REGULARIZATION OF CERTAIN DIVERGENT SERIES OF POLYNOMIALS
|
|
- Russell Cox
- 5 years ago
- Views:
Transcription
1 REGULARIZATION OF CERTAIN DIVERGENT SERIES OF POLYNOMIALS LIVIU I. NICOLAESCU ABSTRACT. We ivestigate the geeralized covergece ad sums of series of the form P at P (x, where P R[x], a R,, ad T : R[x] R[x] is a liear operator that commutes with the differetiatio d : R[x] R[x]. dx CONTENTS. The mai result. Some applicatios 3 Refereces 9 We cosider series of the form. THE MAIN RESULT a T P (x, ( where P R[x], ad T : R[x] R[x] is a liear operator such that T D = DT, where D is the differetiatio operator D = d dx. The coditio ( is equivalet with the traslatio ivariace of T, i.e., T U h = U h T, h R, (I where U h : R[x] R[x] is the traslatio operator R[x] p(x p(x + h R[x]. For simplicity we set U := U. Clearly U h O so a special case of the series ( is the series a P (x + h, h R, ( h a U h P (x = which is typically diverget. We deote by O the R-algebra of traslatio ivariat operators. We have a atural map Q : R[[t]] O, R[[t] t c! c! D. It is ow (see [, Prop. 3.47] that this map is a isomorphism of rigs. We deote by σ the iverse of Q σ : O R[[t]], O T σ T R[[t]]. Mathematics Subject Classificatio. 5A, 5A9, B83, 4A3, 4G5, 4G. Key words ad phrases. traslatio ivariat operators, diverget series, summability. Last modified o July 9, 9. (
2 LIVIU I. NICOLAESCU For T O we will refer to the formal power series σ T as the symbol of the operator T. More explicitely σ T (t = c (T t, c (T = (T x x= R.! We deote by N the set of oegative itegers, ad by Seq the vector space of real sequeces, i.e., maps a : N R. Let Seq c the vector subspace of Seq cosistig of all coverget sequeces. A geeralized otio of covergece or regularizatio method is a pair µ = ( µ lim, Seq µ, where Seq µ is a vector subspace of Seq cotaiig Seq c ad, µ lim is a liear map µ lim : Seq µ R, Seq µ a µ lim a( R such that for ay a Seq c we have µ lim a = lim a(. The sequeces i Seq µ are called µ-coverget ad µ lim is called the µ-limit. To ay sequece a Seq we associate the sequece S[a] of partial sums S[a]( = σ= a(. (. A series a( is said to by µ-coverget if the sequece S[a] is µ-coverget. We set µ a( := µ lim S[a](. We say that µ a( is the µ-sum of the series. The regularizatio method is said to be shift ivariat if it satisfies the additioal coditio µ a( = a( + µ a(. (. We refer to the classic [3] for a large collectio of regularizatio methods. For x R ad N we set { ( i= (x i, x [x] :=, =,, := [x]!. We ca ow state the mai result of this paper. Theorem.. Let µ be a regularizatio method, T O ad f(t = a t R[[t]]. Set c := c (T = T. Suppose that f is µ-regular at t = c, i.e., We deote by f ( (c µ its µ-sum for every N the series a [] c is µ-coverget. (µ f ( (c µ := µ a [] c. The for every P R[x] the series a (T P (x is µ-coverget ad its µ-sum is µ a (T P (x = f(t µ P (x, Hardy refers to such a otio of covergece as covergece i some Picwicia sese.
3 where f(t µ O is the operator REGULARIZATION OF CERTAIN DIVERGENT SERIES OF POLYNOMIALS 3 f(t µ := f ( (c µ (T c. (.3! Proof. Set R := T c ad let P R[x]. The so that R = c (T D! I particular this shows that f(t µ is well defied. We have a T P = a (c + R P = a R P =, > deg P. (.4 = ( deg P c R P = = At the last step we used (.4 ad the fact that ( =, if >. ( c R P. This shows that the formal series a (T P (x ca be writte as a fiite liear combiatio of formal series deg P a (T R P (x P (x = a [] c.! From the liearity of the µ-summatio operator we deduce deg P a (T R P (x P (x = µ a [] c! µ = ( deg P = = = f ( (c µ R P (x = f(t µ P (x!. SOME APPLICATIONS To describe some cosequeces of Theorem. we eed to first describe some classical facts about regularizatio methods. For ay sequece a Seq we deote by G a (t R[[t]] its geeratig series. We regard the partial sum costructio S i (. as a liear operator S : Seq Seq. Observe that G S[a] (t = t G a(t. We say that a regularizatio method µ = ( µ lim, Seq µ is stroger tha the regularizatio method µ = ( µ lim, Seq µ, ad we write this µ µ, if Seq µ Seq µ ad µ lim a( = µ lim a(, a Seq µ.
4 4 LIVIU I. NICOLAESCU The Abel regularizatio method A is defied as follows. We say that a sequece a is A coverget if the radius of covergece of the series a t is at least ad the fuctio t ( t a t has a fiite limit as t. Hece A lim a( = lim t ( t a t, ad Seq A cosists of sequece for which the above limit exists ad it is fiite. Usig ( we deduce that a series a( is A-coverget if ad oly if the limit lim a t t exists ad it is fiite. We have the followig immediate result. Propositio.. Suppose that f(z is a holomorphic fuctio defied i a ope eighborhood of the set {} { z } C. If a z is the Taylor series expasio of f at z = the the correspodig formal power series [f] = a t is A-regular at t =, [f] ( ( A = f (, ad the series [f](r A = [f] ( ( A r! coicides with the Taylor expasio of f at z =, ad it coverges to f( + r. Corollary.. Suppose that f(z is a holomorphic fuctio defied i a ope eighborhood of the set {} { z } C ad a z is the Taylor series expasio of f at z =. The for every T i O such that c (T =, ay P R[x], ad ay x R we have A a T P (x = f ( (T P (x.! Let N. A sequece a Seq is said to be C -coverget (or Cesàro coverget of order if the limit S [a]( lim ( + exists ad it is fiite. We deote this limit by C lim a(. A series a( is said to be C - coverget if the sequece of partial sums S[a] is C coverget. Thus the C -sum of this series is C More explicitly, we have (see [3, Eq.(5.4.5] C a( = lim a( = lim ( + S + [a](. ( + ( ( ν + ν= a( ν This was apparetly ow ad used by Euler.
5 Hece REGULARIZATION OF CERTAIN DIVERGENT SERIES OF POLYNOMIALS 5 C ( + a( S + [a]( A A!, where a( a b lim b( =, if a(, b(, for. The C covergece is equivalet with the classical covergece ad it is ow (see [3, Thm. 43, 55] that C C A, <. Give this fact, we defie a sequece to be C-coverget (Cesàro coverget if it is C -coverget for some N. Note that C A. Both the C ad A methods are shift ivariat, i.e., they satisfy the coditio (.. We wat to commet a bit about possible methods of establishig C-covergece. To formulate a geeral strategy we eed to itroduce a classical otatio. More precisely, if f(t = a t is a formal power series we let [t ]f(t deote the coefficiet of t i this power series, i.e. [t ]f(t = a. Let f(t = a t. The the series a t C-coverges to A if ad oly if there exists a oegative real umber α such that [t ] ( ( t (α+ f(t A α Γ(α +, where Γ is Euler s Gamma fuctio. For a proof we refer to [3, Thm. 43]. Defiitio.3. We say that a power series f(t = a t is Cesàro coveiet (or C-coveiet at if the followig hold. (i The radius of covergeces of the series is (ii The fuctio f is regular at z = ad has fiitely may sigularities ζ,..., ζ ν o the uit circle { z = }. (iii There exist ε > ad θ (, π such that f admits a cotiuatio to the dimpled dis { } z ε,θ := z C; z < + ε, arg( > θ, j =,..., ν. ζ j (iv For every sigular poit ζ j there exists a positive iteger m j such that f(z = O ( (z ζ j m j as z ζ j, z. The results i [, Chap. VI] implies that the collectio R C of C-coveiet power series is a rig satisfyig f R C df dt R C. Ivoig [, Thm VI.5] we deduce the followig useful cosequece. Corollary.4. Let f R[[t]] be a power series C-coveiet at. The f is C-regular at ad f ( C = f ( ( A = f ( (. Usig [, VII.7] we obtai the followig useful result.
6 6 LIVIU I. NICOLAESCU Corollary.5. (a The power series ( + t m = ( + m ( t, m, log( + t = ( + t are C-regular at. (b If f(z is a algebraic fuctio defied o the uit dis z < ad regular at z = the the Taylor series of f at z = is C-regular at. Recall that the Cauchy product of two sequeces a, b Seq is the sequece a b, a b( = a( ib(i, N. i= A regularizatio method is said to be multiplicative if ( µ a b( = µ for ay µ-coverget series a( ad b(. The results of [3, Chap.X] show that the C ad A methods are multiplicative. For ay regularizatio method µ ad c R we deote by R[[t]] µ the set of series that are µ-regular at t =. Propositio.6. Let µ be a multiplicative regularizatio method. The R[[t]] µ is a commutative rig with oe ad we have the product rule ( (f g ( ( µ = f ( ( µ g ( ( µ. = a( Moreover, if T O is such that c (T = the the map is a rig morphism. ( µ R[[t]] µ f f(t µ O b( Proof. The product formula follows from the iterated applicatio of the equalities D t (fg = (D t fg + f(d t g, (fg( µ = f( µ g( µ, f ( µ = (D t f( µ, where D t : R[[t]] R[[t]] is the formal differetiatio operator d dt. The last statemet is a immediate applicatio of the above product rule. Remar.7. The iclusio R[[t]] C R[[t]] A is strict. For example, the power series f(z = e /(+z satisfies the assumptio of Propositio. so that the associated formal power series [f] is A-regular at. O the other had, the argumets i [3, 5.] show that [f] is ot C-regular at. Cosider the traslatio operator U h O. From Taylor s formula p(x + h = h! D p(x we deduce that σ U h(t = e th. Set h := U h. Usig Corollary.5 ad Theorem. we deduce the followig result.,
7 REGULARIZATION OF CERTAIN DIVERGENT SERIES OF POLYNOMIALS 7 Corollary.8. For ay P R[x] we have C ( P (x + h = ( h P (x. (. Observe that ( + ( h h = so that ( h is the iverse of the operator + h. We thus have C ( P (x + h = ( + h P (x = ( + U h P (x. (. Remar.9. Here is a heuristic explaatio of the equality (. assumig the Cesàro covergece of the series ( P (x + h. Deote by S(x the Cesàro sum of this series. The S(x + h = C ( P ( x + ( + h Hece (. = C ( P (x + h + P (x = S(x + P (x. S(x + h + S(x = P (x, x R. If we ew that S(x is a polyomial we would the deduce S(x = ( + U h P (x. The iverse of + U h ca be explicitly expressed usig Euler umbers ad polyomials, [4, Eq. (4, p.34]. The Euler umbers E are defied by the Taylor expasio cosh t = e t + e t = E! t. Sice cosh t is a eve fuctio we deduce that E = for odd. They satisfy the recurrece relatio ( ( E + E + E 4 + =,. (.3 4 Here are the first few Euler umbers The Hece E 5 6, 385 5, 5, 7, , 36, 98 9, 39, 5, 45 h + U h = U U h + U h = U e D + e D C ( P (x + h = = U h cosh hd = U h E h! D. E h (! P ( x h. (.4
8 8 LIVIU I. NICOLAESCU Whe P (x = x m, h =, we have C ( (x + m = ( ( m E x m. (.5 Settig x = ad usig the equality E j+ =, j we coclude that C ( m = ( m m+ ( m E = ( m ( m m+ E. (.6 Usig (.3 we deduce that whe m is eve, m = m, m > we have C ( m =. (.7 For example + + C=, ( C= 4, ( C= 8, ( C= 4. ( 5 Whe P (x = ( x m, x =, h = the it is more coveiet to use (. because ( ( x x =,, x. We deduce ( C ( = m m ( ( = ( m. (.8 m m+ = Example.. Cosider the traslatio ivariat operator T : R[x] R[x], P (x e s P (x + sdx. Set R = T. As explaied i [, II.3.B], the operators T ad R are itimately related to the Laguerre polyomials. We have R = DT = T D ad 3 σ T (t = t σ R(t = t = t t. If P R[x] is a polyomial of degree m the T P (x x= = ( + D + + D m P (x x= = e (s +s + +s P (s + + s ds ds. R For t we deote by (t the ( simplex (t := { (s,..., s R ; s + + s = t }., 3 We ca write formally T = R e s U s ds = R e s( D ds = ( + D, so that σ T (t = t.
9 REGULARIZATION OF CERTAIN DIVERGENT SERIES OF POLYNOMIALS 9 ad by dv (t the Euclidea volume elemet o (t. Itegratig alog the fibers of the fuctio f : R [,, f(s,..., s = s + + s we deduce ( e (s +s + +s P (s + + s ds ds = f dv (t e t P (tdt R = v e s s P (sds, (t where v is the ( -dimesioal volume of the ( -simplex = (t t=. To compute the volume v we view is a regular -simplex with distiguished base, ad distiguished vertex (,...,, R +. The distace d from the vertex to the base is the distace from the vertex to the ceter of the base. We have d = + +, d =, v = ( + / d v = 3 v. Sice v = we deduce v = ( + /, T P (x x= =! (! e s s P (sds, ad R P (x x= = e s s P ( (sds. (! Usig Theorem. ad Corollary.4 with the C-coveiet series f(t = ( + t we deduce C ( T P (x x= = C ( e s s P (sds (! ( deg P ( = + (! s P ( (s ds. If we let P (s = s m we deduce ad = e s s P (sds = (m +!, ( C m + ( = m e s s P ( (sds = [m] (m! = [m ] m!, m = ( + ( m. (.9 Let us poit out that (.9 ca be obtaied from (.8 usig the shift-ivariace of the Cesàro regularizatio method. REFERENCES [] M. Aiger: Combiatorial Theory, Spriger Verlag, 997. [] P. Flajolet, R. Sedgewic: Aalytic Combiatorics, Cambridge Uiversity Press, 9. [3] G.H. Hardy: Diverget Series, Chelsea Publishig Co. 99. [4] N.E. Nörlud: Mémoire sur les polyomes de Beroulli, Acta Math. 43(9, -94. DEPARTMENT OF MATHEMATICS, UNIVERSITY OF NOTRE DAME, NOTRE DAME, IN address: icolaescu.@d.edu URL: licolae/
62. 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 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 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 informationMath 155 (Lecture 3)
Math 55 (Lecture 3) September 8, I this lecture, we ll cosider the aswer to oe of the most basic coutig problems i combiatorics Questio How may ways are there to choose a -elemet subset of the set {,,,
More informationABOUT CHAOS AND SENSITIVITY IN TOPOLOGICAL DYNAMICS
ABOUT CHAOS AND SENSITIVITY IN TOPOLOGICAL DYNAMICS EDUARD KONTOROVICH Abstract. I this work we uify ad geeralize some results about chaos ad sesitivity. Date: March 1, 005. 1 1. Symbolic Dyamics Defiitio
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 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 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 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 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 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 informationThe Gamma function Michael Taylor. Abstract. This material is excerpted from 18 and Appendix J of [T].
The Gamma fuctio Michael Taylor Abstract. This material is excerpted from 8 ad Appedix J of [T]. The Gamma fuctio has bee previewed i 5.7 5.8, arisig i the computatio of a atural Laplace trasform: 8. ft
More informationCHAPTER 1 SEQUENCES AND INFINITE SERIES
CHAPTER SEQUENCES AND INFINITE SERIES SEQUENCES AND INFINITE SERIES (0 meetigs) Sequeces ad limit of a sequece Mootoic ad bouded sequece Ifiite series of costat terms Ifiite series of positive terms Alteratig
More informationMetric Space Properties
Metric Space Properties Math 40 Fial Project Preseted by: Michael Brow, Alex Cordova, ad Alyssa Sachez We have already poited out ad will recogize throughout this book the importace of compact sets. All
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 informationWe are mainly going to be concerned with power series in x, such as. (x)} converges - that is, lims N n
Review of Power Series, Power Series Solutios A power series i x - a is a ifiite series of the form c (x a) =c +c (x a)+(x a) +... We also call this a power series cetered at a. Ex. (x+) is cetered at
More informationMAS111 Convergence and Continuity
MAS Covergece ad Cotiuity Key Objectives At the ed of the course, studets should kow the followig topics ad be able to apply the basic priciples ad theorems therei to solvig various problems cocerig covergece
More informationSome p-adic congruences for p q -Catalan numbers
Some p-adic cogrueces for p q -Catala umbers Floria Luca Istituto de Matemáticas Uiversidad Nacioal Autóoma de México C.P. 58089, Morelia, Michoacá, México fluca@matmor.uam.mx Paul Thomas Youg Departmet
More information3. Z Transform. Recall that the Fourier transform (FT) of a DT signal xn [ ] is ( ) [ ] = In order for the FT to exist in the finite magnitude sense,
3. Z Trasform Referece: Etire Chapter 3 of text. Recall that the Fourier trasform (FT) of a DT sigal x [ ] is ω ( ) [ ] X e = j jω k = xe I order for the FT to exist i the fiite magitude sese, S = x [
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 information7.1 Convergence of sequences of random variables
Chapter 7 Limit Theorems Throughout this sectio we will assume a probability space (, F, P), i which is defied a ifiite sequece of radom variables (X ) ad a radom variable X. The fact that for every ifiite
More informationCourse : Algebraic Combinatorics
Course 8.32: Algebraic Combiatorics Lecture Notes # Addedum by Gregg Musier February 4th - 6th, 2009 Recurrece Relatios ad Geeratig Fuctios Give a ifiite sequece of umbers, a geeratig fuctio is a compact
More informationSolutions to Homework 1
Solutios to Homework MATH 36. Describe geometrically the sets of poits z i the complex plae defied by the followig relatios /z = z () Re(az + b) >, where a, b (2) Im(z) = c, with c (3) () = = z z = z 2.
More informationHarmonic Number Identities Via Euler s Transform
1 2 3 47 6 23 11 Joural of Iteger Sequeces, Vol. 12 2009), Article 09.6.1 Harmoic Number Idetities Via Euler s Trasform Khristo N. Boyadzhiev Departmet of Mathematics Ohio Norther Uiversity Ada, Ohio 45810
More informationON THE EXTENDED AND ALLAN SPECTRA AND TOPOLOGICAL RADII. Hugo Arizmendi-Peimbert, Angel Carrillo-Hoyo, and Jairo Roa-Fajardo
Opuscula Mathematica Vol. 32 No. 2 2012 http://dx.doi.org/10.7494/opmath.2012.32.2.227 ON THE EXTENDED AND ALLAN SPECTRA AND TOPOLOGICAL RADII Hugo Arizmedi-Peimbert, Agel Carrillo-Hoyo, ad Jairo Roa-Fajardo
More informationLECTURE SERIES WITH NONNEGATIVE TERMS (II). SERIES WITH ARBITRARY TERMS
LECTURE 4 SERIES WITH NONNEGATIVE TERMS II). SERIES WITH ARBITRARY TERMS Series with oegative terms II) Theorem 4.1 Kummer s Test) Let x be a series with positive terms. 1 If c ) N i 0, + ), r > 0 ad 0
More informationA solid Foundation for q-appell Polynomials
Advaces i Dyamical Systems ad Applicatios ISSN 0973-5321, Volume 10, Number 1, pp. 27 35 2015) http://campus.mst.edu/adsa A solid Foudatio for -Appell Polyomials Thomas Erst Uppsala Uiversity Departmet
More informationInfinite Sequences and Series
Chapter 6 Ifiite Sequeces ad Series 6.1 Ifiite Sequeces 6.1.1 Elemetary Cocepts Simply speakig, a sequece is a ordered list of umbers writte: {a 1, a 2, a 3,...a, a +1,...} where the elemets a i represet
More informationPLEASE MARK YOUR ANSWERS WITH AN X, not a circle! 1. (a) (b) (c) (d) (e) 3. (a) (b) (c) (d) (e) 5. (a) (b) (c) (d) (e) 7. (a) (b) (c) (d) (e)
Math 0560, Exam 3 November 6, 07 The Hoor Code is i effect for this examiatio. All work is to be your ow. No calculators. The exam lasts for hour ad 5 mi. Be sure that your ame is o every page i case pages
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 informationSUMMARY OF SEQUENCES AND SERIES
SUMMARY OF SEQUENCES AND SERIES Importat Defiitios, Results ad Theorems for Sequeces ad Series Defiitio. A sequece {a } has a limit L ad we write lim a = L if for every ɛ > 0, there is a correspodig iteger
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 informationBenaissa Bernoussi Université Abdelmalek Essaadi, ENSAT de Tanger, B.P. 416, Tanger, Morocco
EXTENDING THE BERNOULLI-EULER METHOD FOR FINDING ZEROS OF HOLOMORPHIC FUNCTIONS Beaissa Beroussi Uiversité Abdelmalek Essaadi, ENSAT de Tager, B.P. 416, Tager, Morocco e-mail: Beaissa@fstt.ac.ma Mustapha
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 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 informationMASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.436J/15.085J Fall 2008 Lecture 19 11/17/2008 LAWS OF LARGE NUMBERS II THE STRONG LAW OF LARGE NUMBERS
MASSACHUSTTS INSTITUT OF TCHNOLOGY 6.436J/5.085J Fall 2008 Lecture 9 /7/2008 LAWS OF LARG NUMBRS II Cotets. The strog law of large umbers 2. The Cheroff boud TH STRONG LAW OF LARG NUMBRS While the weak
More informationChapter 8. Uniform Convergence and Differentiation.
Chapter 8 Uiform Covergece ad Differetiatio This chapter cotiues the study of the cosequece of uiform covergece of a series of fuctio I Chapter 7 we have observed that the uiform limit of a sequece of
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 informationAdditional Notes on Power Series
Additioal Notes o Power Series Mauela Girotti MATH 37-0 Advaced Calculus of oe variable Cotets Quick recall 2 Abel s Theorem 2 3 Differetiatio ad Itegratio of Power series 4 Quick recall We recall here
More informationA NOTE ON INVARIANT SETS OF ITERATED FUNCTION SYSTEMS
Acta Math. Hugar., 2007 DOI: 10.1007/s10474-007-7013-6 A NOTE ON INVARIANT SETS OF ITERATED FUNCTION SYSTEMS L. L. STACHÓ ad L. I. SZABÓ Bolyai Istitute, Uiversity of Szeged, Aradi vértaúk tere 1, H-6720
More informationON THE LEHMER CONSTANT OF FINITE CYCLIC GROUPS
ON THE LEHMER CONSTANT OF FINITE CYCLIC GROUPS NORBERT KAIBLINGER Abstract. Results of Lid o Lehmer s problem iclude the value of the Lehmer costat of the fiite cyclic group Z/Z, for 5 ad all odd. By complemetary
More informationMath 61CM - Solutions to homework 3
Math 6CM - Solutios to homework 3 Cédric De Groote October 2 th, 208 Problem : Let F be a field, m 0 a fixed oegative iteger ad let V = {a 0 + a x + + a m x m a 0,, a m F} be the vector space cosistig
More informationFourier Series and their Applications
Fourier Series ad their Applicatios The fuctios, cos x, si x, cos x, si x, are orthogoal over (, ). m cos mx cos xdx = m = m = = cos mx si xdx = for all m, { m si mx si xdx = m = I fact the fuctios satisfy
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 informationSequences, Series, and All That
Chapter Te Sequeces, Series, ad All That. Itroductio Suppose we wat to compute a approximatio of the umber e by usig the Taylor polyomial p for f ( x) = e x at a =. This polyomial is easily see to be 3
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 informationBeurling Integers: Part 2
Beurlig Itegers: Part 2 Isomorphisms Devi Platt July 11, 2015 1 Prime Factorizatio Sequeces I the last article we itroduced the Beurlig geeralized itegers, which ca be represeted as a sequece of real umbers
More informationCOMMON FIXED POINT THEOREMS VIA w-distance
Bulleti of Mathematical Aalysis ad Applicatios ISSN: 1821-1291, URL: http://www.bmathaa.org Volume 3 Issue 3, Pages 182-189 COMMON FIXED POINT THEOREMS VIA w-distance (COMMUNICATED BY DENNY H. LEUNG) SUSHANTA
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 information5. Matrix exponentials and Von Neumann s theorem The matrix exponential. For an n n matrix X we define
5. Matrix expoetials ad Vo Neuma s theorem 5.1. The matrix expoetial. For a matrix X we defie e X = exp X = I + X + X2 2! +... = 0 X!. We assume that the etries are complex so that exp is well defied o
More informationTauberian theorems for the product of Borel and Hölder summability methods
A. Ştiiţ. Uiv. Al. I. Cuza Iaşi. Mat. (N.S.) Tomul LXIII, 2017, f. 1 Tauberia theorems for the product of Borel ad Hölder summability methods İbrahim Çaak Received: Received: 17.X.2012 / Revised: 5.XI.2012
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 informationk-generalized FIBONACCI NUMBERS CLOSE TO THE FORM 2 a + 3 b + 5 c 1. Introduction
Acta Math. Uiv. Comeiaae Vol. LXXXVI, 2 (2017), pp. 279 286 279 k-generalized FIBONACCI NUMBERS CLOSE TO THE FORM 2 a + 3 b + 5 c N. IRMAK ad M. ALP Abstract. The k-geeralized Fiboacci sequece { F (k)
More informationMath Homotopy Theory Spring 2013 Homework 6 Solutions
Math 527 - Homotopy Theory Sprig 2013 Homework 6 Solutios Problem 1. (The Hopf fibratio) Let S 3 C 2 = R 4 be the uit sphere. Stereographic projectio provides a homeomorphism S 2 = CP 1, where the North
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 informationSolutions to Math 347 Practice Problems for the final
Solutios to Math 347 Practice Problems for the fial 1) True or False: a) There exist itegers x,y such that 50x + 76y = 6. True: the gcd of 50 ad 76 is, ad 6 is a multiple of. b) The ifiimum of a set is
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 information1 1 2 = show that: over variables x and y. [2 marks] Write down necessary conditions involving first and second-order partial derivatives for ( x0, y
Questio (a) A square matrix A= A is called positive defiite if the quadratic form waw > 0 for every o-zero vector w [Note: Here (.) deotes the traspose of a matrix or a vector]. Let 0 A = 0 = show that:
More information1. By using truth tables prove that, for all statements P and Q, the statement
Author: Satiago Salazar Problems I: Mathematical Statemets ad Proofs. By usig truth tables prove that, for all statemets P ad Q, the statemet P Q ad its cotrapositive ot Q (ot P) are equivalet. I example.2.3
More informationSCORE. Exam 2. MA 114 Exam 2 Fall 2017
Exam Name: Sectio ad/or TA: Do ot remove this aswer page you will retur the whole exam. You will be allowed two hours to complete this test. No books or otes may be used. You may use a graphig calculator
More informationA Characterization of Compact Operators by Orthogonality
Australia Joural of Basic ad Applied Scieces, 5(6): 253-257, 211 ISSN 1991-8178 A Characterizatio of Compact Operators by Orthogoality Abdorreza Paahi, Mohamad Reza Farmai ad Azam Noorafa Zaai Departmet
More information4 The Sperner property.
4 The Sperer property. I this sectio we cosider a surprisig applicatio of certai adjacecy matrices to some problems i extremal set theory. A importat role will also be played by fiite groups. I geeral,
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 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 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 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 informationChapter 0. Review of set theory. 0.1 Sets
Chapter 0 Review of set theory Set theory plays a cetral role i the theory of probability. Thus, we will ope this course with a quick review of those otios of set theory which will be used repeatedly.
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 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 informationQuiz. Use either the RATIO or ROOT TEST to determine whether the series is convergent or not.
Quiz. Use either the RATIO or ROOT TEST to determie whether the series is coverget or ot. e .6 POWER SERIES Defiitio. A power series i about is a series of the form c 0 c a c a... c a... a 0 c a where
More informationMa 530 Introduction to Power Series
Ma 530 Itroductio to Power Series Please ote that there is material o power series at Visual Calculus. Some of this material was used as part of the presetatio of the topics that follow. What is a Power
More informationIt is often useful to approximate complicated functions using simpler ones. We consider the task of approximating a function by a polynomial.
Taylor Polyomials ad Taylor Series It is ofte useful to approximate complicated fuctios usig simpler oes We cosider the task of approximatig a fuctio by a polyomial If f is at least -times differetiable
More informationSOLUTIONS TO EXAM 3. Solution: Note that this defines two convergent geometric series with respective radii r 1 = 2/5 < 1 and r 2 = 1/5 < 1.
SOLUTIONS TO EXAM 3 Problem Fid the sum of the followig series 2 + ( ) 5 5 2 5 3 25 2 2 This series diverges Solutio: Note that this defies two coverget geometric series with respective radii r 2/5 < ad
More informationChapter 8. Euler s Gamma function
Chapter 8 Euler s Gamma fuctio The Gamma fuctio plays a importat role i the fuctioal equatio for ζ(s) that we will derive i the ext chapter. I the preset chapter we have collected some properties of the
More informationMIDTERM 3 CALCULUS 2. Monday, December 3, :15 PM to 6:45 PM. Name PRACTICE EXAM SOLUTIONS
MIDTERM 3 CALCULUS MATH 300 FALL 08 Moday, December 3, 08 5:5 PM to 6:45 PM Name PRACTICE EXAM S Please aswer all of the questios, ad show your work. You must explai your aswers to get credit. You will
More informationAlgebra of Least Squares
October 19, 2018 Algebra of Least Squares Geometry of Least Squares Recall that out data is like a table [Y X] where Y collects observatios o the depedet variable Y ad X collects observatios o the k-dimesioal
More informationMATH 312 Midterm I(Spring 2015)
MATH 3 Midterm I(Sprig 05) Istructor: Xiaowei Wag Feb 3rd, :30pm-3:50pm, 05 Problem (0 poits). Test for covergece:.. 3.. p, p 0. (coverges for p < ad diverges for p by ratio test.). ( coverges, sice (log
More informationSolution. 1 Solutions of Homework 1. Sangchul Lee. October 27, Problem 1.1
Solutio Sagchul Lee October 7, 017 1 Solutios of Homework 1 Problem 1.1 Let Ω,F,P) be a probability space. Show that if {A : N} F such that A := lim A exists, the PA) = lim PA ). Proof. Usig the cotiuity
More informationMa 530 Infinite Series I
Ma 50 Ifiite Series I Please ote that i additio to the material below this lecture icorporated material from the Visual Calculus web site. The material o sequeces is at Visual Sequeces. (To use this li
More informationMDIV. Multiple divisor functions
MDIV. Multiple divisor fuctios The fuctios τ k For k, defie τ k ( to be the umber of (ordered factorisatios of ito k factors, i other words, the umber of ordered k-tuples (j, j 2,..., j k with j j 2...
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 informationMathematical Methods for Physics and Engineering
Mathematical Methods for Physics ad Egieerig Lecture otes Sergei V. Shabaov Departmet of Mathematics, Uiversity of Florida, Gaiesville, FL 326 USA CHAPTER The theory of covergece. Numerical sequeces..
More informationWeek 5-6: The Binomial Coefficients
Wee 5-6: The Biomial Coefficiets March 6, 2018 1 Pascal Formula Theorem 11 (Pascal s Formula For itegers ad such that 1, ( ( ( 1 1 + 1 The umbers ( 2 ( 1 2 ( 2 are triagle umbers, that is, The petago umbers
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 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 informationMath 299 Supplement: Real Analysis Nov 2013
Math 299 Supplemet: Real Aalysis Nov 203 Algebra Axioms. I Real Aalysis, we work withi the axiomatic system of real umbers: the set R alog with the additio ad multiplicatio operatios +,, ad the iequality
More informationBasic Sets. Functions. MTH299 - Examples. Example 1. Let S = {1, {2, 3}, 4}. Indicate whether each statement is true or false. (a) S = 4. (e) 2 S.
Basic Sets Example 1. Let S = {1, {2, 3}, 4}. Idicate whether each statemet is true or false. (a) S = 4 (b) {1} S (c) {2, 3} S (d) {1, 4} S (e) 2 S. (f) S = {1, 4, {2, 3}} (g) S Example 2. Compute the
More information5.6 Absolute Convergence and The Ratio and Root Tests
5.6 Absolute Covergece ad The Ratio ad Root Tests Bria E. Veitch 5.6 Absolute Covergece ad The Ratio ad Root Tests Recall from our previous sectio that diverged but ( ) coverged. Both of these sequeces
More informationIn number theory we will generally be working with integers, though occasionally fractions and irrationals will come into play.
Number Theory Math 5840 otes. Sectio 1: Axioms. I umber theory we will geerally be workig with itegers, though occasioally fractios ad irratioals will come ito play. Notatio: Z deotes the set of all itegers
More informationSequences and Limits
Chapter Sequeces ad Limits Let { a } be a sequece of real or complex umbers A ecessary ad sufficiet coditio for the sequece to coverge is that for ay ɛ > 0 there exists a iteger N > 0 such that a p a q
More informationPRELIM PROBLEM SOLUTIONS
PRELIM PROBLEM SOLUTIONS THE GRAD STUDENTS + KEN Cotets. Complex Aalysis Practice Problems 2. 2. Real Aalysis Practice Problems 2. 4 3. Algebra Practice Problems 2. 8. Complex Aalysis Practice Problems
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 information1 Lecture 2: Sequence, Series and power series (8/14/2012)
Summer Jump-Start Program for Aalysis, 202 Sog-Yig Li Lecture 2: Sequece, Series ad power series (8/4/202). More o sequeces Example.. Let {x } ad {y } be two bouded sequeces. Show lim sup (x + y ) lim
More informationGENERALIZED HARMONIC NUMBER IDENTITIES AND A RELATED MATRIX REPRESENTATION
J Korea Math Soc 44 (2007), No 2, pp 487 498 GENERALIZED HARMONIC NUMBER IDENTITIES AND A RELATED MATRIX REPRESENTATION Gi-Sag Cheo ad Moawwad E A El-Miawy Reprited from the Joural of the Korea Mathematical
More informationOn a class of convergent sequences defined by integrals 1
Geeral Mathematics Vol. 4, No. 2 (26, 43 54 O a class of coverget sequeces defied by itegrals Dori Adrica ad Mihai Piticari Abstract The mai result shows that if g : [, ] R is a cotiuous fuctio such that
More informationFUNDAMENTALS OF REAL ANALYSIS by
FUNDAMENTALS OF REAL ANALYSIS by Doğa Çömez Backgroud: All of Math 450/1 material. Namely: basic set theory, relatios ad PMI, structure of N, Z, Q ad R, basic properties of (cotiuous ad differetiable)
More informationDupuy Complex Analysis Spring 2016 Homework 02
Dupuy Complex Aalysis Sprig 206 Homework 02. (CUNY, Fall 2005) Let D be the closed uit disc. Let g be a sequece of aalytic fuctios covergig uiformly to f o D. (a) Show that g coverges. Solutio We have
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 informationAdvanced Analysis. Min Yan Department of Mathematics Hong Kong University of Science and Technology
Advaced Aalysis Mi Ya Departmet of Mathematics Hog Kog Uiversity of Sciece ad Techology September 3, 009 Cotets Limit ad Cotiuity 7 Limit of Sequece 8 Defiitio 8 Property 3 3 Ifiity ad Ifiitesimal 8 4
More informationNEW FAST CONVERGENT SEQUENCES OF EULER-MASCHERONI TYPE
UPB Sci Bull, Series A, Vol 79, Iss, 207 ISSN 22-7027 NEW FAST CONVERGENT SEQUENCES OF EULER-MASCHERONI TYPE Gabriel Bercu We itroduce two ew sequeces of Euler-Mascheroi type which have fast covergece
More information