiform ccr~exity of L (Z,u;B!. Aother cosequece is the follwig. C:XLUKY 5. For f r LiZJy;A), E(flZG)llZ 5 llfl1. ( ) ~rzieed rip -cpig j + ~ ( iol f le z ~ar me lie projectio mto L(Zlr~;B : LCF. 1: C 11 = ifo,,fo) = (f - if - fo),fo) = (f,fo), by lemma 4. 5 \~lillf 11, by the Cauchy-Schvartz iequality. S
4 By a.mtigaze fz f i L fz,p;x we mea a ested sequece ' 5... ; Z of sub o-algebras ad a sequece of fuctios T, 5 Li S... d - - f,... with f 6 L fz,b;x, -p'fl,..., f- = 7:fE 1 Z j for a11 m. m (51 which satisfy the mrtirqale corditio As a example ote that for ay ested sequeces of sub a-algebras Z,, E 5... r Z ad ay f c LfZ,p;H, (1,E(fjEj) is a aartigale. To check the martigale coditio observe that for r m ad E c Zm we have J E EfflZm) du = f du, by the defiitio of -f. (., JE = EfflZ d,~, agai by the defiitio of E(-(. ad the JE fact that E r..?yc,~ of t& Mai Theorem. Ue are ow i a positio to prove Theorem 1..;/ A proof of this result for geeral p c [?,- is the follwig. By their desity we may assume that f is a shple fuctio, the r Jr: Eff Z h.11 XE 1 (1 du, by Jese's iequality. G- 1 t t = 1, f lzill X ) dp, by defiitio of.ei. /zo) ad the fact i=? Ei il c r. 't, Settig ufe = je f dp, for a11 E E Z, the m.artig.de coditio may be re-expressed as: For m the restrictio of u to Z is pm.
5 , Let ('i f l be a martigale i L(Z,p;H) with lf U 5 K for all. Let = L,!E,;i;Hl, the we have the follovig structure.. - A ested sequece of closed subspaces M < M 5... s M 5... ad a szifcrmly bouded sequece fo,fl,...,fj... - with f e M. By the martigale coditio ad corollary 5 we have llfuz 5 forsm.?"us.soll, ]fl 11,...,llfll,... is a icreasig sequece of real ders bouded aowe by K, ad so coverget from below to some real umber. Further, for < m r ile(f + ~;lr/z~lll~ = ~ l l f 1 lagai, by corollary 5 ad the martigale coditio), so + -- ' 'm li k as m, -t -.?ece, by the uiform covexity of L (1, p;h'), fo, fl,...,f,... is a Cauchy sequece zr;d sc by the completeess of the space, coverges to some f- e LiZ,;?/. - f Z_ deotes the o-algebra geerated by u, it is trivial to check that? -- t J.4 = L(Z, p;h) - the proper home for our martigzle limit. - O the basis of our earlier observatios, the proof for arbitrary? E a d uiformly covex Baach space X is merely a paraphrase of the above. (,-) REFERENCES ;11 X.X. Day, "Some more uiformly covex spaces", Bull. her. Math. Soc. 9 (1941) J. Diestel ad J.J. Uhl Jr., "Vector measures", ilmer. Hcth. Soc., Xath. Survey No. 15 (1977). C31 N. Duford ad J.T. Schvartz, art. tersciece. N.Y. (1956). THE LARGEST RME Readers of the h a Ageles Times (November 16, 1978) vill o doubt kziov cf che exploits of two freshme studets of CSC Hayvard, Laura Hichel ad Curt Soil, L ~ O, for a high school computig project, shoved that (5553 digits) is prie. holl cotiued this work ad i February 1979 foud that '"" (f567 digits) is ~rhe. A more recet aoucemet of the L.A. Times (Xay 31, 1979) reveals that the is :4497 tbe ext Hersere prime - (;3:35 digits) a result due tc rry tielso d T'avid Sluwisiii
2 Banach spaces and Hilbert spaces
2 Baach spaces ad Hilbert spaces Tryig to do aalysis i the ratioal umbers is difficult for example cosider the set {x Q : x 2 2}. This set is o-empty ad bouded above but does ot have a least upper boud
More informationChapter 6 Infinite Series
Chapter 6 Ifiite Series I the previous chapter we cosidered itegrals which were improper i the sese that the iterval of itegratio was ubouded. I this chapter we are goig to discuss a topic which is somewhat
More informationMATH301 Real Analysis (2008 Fall) Tutorial Note #7. k=1 f k (x) converges pointwise to S(x) on E if and
MATH01 Real Aalysis (2008 Fall) Tutorial Note #7 Sequece ad Series of fuctio 1: Poitwise Covergece ad Uiform Covergece Part I: Poitwise Covergece Defiitio of poitwise covergece: A sequece of fuctios f
More informationDefinition 4.2. (a) A sequence {x n } in a Banach space X is a basis for X if. unique scalars a n (x) such that x = n. a n (x) x n. (4.
4. BASES I BAACH SPACES 39 4. BASES I BAACH SPACES Sice a Baach space X is a vector space, it must possess a Hamel, or vector space, basis, i.e., a subset {x γ } γ Γ whose fiite liear spa is all of X ad
More informationMath Solutions to homework 6
Math 175 - Solutios to homework 6 Cédric De Groote November 16, 2017 Problem 1 (8.11 i the book): Let K be a compact Hermitia operator o a Hilbert space H ad let the kerel of K be {0}. Show that there
More informationAxioms of Measure Theory
MATH 532 Axioms of Measure Theory Dr. Neal, WKU I. The Space Throughout the course, we shall let X deote a geeric o-empty set. I geeral, we shall ot assume that ay algebraic structure exists o X so that
More informationIntegrable Functions. { f n } is called a determining sequence for f. If f is integrable with respect to, then f d does exist as a finite real number
MATH 532 Itegrable Fuctios Dr. Neal, WKU We ow shall defie what it meas for a measurable fuctio to be itegrable, show that all itegral properties of simple fuctios still hold, ad the give some coditios
More informationMeasure and Measurable Functions
3 Measure ad Measurable Fuctios 3.1 Measure o a Arbitrary σ-algebra Recall from Chapter 2 that the set M of all Lebesgue measurable sets has the followig properties: R M, E M implies E c M, E M for N implies
More informationOn equivalent strictly G-convex renormings of Banach spaces
Cet. Eur. J. Math. 8(5) 200 87-877 DOI: 0.2478/s533-00-0050-3 Cetral Europea Joural of Mathematics O equivalet strictly G-covex reormigs of Baach spaces Research Article Nataliia V. Boyko Departmet of
More informationHomework 4. x n x X = f(x n x) +
Homework 4 1. Let X ad Y be ormed spaces, T B(X, Y ) ad {x } a sequece i X. If x x weakly, show that T x T x weakly. Solutio: We eed to show that g(t x) g(t x) g Y. It suffices to do this whe g Y = 1.
More informationProduct measures, Tonelli s and Fubini s theorems For use in MAT3400/4400, autumn 2014 Nadia S. Larsen. Version of 13 October 2014.
Product measures, Toelli s ad Fubii s theorems For use i MAT3400/4400, autum 2014 Nadia S. Larse Versio of 13 October 2014. 1. Costructio of the product measure The purpose of these otes is to preset the
More informationIntroduction to Optimization Techniques
Itroductio to Optimizatio Techiques Basic Cocepts of Aalysis - Real Aalysis, Fuctioal Aalysis 1 Basic Cocepts of Aalysis Liear Vector Spaces Defiitio: A vector space X is a set of elemets called vectors
More information62. Power series Definition 16. (Power series) Given a sequence {c n }, the series. c n x n = c 0 + c 1 x + c 2 x 2 + c 3 x 3 +
62. Power series Defiitio 16. (Power series) Give a sequece {c }, the series c x = c 0 + c 1 x + c 2 x 2 + c 3 x 3 + is called a power series i the variable x. The umbers c are called the coefficiets of
More informationSequences and Series of Functions
Chapter 6 Sequeces ad Series of Fuctios 6.1. Covergece of a Sequece of Fuctios Poitwise Covergece. Defiitio 6.1. Let, for each N, fuctio f : A R be defied. If, for each x A, the sequece (f (x)) coverges
More informationConvergence of random variables. (telegram style notes) P.J.C. Spreij
Covergece of radom variables (telegram style otes).j.c. Spreij this versio: September 6, 2005 Itroductio As we kow, radom variables are by defiitio measurable fuctios o some uderlyig measurable space
More informationINFINITE SEQUENCES AND SERIES
INFINITE SEQUENCES AND SERIES INFINITE SEQUENCES AND SERIES I geeral, it is difficult to fid the exact sum of a series. We were able to accomplish this for geometric series ad the series /[(+)]. This is
More informationA Proof of Birkhoff s Ergodic Theorem
A Proof of Birkhoff s Ergodic Theorem Joseph Hora September 2, 205 Itroductio I Fall 203, I was learig the basics of ergodic theory, ad I came across this theorem. Oe of my supervisors, Athoy Quas, showed
More information(A sequence also can be thought of as the list of function values attained for a function f :ℵ X, where f (n) = x n for n 1.) x 1 x N +k x N +4 x 3
MATH 337 Sequeces Dr. Neal, WKU Let X be a metric space with distace fuctio d. We shall defie the geeral cocept of sequece ad limit i a metric space, the apply the results i particular to some special
More informationChapter 3 Inner Product Spaces. Hilbert Spaces
Chapter 3 Ier Product Spaces. Hilbert Spaces 3. Ier Product Spaces. Hilbert Spaces 3.- Defiitio. A ier product space is a vector space X with a ier product defied o X. A Hilbert space is a complete ier
More informationIntroduction to Probability. Ariel Yadin
Itroductio to robability Ariel Yadi Lecture 2 *** Ja. 7 ***. Covergece of Radom Variables As i the case of sequeces of umbers, we would like to talk about covergece of radom variables. There are may ways
More informationLecture 3 : Random variables and their distributions
Lecture 3 : Radom variables ad their distributios 3.1 Radom variables Let (Ω, F) ad (S, S) be two measurable spaces. A map X : Ω S is measurable or a radom variable (deoted r.v.) if X 1 (A) {ω : X(ω) A}
More informationANSWERS TO MIDTERM EXAM # 2
MATH 03, FALL 003 ANSWERS TO MIDTERM EXAM # PENN STATE UNIVERSITY Problem 1 (18 pts). State ad prove the Itermediate Value Theorem. Solutio See class otes or Theorem 5.6.1 from our textbook. Problem (18
More informationLecture Notes for Analysis Class
Lecture Notes for Aalysis Class Topological Spaces A topology for a set X is a collectio T of subsets of X such that: (a) X ad the empty set are i T (b) Uios of elemets of T are i T (c) Fiite itersectios
More informationLecture 19. sup y 1,..., yn B d n
STAT 06A: Polyomials of adom Variables Lecture date: Nov Lecture 19 Grothedieck s Iequality Scribe: Be Hough The scribes are based o a guest lecture by ya O Doell. I this lecture we prove Grothedieck s
More informationMATH 413 FINAL EXAM. f(x) f(y) M x y. x + 1 n
MATH 43 FINAL EXAM Math 43 fial exam, 3 May 28. The exam starts at 9: am ad you have 5 miutes. No textbooks or calculators may be used durig the exam. This exam is prited o both sides of the paper. Good
More information1 Convergence in Probability and the Weak Law of Large Numbers
36-752 Advaced Probability Overview Sprig 2018 8. Covergece Cocepts: i Probability, i L p ad Almost Surely Istructor: Alessadro Rialdo Associated readig: Sec 2.4, 2.5, ad 4.11 of Ash ad Doléas-Dade; Sec
More informationVECTOR SEMINORMS, SPACES WITH VECTOR NORM, AND REGULAR OPERATORS
Dedicated to Professor Philippe G. Ciarlet o his 70th birthday VECTOR SEMINORMS, SPACES WITH VECTOR NORM, AND REGULAR OPERATORS ROMULUS CRISTESCU The rst sectio of this paper deals with the properties
More informationM17 MAT25-21 HOMEWORK 5 SOLUTIONS
M17 MAT5-1 HOMEWORK 5 SOLUTIONS 1. To Had I Cauchy Codesatio Test. Exercise 1: Applicatio of the Cauchy Codesatio Test Use the Cauchy Codesatio Test to prove that 1 diverges. Solutio 1. Give the series
More informationTheorem 3. A subset S of a topological space X is compact if and only if every open cover of S by open sets in X has a finite subcover.
Compactess Defiitio 1. A cover or a coverig of a topological space X is a family C of subsets of X whose uio is X. A subcover of a cover C is a subfamily of C which is a cover of X. A ope cover of X is
More informationSEMIGROUPS. D. Pfeifer. Communicated by Jerome A. Goldstein Dedicated to E.S. Lyapin on his 70th Birthday
A NOTE ON ~ROBABILISTIC REPRESENTATIONS OF OPERATOR SEMIGROUPS D. Pfeifer Commuicated by Jerome A. Goldstei Dedicated to E.S. Lyapi o his 70th Birthday I the theory of strogly cotiuous semigroups of bouded
More informationsin(n) + 2 cos(2n) n 3/2 3 sin(n) 2cos(2n) n 3/2 a n =
60. Ratio ad root tests 60.1. Absolutely coverget series. Defiitio 13. (Absolute covergece) A series a is called absolutely coverget if the series of absolute values a is coverget. The absolute covergece
More information6.3 Testing Series With Positive Terms
6.3. TESTING SERIES WITH POSITIVE TERMS 307 6.3 Testig Series With Positive Terms 6.3. Review of what is kow up to ow I theory, testig a series a i for covergece amouts to fidig the i= sequece of partial
More informationLECTURE 11 LINEAR PROCESSES III: ASYMPTOTIC RESULTS
PRIL 7, 9 where LECTURE LINER PROCESSES III: SYMPTOTIC RESULTS (Phillips ad Solo (99) ad Phillips Lecture Notes o Statioary ad Nostatioary Time Series) I this lecture, we discuss the LLN ad CLT for a liear
More informationA NOTE ON LEBESGUE SPACES
Volume 6, 1981 Pages 363 369 http://topology.aubur.edu/tp/ A NOTE ON LEBESGUE SPACES by Sam B. Nadler, Jr. ad Thelma West Topology Proceedigs Web: http://topology.aubur.edu/tp/ Mail: Topology Proceedigs
More information1 The Haar functions and the Brownian motion
1 The Haar fuctios ad the Browia motio 1.1 The Haar fuctios ad their completeess The Haar fuctios The basic Haar fuctio is 1 if x < 1/2, ψx) = 1 if 1/2 x < 1, otherwise. 1.1) It has mea zero 1 ψx)dx =,
More informationThe Wasserstein distances
The Wasserstei distaces March 20, 2011 This documet presets the proof of the mai results we proved o Wasserstei distaces themselves (ad ot o curves i the Wasserstei space). I particular, triagle iequality
More informationCommon Coupled Fixed Point of Mappings Satisfying Rational Inequalities in Ordered Complex Valued Generalized Metric Spaces
IOSR Joural of Mathematics (IOSR-JM) e-issn: 78-578, p-issn:319-765x Volume 10, Issue 3 Ver II (May-Ju 014), PP 69-77 Commo Coupled Fixed Poit of Mappigs Satisfyig Ratioal Iequalities i Ordered Complex
More informationEquivalent Banach Operator Ideal Norms 1
It. Joural of Math. Aalysis, Vol. 6, 2012, o. 1, 19-27 Equivalet Baach Operator Ideal Norms 1 Musudi Sammy Chuka Uiversity College P.O. Box 109-60400, Keya sammusudi@yahoo.com Shem Aywa Maside Muliro Uiversity
More informationON MEAN ERGODIC CONVERGENCE IN THE CALKIN ALGEBRAS
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 00, Number 0, Pages 000 000 S 0002-9939(XX0000-0 ON MEAN ERGODIC CONVERGENCE IN THE CALKIN ALGEBRAS MARCH T. BOEDIHARDJO AND WILLIAM B. JOHNSON 2
More informationSolutions to home assignments (sketches)
Matematiska Istitutioe Peter Kumli 26th May 2004 TMA401 Fuctioal Aalysis MAN670 Applied Fuctioal Aalysis 4th quarter 2003/2004 All documet cocerig the course ca be foud o the course home page: http://www.math.chalmers.se/math/grudutb/cth/tma401/
More informationThe Boolean Ring of Intervals
MATH 532 Lebesgue Measure Dr. Neal, WKU We ow shall apply the results obtaied about outer measure to the legth measure o the real lie. Throughout, our space X will be the set of real umbers R. Whe ecessary,
More informationf n (x) f m (x) < ɛ/3 for all x A. By continuity of f n and f m we can find δ > 0 such that d(x, x 0 ) < δ implies that
Lecture 15 We have see that a sequece of cotiuous fuctios which is uiformly coverget produces a limit fuctio which is also cotiuous. We shall stregthe this result ow. Theorem 1 Let f : X R or (C) be a
More informationMath 451: Euclidean and Non-Euclidean Geometry MWF 3pm, Gasson 204 Homework 3 Solutions
Math 451: Euclidea ad No-Euclidea Geometry MWF 3pm, Gasso 204 Homework 3 Solutios Exercises from 1.4 ad 1.5 of the otes: 4.3, 4.10, 4.12, 4.14, 4.15, 5.3, 5.4, 5.5 Exercise 4.3. Explai why Hp, q) = {x
More informationArkansas Tech University MATH 2924: Calculus II Dr. Marcel B. Finan
Arkasas Tech Uiversity MATH 94: Calculus II Dr Marcel B Fia 85 Power Series Let {a } =0 be a sequece of umbers The a power series about x = a is a series of the form a (x a) = a 0 + a (x a) + a (x a) +
More informationDANIELL AND RIEMANN INTEGRABILITY
DANIELL AND RIEMANN INTEGRABILITY ILEANA BUCUR We itroduce the otio of Riema itegrable fuctio with respect to a Daiell itegral ad prove the approximatio theorem of such fuctios by a mootoe sequece of Jorda
More informationON WELLPOSEDNESS QUADRATIC FUNCTION MINIMIZATION PROBLEM ON INTERSECTION OF TWO ELLIPSOIDS * M. JA]IMOVI], I. KRNI] 1.
Yugoslav Joural of Operatios Research 1 (00), Number 1, 49-60 ON WELLPOSEDNESS QUADRATIC FUNCTION MINIMIZATION PROBLEM ON INTERSECTION OF TWO ELLIPSOIDS M. JA]IMOVI], I. KRNI] Departmet of Mathematics
More informationSequence A sequence is a function whose domain of definition is the set of natural numbers.
Chapter Sequeces Course Title: Real Aalysis Course Code: MTH3 Course istructor: Dr Atiq ur Rehma Class: MSc-I Course URL: wwwmathcityorg/atiq/fa8-mth3 Sequeces form a importat compoet of Mathematical Aalysis
More informationThe Positivity of a Sequence of Numbers and the Riemann Hypothesis
joural of umber theory 65, 325333 (997) article o. NT97237 The Positivity of a Sequece of Numbers ad the Riema Hypothesis Xia-Ji Li The Uiversity of Texas at Austi, Austi, Texas 7872 Commuicated by A.
More informationNotes #3 Sequences Limit Theorems Monotone and Subsequences Bolzano-WeierstraßTheorem Limsup & Liminf of Sequences Cauchy Sequences and Completeness
Notes #3 Sequeces Limit Theorems Mootoe ad Subsequeces Bolzao-WeierstraßTheorem Limsup & Limif of Sequeces Cauchy Sequeces ad Completeess This sectio of otes focuses o some of the basics of sequeces of
More informationOn Involutions which Preserve Natural Filtration
Proceedigs of Istitute of Mathematics of NAS of Ukraie 00, Vol. 43, Part, 490 494 O Ivolutios which Preserve Natural Filtratio Alexader V. STRELETS Istitute of Mathematics of the NAS of Ukraie, 3 Tereshchekivska
More information1+x 1 + α+x. x = 2(α x2 ) 1+x
Math 2030 Homework 6 Solutios # [Problem 5] For coveiece we let α lim sup a ad β lim sup b. Without loss of geerality let us assume that α β. If α the by assumptio β < so i this case α + β. By Theorem
More informationGlivenko-Cantelli Classes
CS28B/Stat24B (Sprig 2008 Statistical Learig Theory Lecture: 4 Gliveko-Catelli Classes Lecturer: Peter Bartlett Scribe: Michelle Besi Itroductio This lecture will cover Gliveko-Catelli (GC classes ad itroduce
More informationFall 2013 MTH431/531 Real analysis Section Notes
Fall 013 MTH431/531 Real aalysis Sectio 8.1-8. Notes Yi Su 013.11.1 1. Defiitio of uiform covergece. We look at a sequece of fuctios f (x) ad study the coverget property. Notice we have two parameters
More informationA LIMITED ARITHMETIC ON SIMPLE CONTINUED FRACTIONS - II 1. INTRODUCTION
A LIMITED ARITHMETIC ON SIMPLE CONTINUED FRACTIONS - II C. T. LONG J. H. JORDAN* Washigto State Uiversity, Pullma, Washigto 1. INTRODUCTION I the first paper [2 ] i this series, we developed certai properties
More informationINFINITE SEQUENCES AND SERIES
11 INFINITE SEQUENCES AND SERIES INFINITE SEQUENCES AND SERIES 11.4 The Compariso Tests I this sectio, we will lear: How to fid the value of a series by comparig it with a kow series. COMPARISON TESTS
More informationBoundaries and the James theorem
Boudaries ad the James theorem L. Vesely 1. Itroductio The followig theorem is importat ad well kow. All spaces cosidered here are real ormed or Baach spaces. Give a ormed space X, we deote by B X ad S
More informationMATH4822E FOURIER ANALYSIS AND ITS APPLICATIONS
MATH48E FOURIER ANALYSIS AND ITS APPLICATIONS 7.. Cesàro summability. 7. Summability methods Arithmetic meas. The followig idea is due to the Italia geometer Eresto Cesàro (859-96). He shows that eve if
More informationMath 210A Homework 1
Math 0A Homework Edward Burkard Exercise. a) State the defiitio of a aalytic fuctio. b) What are the relatioships betwee aalytic fuctios ad the Cauchy-Riema equatios? Solutio. a) A fuctio f : G C is called
More informationd) If the sequence of partial sums converges to a limit L, we say that the series converges and its
Ifiite Series. Defiitios & covergece Defiitio... Let {a } be a sequece of real umbers. a) A expressio of the form a + a +... + a +... is called a ifiite series. b) The umber a is called as the th term
More informationMath 525: Lecture 5. January 18, 2018
Math 525: Lecture 5 Jauary 18, 2018 1 Series (review) Defiitio 1.1. A sequece (a ) R coverges to a poit L R (writte a L or lim a = L) if for each ǫ > 0, we ca fid N such that a L < ǫ for all N. If the
More information} is said to be a Cauchy sequence provided the following condition is true.
Math 4200, Fial Exam Review I. Itroductio to Proofs 1. Prove the Pythagorea theorem. 2. Show that 43 is a irratioal umber. II. Itroductio to Logic 1. Costruct a truth table for the statemet ( p ad ~ r
More informationRiesz-Fischer Sequences and Lower Frame Bounds
Zeitschrift für Aalysis ud ihre Aweduge Joural for Aalysis ad its Applicatios Volume 1 (00), No., 305 314 Riesz-Fischer Sequeces ad Lower Frame Bouds P. Casazza, O. Christese, S. Li ad A. Lider Abstract.
More informationThe log-behavior of n p(n) and n p(n)/n
Ramauja J. 44 017, 81-99 The log-behavior of p ad p/ William Y.C. Che 1 ad Ke Y. Zheg 1 Ceter for Applied Mathematics Tiaji Uiversity Tiaji 0007, P. R. Chia Ceter for Combiatorics, LPMC Nakai Uivercity
More informationChapter IV Integration Theory
Chapter IV Itegratio Theory Lectures 32-33 1. Costructio of the itegral I this sectio we costruct the abstract itegral. As a matter of termiology, we defie a measure space as beig a triple (, A, µ), where
More informationIntroduction to Functional Analysis
MIT OpeCourseWare http://ocw.mit.edu 18.10 Itroductio to Fuctioal Aalysis Sprig 009 For iformatio about citig these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. LECTURE OTES FOR 18.10,
More informationlim za n n = z lim a n n.
Lecture 6 Sequeces ad Series Defiitio 1 By a sequece i a set A, we mea a mappig f : N A. It is customary to deote a sequece f by {s } where, s := f(). A sequece {z } of (complex) umbers is said to be coverget
More informationMath 341 Lecture #31 6.5: Power Series
Math 341 Lecture #31 6.5: Power Series We ow tur our attetio to a particular kid of series of fuctios, amely, power series, f(x = a x = a 0 + a 1 x + a 2 x 2 + where a R for all N. I terms of a series
More informationPAPER : IIT-JAM 2010
MATHEMATICS-MA (CODE A) Q.-Q.5: Oly oe optio is correct for each questio. Each questio carries (+6) marks for correct aswer ad ( ) marks for icorrect aswer.. Which of the followig coditios does NOT esure
More informationn=1 a n is the sequence (s n ) n 1 n=1 a n converges to s. We write a n = s, n=1 n=1 a n
Series. Defiitios ad first properties A series is a ifiite sum a + a + a +..., deoted i short by a. The sequece of partial sums of the series a is the sequece s ) defied by s = a k = a +... + a,. k= Defiitio
More informationA Fixed Point Result Using a Function of 5-Variables
Joural of Physical Scieces, Vol., 2007, 57-6 Fixed Poit Result Usig a Fuctio of 5-Variables P. N. Dutta ad Biayak S. Choudhury Departmet of Mathematics Begal Egieerig ad Sciece Uiversity, Shibpur P.O.:
More informationCARLEMAN INTEGRAL OPERATORS AS MULTIPLICATION OPERATORS AND PERTURBATION THEORY
Kragujevac Joural of Mathematics Volume 41(1) (2017), Pages 71 80. CARLEMAN INTEGRAL OPERATORS AS MULTIPLICATION OPERATORS AND PERTURBATION THEORY S. M. BAHRI 1 Abstract. I this paper we itroduce a multiplicatio
More informationReal Numbers R ) - LUB(B) may or may not belong to B. (Ex; B= { y: y = 1 x, - Note that A B LUB( A) LUB( B)
Real Numbers The least upper boud - Let B be ay subset of R B is bouded above if there is a k R such that x k for all x B - A real umber, k R is a uique least upper boud of B, ie k = LUB(B), if () k is
More informationMATH 112: HOMEWORK 6 SOLUTIONS. Problem 1: Rudin, Chapter 3, Problem s k < s k < 2 + s k+1
MATH 2: HOMEWORK 6 SOLUTIONS CA PRO JIRADILOK Problem. If s = 2, ad Problem : Rudi, Chapter 3, Problem 3. s + = 2 + s ( =, 2, 3,... ), prove that {s } coverges, ad that s < 2 for =, 2, 3,.... Proof. The
More informationSome vector-valued statistical convergent sequence spaces
Malaya J. Mat. 32)205) 6 67 Some vector-valued statistical coverget sequece spaces Kuldip Raj a, ad Suruchi Padoh b a School of Mathematics, Shri Mata Vaisho Devi Uiversity, Katra-82320, J&K, Idia. b School
More informationLecture 10: Bounded Linear Operators and Orthogonality in Hilbert Spaces
Lecture : Bouded Liear Operators ad Orthogoality i Hilbert Spaces 34 Bouded Liear Operator Let ( X, ), ( Y, ) i i be ored liear vector spaces ad { } X Y The, T is said to be bouded if a real uber c such
More informationThe value of Banach limits on a certain sequence of all rational numbers in the interval (0,1) Bao Qi Feng
The value of Baach limits o a certai sequece of all ratioal umbers i the iterval 0, Bao Qi Feg Departmet of Mathematical Scieces, Ket State Uiversity, Tuscarawas, 330 Uiversity Dr. NE, New Philadelphia,
More informationDirichlet s Theorem on Arithmetic Progressions
Dirichlet s Theorem o Arithmetic Progressios Athoy Várilly Harvard Uiversity, Cambridge, MA 0238 Itroductio Dirichlet s theorem o arithmetic progressios is a gem of umber theory. A great part of its beauty
More informationFinal Solutions. 1. (25pts) Define the following terms. Be as precise as you can.
Mathematics H104 A. Ogus Fall, 004 Fial Solutios 1. (5ts) Defie the followig terms. Be as recise as you ca. (a) (3ts) A ucoutable set. A ucoutable set is a set which ca ot be ut ito bijectio with a fiite
More informationSOME SEQUENCE SPACES DEFINED BY ORLICZ FUNCTIONS
ARCHIVU ATHEATICU BRNO Tomus 40 2004, 33 40 SOE SEQUENCE SPACES DEFINED BY ORLICZ FUNCTIONS E. SAVAŞ AND R. SAVAŞ Abstract. I this paper we itroduce a ew cocept of λ-strog covergece with respect to a Orlicz
More informationMAT1026 Calculus II Basic Convergence Tests for Series
MAT026 Calculus II Basic Covergece Tests for Series Egi MERMUT 202.03.08 Dokuz Eylül Uiversity Faculty of Sciece Departmet of Mathematics İzmir/TURKEY Cotets Mootoe Covergece Theorem 2 2 Series of Real
More informationCommutativity in Permutation Groups
Commutativity i Permutatio Groups Richard Wito, PhD Abstract I the group Sym(S) of permutatios o a oempty set S, fixed poits ad trasiet poits are defied Prelimiary results o fixed ad trasiet poits are
More informationMASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.265/15.070J Fall 2013 Lecture 3 9/11/2013. Large deviations Theory. Cramér s Theorem
MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.265/5.070J Fall 203 Lecture 3 9//203 Large deviatios Theory. Cramér s Theorem Cotet.. Cramér s Theorem. 2. Rate fuctio ad properties. 3. Chage of measure techique.
More informationb i u x i U a i j u x i u x j
M ath 5 2 7 Fall 2 0 0 9 L ecture 1 9 N ov. 1 6, 2 0 0 9 ) S ecod- Order Elliptic Equatios: Weak S olutios 1. Defiitios. I this ad the followig two lectures we will study the boudary value problem Here
More informationCouncil for Innovative Research
ABSTRACT ON ABEL CONVERGENT SERIES OF FUNCTIONS ERDAL GÜL AND MEHMET ALBAYRAK Yildiz Techical Uiversity, Departmet of Mathematics, 34210 Eseler, Istabul egul34@gmail.com mehmetalbayrak12@gmail.com I this
More informationn p (Ω). This means that the
Sobolev s Iequality, Poicaré Iequality ad Compactess I. Sobolev iequality ad Sobolev Embeddig Theorems Theorem (Sobolev s embeddig theorem). Give the bouded, ope set R with 3 ad p
More informationTopics. Homework Problems. MATH 301 Introduction to Analysis Chapter Four Sequences. 1. Definition of convergence of sequences.
MATH 301 Itroductio to Aalysis Chapter Four Sequeces Topics 1. Defiitio of covergece of sequeces. 2. Fidig ad provig the limit of sequeces. 3. Bouded covergece theorem: Theorem 4.1.8. 4. Theorems 4.1.13
More informationMath 132, Fall 2009 Exam 2: Solutions
Math 3, Fall 009 Exam : Solutios () a) ( poits) Determie for which positive real umbers p, is the followig improper itegral coverget, ad for which it is diverget. Evaluate the itegral for each value of
More informationA REFINEMENT OF JENSEN S INEQUALITY WITH APPLICATIONS. S. S. Dragomir 1. INTRODUCTION
TAIWANESE JOURNAL OF MATHEMATICS Vol. 14, No. 1,. 153-164, February 2010 This aer is available olie at htt://www.tjm.sysu.edu.tw/ A REFINEMENT OF JENSEN S INEQUALITY WITH APPLICATIONS FOR f-divergence
More information2.4.2 A Theorem About Absolutely Convergent Series
0 Versio of August 27, 200 CHAPTER 2. INFINITE SERIES Add these two series: + 3 2 + 5 + 7 4 + 9 + 6 +... = 3 l 2. (2.20) 2 Sice the reciprocal of each iteger occurs exactly oce i the last series, we would
More informationRead carefully the instructions on the answer book and make sure that the particulars required are entered on each answer book.
THE UNIVERSITY OF WARWICK FIRST YEAR EXAMINATION: Jauary 2009 Aalysis I Time Allowed:.5 hours Read carefully the istructios o the aswer book ad make sure that the particulars required are etered o each
More informationNotes on Snell Envelops and Examples
Notes o Sell Evelops ad Examples Example (Secretary Problem): Coside a pool of N cadidates whose qualificatios are represeted by ukow umbers {a > a 2 > > a N } from best to last. They are iterviewed sequetially
More informationIf a subset E of R contains no open interval, is it of zero measure? For instance, is the set of irrationals in [0, 1] is of measure zero?
2 Lebesgue Measure I Chapter 1 we defied the cocept of a set of measure zero, ad we have observed that every coutable set is of measure zero. Here are some atural questios: If a subset E of R cotais a
More informationA General Iterative Scheme for Variational Inequality Problems and Fixed Point Problems
A Geeral Iterative Scheme for Variatioal Iequality Problems ad Fixed Poit Problems Wicha Khogtham Abstract We itroduce a geeral iterative scheme for fidig a commo of the set solutios of variatioal iequality
More informationA REMARK ON A PROBLEM OF KLEE
C O L L O Q U I U M M A T H E M A T I C U M VOL. 71 1996 NO. 1 A REMARK ON A PROBLEM OF KLEE BY N. J. K A L T O N (COLUMBIA, MISSOURI) AND N. T. P E C K (URBANA, ILLINOIS) This paper treats a property
More informationAssignment 5: Solutions
McGill Uiversity Departmet of Mathematics ad Statistics MATH 54 Aalysis, Fall 05 Assigmet 5: Solutios. Let y be a ubouded sequece of positive umbers satisfyig y + > y for all N. Let x be aother sequece
More informationMath 116 Practice for Exam 3
Math 6 Practice for Exam Geerated October 0, 207 Name: SOLUTIONS Istructor: Sectio Number:. This exam has 7 questios. Note that the problems are ot of equal difficulty, so you may wat to skip over ad retur
More information1.3 Convergence Theorems of Fourier Series. k k k k. N N k 1. With this in mind, we state (without proof) the convergence of Fourier series.
.3 Covergece Theorems of Fourier Series I this sectio, we preset the covergece of Fourier series. A ifiite sum is, by defiitio, a limit of partial sums, that is, a cos( kx) b si( kx) lim a cos( kx) b si(
More informationAnalytic Continuation
Aalytic Cotiuatio The stadard example of this is give by Example Let h (z) = 1 + z + z 2 + z 3 +... kow to coverge oly for z < 1. I fact h (z) = 1/ (1 z) for such z. Yet H (z) = 1/ (1 z) is defied for
More informationAn alternative proof of a theorem of Aldous. concerning convergence in distribution for martingales.
A alterative proof of a theorem of Aldous cocerig covergece i distributio for martigales. Maurizio Pratelli Dipartimeto di Matematica, Uiversità di Pisa. Via Buoarroti 2. I-56127 Pisa, Italy e-mail: pratelli@dm.uipi.it
More informationSingular Continuous Measures by Michael Pejic 5/14/10
Sigular Cotiuous Measures by Michael Peic 5/4/0 Prelimiaries Give a set X, a σ-algebra o X is a collectio of subsets of X that cotais X ad ad is closed uder complemetatio ad coutable uios hece, coutable
More information