Orthogonal Dirichlet Polynomials with Arctangent Density
|
|
- Brianne Wilkins
- 5 years ago
- Views:
Transcription
1 Orthogoal Dirichlet Polyomials with Arctaget Desity Doro S. Lubisky School of Mathematics, Georgia Istitute of Techology, Atlata, GA USA. Abstract Let { j } j= be a strictly icreasig sequece of positive umbers with =. We fid a simple explicit formula for the orthogoal { } Dirchlet polyomials {φ } formed from liear combiatios of, associated with the arctaget desity. Thus it j j= dt φ tφ m t π + t = δ m. We obtai formulae for their Christoffel fuctios, ad deduce their asymptotics, as well as uiversality limits, ad spacig of zeros for their reproducig kerels. We also ivestigate the relatioship betwee ordiary Dirichlet series, ad orthogoal expasios ivolvig the {φ }, ad establish Markov- Berstei iequalities. Key words: Dirichlet polyomials, orthogoal polyomials. Itroductio Throughout, let = < < 3 <.. A Dirichlet series associated with this sequece of expoets has the form a it = a e ilog t. Research supported by NSF grat DMS008 ad US-Israel BSF grat addresses: lubisky@math.gatech.edu Doro S. Lubisky Preprit submitted to Joural of Approximatio Theory October 4, 03
2 I particular, whe j = j, j, we obtai the stadard Dirichlet series, of which the Riema zeta fuctio is a special case. It was Harald Bohr [] who developed much of the theory of almostperiodic fuctios. Oe of its basic tools is that if 0 < α β <, the α it ad β it are orthoormal o, i the mea, that is lim T T T T α it β it dt = δ αβ.. Cosequetly, if {a } ad {b } are square summable, ad the f t = lim T T a it ad g t = T T f t g tdt = b it, a b. Thus oe ca idetify spaces of Dirichlet series with the sequece space l. The mai results of the theory iclude existece ad uiqueess of o-harmoic Fourier series for almost periodic fuctios, ad their approximability by oharmoic trigoometric polyomials. Notable cotributors, i additio to Bohr, iclude Bocher, Stepaov, ad Besicovitch [], []. This has led to a very rich theory, i which Dirichlet polyomials { m } L m = a it : a,a,...,a m C, m,.3 have also bee extesively studied [6], [7]. It is the purpose of this paper to ivestigate various properties of Dirichlet polyomials, usig the arctaget desity,t,. Our π+t hope is that a more direct orthoormality relatio tha., might have some advatages. Our aalysis uses the orthoormal polyomials {φ } formed by applyig the Gram-Schmidt process to { it } with respect to the arctaget desity. Thus φ L, has positive leadig coefficiet, ad dt φ t φ m t π + t = δ m, m,..4 These Dirichlet orthogoal polyomials admit a very simple explicit expressio:
3 Theorem.. For =, φ =, ad for, φ t = it it..5 The author has searched the extesive literature of Dirichlet polyomials, ad ot foud this, eve i the special case j = j. There of course, it it arises i oe of the stadard ways of aalytically cotiuig the Riema zeta fuctio, via summatio by parts. We believe that eve if.5 is kow, at least the applicatios below are ew. Some elemetary properties of {φ } are give i the followig propositio. I it, ad i the sequel, we use the covetio 0 = 0. Propositio.. Let. a sup φ t = t R b The zeros of φ are simple ad have the form i kπ, k Z..7 log / c d e f sup φ t log + log =..8 t R it m = m j= φ t π + t dt = log + if c,c,...,c C it j= j j φ j t..9 c j it j log..0 dt π + t =.. 3
4 The machiery of orthogoal fuctios ad the simplicity of the formula above allow us to aalyze reproducig kerels, Christoffel fuctios, ad Markov-Berstei iequalities. The mth reproducig kerel is K m x,t = ad mth Christoffel fuctio is Λ m x = It admits the extremal property Λ m x = if φ xφ t, K m x,x = m φ x.. { } dt P t π+t P x : P L m..3 Christoffel fuctios are a essetial tool i aalysis of orthogoal polyomials [5]. We eed the sic ad hyperbolic sic kerels S z = siπz πz ; i describig limits of Christoffel fuctios. S z = sih πz πz.4 Theorem.3. a For s,t C, ad m, K m s,t = i s t = [ s + t si log 4 b For real x, ad m, si [ t s 4 + i ] log ]..5 K m x,x = + = [ x + 4 si log ]..6 4
5 c For all real x, ad m, K m x,x + d Moreover, if as m, = + x..7 + m ad m m = + o,.8 the as m, uiformly for s,t, i compact subsets of the complex plae, K m s,t = s t + is it is t/ m log m S π log m + o Ims t / m log m S Ims t log m. π.9 e As m, uiformly for x i compact subsets of the real lie, K m x,x = + x log m + o..0 We ca deduce uiversality limits cf. [4] for the reproducig kerels, ad asymptotics for their zeros: Theorem.4. Assume.8. a We have, uiformly for α,β i compact subsets of C, ad x i compact subsets of the real lie, lim K m x + α,x + β m log m log m log m = [ + x ] α β e iα β/ S.. π b Let x R. The for each fixed iteger j = ±, ±, ±3,..., ad large eough m, K m x,t has a simple zero t m,j, which satisfies lim t m,j x log m = jπ.. m Moreover, { give r > 0, } for large eough m, the oly zeros of K m x,t i z : z x r log m are the zeros {t m,j }. 5
6 Next, we tur to Markov-Berstei iequalities, which estimate derivatives of Dirichlet polyomials. There is a substatial literature for such iequalities for Műtz polyomials [3], but the author has ot foud ay such results for Dirichlet polyomials. Theorem.5 Markov-Berstei Iequality. For P L m, I particular, P t / π + t dt P t / π + t dt / log m + j j..3 j + j P t / π + t dt P t / π + t dt log m + log m /..4 Propositio. e shows that this is essetially sharp with respect to the order of log m, ad moreover, just a growth factor of log m is isufficiet we eed a extra smaller term i the last right-had side. Fially, we tur to orthoormal expasios. Let { } H = f = a φ : a <..5 This is a subspace of the weighted L space cosistig of measurable fuctios f : R R with f = j= f t / π + t dt <, which we deote by G. For f H ad m, we deote the mth partial sum of its orthoormal expasio by S m [f] = a φ,.6 6
7 where for, a = a [f] = f tφ t π + t dt..7 The relatioship betwee formal orthoormal expasios ad formal Dirichlet series is give i: Theorem.6. expasio For, let The for m, a Let {a } C ad fdeote the formal orthoormal f = a a φ..8 b = S m [f] = m a b it + a m m m m it m..30 b Coversely, let {b } C. Choose a C, ad for m, let m a m = m b m a..3 Defie f by the formal orthoormal expasio.8. The the partial sums S m [f] satisfy.30 for m. Uder additioal coditios, we ca give aalytic meaig to these formal idetities: Theorem.7. a Let {b } C be a sequece for which b.3 coverges. Defie a m for m by a m = m m =m b,.33 ad f by.8. The the coclusio.30 of Propositio.6a remais valid. 7
8 b If i additio m m b m= =m <,.34 the f defied by.8 has f H, ad this last sum equals f. c If i additio lim m a m m m m = 0,.35 we have lim m ad as fuctios i G, I particular, if S m [f] f t = P t = the Theorem.7b implies that P t l π + t dt =. Proofs of Theorems.-.4 m= m l b it L R = 0,.36 b it..37 b it, m m l b =m..38 Proof of Theorem.. Let φ # it t = it.. We use the followig simple cosequece of the residue theorem: for real µ, e iµt π + t dt = e µ.. 8
9 The for µ log, e iµt φ # t π + t dt = e iµ log t e iµ log t dt π + t = e µ log e µ log = e µ e µ = 0. For log < µ < log, istead φ # e iµt π + t dt = e µ log e µ log For µ log, istead φ # I summary, φ # = e µ e µ. e iµt π + t dt = e µ log e µ log e µ e µ. e iµt π + t dt = = 0, µ log e µ e µ, log µ < log. e µ, µ log.3 This immediately yields the desired orthogoality relatios for φ = φ #. Fially,.3 shows that φ # t dt π + t = φ # t ilog t e π + t dt =. 9
10 Proof of Propositio.. a φ t = / it + = +, with equality if t = π/log /. b This is immediate. c φ t = i it log it log,.4 so the result follows as i a. d Usig our covetio 0 = 0, j= e From.4 ad d, j j φ j t = j= it j it j = it m. φ t i = log j j φ j t log j j φ j t j= j= i = j j φ j tlog log ilog φ t. j= So by orthoormality, φ t π + t dt = j= j j log log + log. This telescopes to the right-had side of.0. 0
11 f Let ψ t = it = it φ t deote the th moic Dirichlet orthogoal polyomial. The orthoormality relatios show that for ay moic Dirichlet polyomial P t = ψ t + j= a jφ j t, we have P t π + t dt = + a j. j= Thus, the if over such moic polyomials P is, with equality iff P t = ψ t. Proof of Theorem.3. a Let ad τ = log. Elemetary trigoometric idetities give φ tφ s = it it +is +is [ s t +i i t ] i t = [ +i s ] +i s s t +i = 4 si t + i τ si s i τ [ s t +i = cos t s + i τ cos s + t τ ] s t log +i = 4 [si s + t 4 ] log si t s + i. 4 Now add for =,3,...,m, ad recall φ t =.
12 b Whe s = t = x, the above idetity simplifies to φ x = 4 [ si x log si i log = + 4 si x log Now add over =,3,...,m, ad recall φ t =. c Usig first siu u, for all real u, ad the log + u u for u 0, K m x,x + + = + = = =. ] [ [ x ] ] + 4 log / [ [ x + 4 [ [ ] ] x +. ] ] d Usig si t = t + o as t 0, we see that as m, with the o term below havig limit 0 as, K m s,t +is t + o = 4 = log [ s + t t s + i + o + o] 4 4 +is t + o = 4 = [ s + t t s + i + o + o] 4 4 is t = = [ + is t + st + o] is t = [ + is it + o], + =
13 uiformly for s, t i compact subsets of the plae. Agai usig.8, we cotiue this as u is t = + o [ + is it + o] du = u = m + is it u is t du + o u Imt s du = = m + is it u is t du + o = m u Imt s du. Here we are also usig that m as m, so that the o term grows at least as fast as log m. Simple calculatios show that for complex α, real oegative β, ad for T, T α u iα du = T iα/ log TS π log T ; T u β du = T β/ log T S β π log T. Hece K m s,t = + o + is it is t/ m Ims t / m log m S s t log m S Im s t π π log m log m e Settig s = t = x, we also obtai.0. Proof of Theorem.4. a We choose s = x + ad α log m ad t = x + β log m i.9. We see that + is it = + x + o, s t is t/ m S π log m = e iα β/ S The. follows from.9. α β b This follows directly from a, from Hurwitz theorem, ad the fact that the oly zeros of S z are the o-zero itegers. 3 π.
14 3. Proof of Theorem.5 Proof of Theorem.5. Now for j,.5 ad.4 show that φ j t + ilog j φ j t = i log j log j it j j, 3. j so φ j t + ilog j φ j t j = j j j / log j j. 3. Next if P t = we recall that φ t = ad write P t = a j φ j t, j= [ a j φ j t + ilog j φ j t ] j= =: T t + T t. a j ilog j φ j t Here, usig Cauchy-Schwarz, 3. ad the iequality log + u u,u 0, T t a j / j= j= j j P t / π + t dt j= j j j j j + j j= / j j so the triagle iequality ad orthoormality, ad our boud o T give P t / π + t dt T t / π + t dt + P t / π + t dt j=, T t / π + t dt j j j + j / / 4
15 + a j log j j= / / P t π + t dt / log m + j j. j + j j= Here j= j j j + j j= j j j j= j dt j t = log m. We ote that usig our explicit expressio for φ j, it is possible to obtai a explicit orthoormal expasio for P i terms of the {φ j }. However, estimatio of that does ot seem to lead to a better estimate tha that i.3/ Proof of Theorems.6 ad.7 Proof of Theorem.6. a From Theorem., S m [f] = = a m it [ it a it ] a a m m m by a summatio by parts. Our defiitio.9 of {b } gives the result. b It is easily see from.3 that b satisfies.9 for, so the result follows from a. Proof of Theorem.7. it m, 5
16 a If {a m } are defied by.33 for m, the it is easily see that.9 is satisfied for, ad Theorem.6a yields the result. b Our hypothesis.34 asserts that a <, so ideed f = a φ H, ad f = a = m m m= b =m. c From.30, S m [f]t m b it L R = a m m. m m The.36 follows from.35. Moreover, the m f t b it f S m [f] + S m [f]t m / a + S m [f]t =m 0, m. b it m b it L R The.37 follows. Refereces [] A. S. Besicovitch, Almost Periodic Fuctios, Dover, New York, 954. [] H. Bohr, Almost Periodic Fuctios, Chelsea, New York, 947. [3] P. Borwei ad T. Erdelyi, Polyomials ad Polyomial Iequalities, Spriger, New York, 995. [4] P. Deift, Orthogoal Polyomials ad Radom Matrices: A Riema- Hilbert Approach, Courat Lecture Notes, Vol. 3, New York Uiversity,
17 [5] P. Nevai, Geza Freud, Orthogoal Polyomials ad Christoffel Fuctios: A Case Study, J. Approx. Theory, , [6] K. M. Seip, Estimates for Dirichlet Polyomials, CRM Notes, 0, olie at [7] M. Weber, Dirichlet polyomials: some old ad recet results, ad their iterplay i umber theory, i Depedece i probability, aalysis ad umber theory, 00, Kedrick Press, Heber City, UT, pp
Marcinkiwiecz-Zygmund Type Inequalities for all Arcs of the Circle
Marcikiwiecz-ygmud Type Iequalities for all Arcs of the Circle C.K. Kobidarajah ad D. S. Lubisky Mathematics Departmet, Easter Uiversity, Chekalady, Sri Laka; Mathematics Departmet, Georgia Istitute of
More informationThe Degree of Shape Preserving Weighted Polynomial Approximation
The Degree of Shape Preservig eighted Polyomial Approximatio Day Leviata School of Mathematical Scieces, Tel Aviv Uiversity, Tel Aviv, Israel Doro S Lubisky School of Mathematics, Georgia Istitute of Techology,
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 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 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 informationMa 4121: Introduction to Lebesgue Integration Solutions to Homework Assignment 5
Ma 42: Itroductio to Lebesgue Itegratio Solutios to Homework Assigmet 5 Prof. Wickerhauser Due Thursday, April th, 23 Please retur your solutios to the istructor by the ed of class o the due date. You
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 informationTR/46 OCTOBER THE ZEROS OF PARTIAL SUMS OF A MACLAURIN EXPANSION A. TALBOT
TR/46 OCTOBER 974 THE ZEROS OF PARTIAL SUMS OF A MACLAURIN EXPANSION by A. TALBOT .. Itroductio. A problem i approximatio theory o which I have recetly worked [] required for its solutio a proof that the
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 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 informationRational Bounds for the Logarithm Function with Applications
Ratioal Bouds for the Logarithm Fuctio with Applicatios Robert Bosch Abstract We fid ratioal bouds for the logarithm fuctio ad we show applicatios to problem-solvig. Itroductio Let a = + solvig the problem
More informationREGULARIZATION OF CERTAIN DIVERGENT SERIES OF POLYNOMIALS
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]
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 informationarxiv: v1 [math.fa] 3 Apr 2016
Aticommutator Norm Formula for Proectio Operators arxiv:164.699v1 math.fa] 3 Apr 16 Sam Walters Uiversity of Norther British Columbia ABSTRACT. We prove that for ay two proectio operators f, g o Hilbert
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 informationECE 330:541, Stochastic Signals and Systems Lecture Notes on Limit Theorems from Probability Fall 2002
ECE 330:541, Stochastic Sigals ad Systems Lecture Notes o Limit Theorems from robability Fall 00 I practice, there are two ways we ca costruct a ew sequece of radom variables from a old sequece of radom
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 informationAppendix to Quicksort Asymptotics
Appedix to Quicksort Asymptotics James Alle Fill Departmet of Mathematical Scieces The Johs Hopkis Uiversity jimfill@jhu.edu ad http://www.mts.jhu.edu/~fill/ ad Svate Jaso Departmet of Mathematics Uppsala
More informationA Hadamard-type lower bound for symmetric diagonally dominant positive matrices
A Hadamard-type lower boud for symmetric diagoally domiat positive matrices Christopher J. Hillar, Adre Wibisoo Uiversity of Califoria, Berkeley Jauary 7, 205 Abstract We prove a ew lower-boud form of
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 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 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 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 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 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 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 informationA collocation method for singular integral equations with cosecant kernel via Semi-trigonometric interpolation
Iteratioal Joural of Mathematics Research. ISSN 0976-5840 Volume 9 Number 1 (017) pp. 45-51 Iteratioal Research Publicatio House http://www.irphouse.com A collocatio method for sigular itegral equatios
More informationThis section is optional.
4 Momet Geeratig Fuctios* This sectio is optioal. The momet geeratig fuctio g : R R of a radom variable X is defied as g(t) = E[e tx ]. Propositio 1. We have g () (0) = E[X ] for = 1, 2,... Proof. Therefore
More informationSequences of Definite Integrals, Factorials and Double Factorials
47 6 Joural of Iteger Sequeces, Vol. 8 (5), Article 5.4.6 Sequeces of Defiite Itegrals, Factorials ad Double Factorials Thierry Daa-Picard Departmet of Applied Mathematics Jerusalem College of Techology
More informationPrecise Rates in Complete Moment Convergence for Negatively Associated Sequences
Commuicatios of the Korea Statistical Society 29, Vol. 16, No. 5, 841 849 Precise Rates i Complete Momet Covergece for Negatively Associated Sequeces Dae-Hee Ryu 1,a a Departmet of Computer Sciece, ChugWoo
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 informationSelf-normalized deviation inequalities with application to t-statistic
Self-ormalized deviatio iequalities with applicatio to t-statistic Xiequa Fa Ceter for Applied Mathematics, Tiaji Uiversity, 30007 Tiaji, Chia Abstract Let ξ i i 1 be a sequece of idepedet ad symmetric
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 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 informationComplex Analysis Spring 2001 Homework I Solution
Complex Aalysis Sprig 2001 Homework I Solutio 1. Coway, Chapter 1, sectio 3, problem 3. Describe the set of poits satisfyig the equatio z a z + a = 2c, where c > 0 ad a R. To begi, we see from the triagle
More informationFLOOR AND ROOF FUNCTION ANALOGS OF THE BELL NUMBERS. H. W. Gould Department of Mathematics, West Virginia University, Morgantown, WV 26506, USA
INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 7 (2007), #A58 FLOOR AND ROOF FUNCTION ANALOGS OF THE BELL NUMBERS H. W. Gould Departmet of Mathematics, West Virgiia Uiversity, Morgatow, WV
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 informationLecture 3 The Lebesgue Integral
Lecture 3: The Lebesgue Itegral 1 of 14 Course: Theory of Probability I Term: Fall 2013 Istructor: Gorda Zitkovic Lecture 3 The Lebesgue Itegral The costructio of the itegral Uless expressly specified
More informationOn Orlicz N-frames. 1 Introduction. Renu Chugh 1,, Shashank Goel 2
Joural of Advaced Research i Pure Mathematics Olie ISSN: 1943-2380 Vol. 3, Issue. 1, 2010, pp. 104-110 doi: 10.5373/jarpm.473.061810 O Orlicz N-frames Reu Chugh 1,, Shashak Goel 2 1 Departmet of Mathematics,
More informationSolutions to Tutorial 5 (Week 6)
The Uiversity of Sydey School of Mathematics ad Statistics Solutios to Tutorial 5 (Wee 6 MATH2962: Real ad Complex Aalysis (Advaced Semester, 207 Web Page: http://www.maths.usyd.edu.au/u/ug/im/math2962/
More informationMASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.265/15.070J Fall 2013 Lecture 2 9/9/2013. Large Deviations for i.i.d. Random Variables
MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.265/15.070J Fall 2013 Lecture 2 9/9/2013 Large Deviatios for i.i.d. Radom Variables Cotet. Cheroff boud usig expoetial momet geeratig fuctios. Properties of a momet
More informationThe Arakawa-Kaneko Zeta Function
The Arakawa-Kaeko Zeta Fuctio Marc-Atoie Coppo ad Berard Cadelpergher Nice Sophia Atipolis Uiversity Laboratoire Jea Alexadre Dieudoé Parc Valrose F-0608 Nice Cedex 2 FRANCE Marc-Atoie.COPPO@uice.fr Berard.CANDELPERGHER@uice.fr
More informationLecture 3: Convergence of Fourier Series
Lecture 3: Covergece of Fourier Series Himashu Tyagi Let f be a absolutely itegrable fuctio o T : [ π,π], i.e., f L (T). For,,... defie ˆf() f(θ)e i θ dθ. π T The series ˆf()e i θ is called the Fourier
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 informationOn the Weak Localization Principle of the Eigenfunction Expansions of the Laplace-Beltrami Operator by Riesz Method ABSTRACT 1.
Malaysia Joural of Mathematical Scieces 9(): 337-348 (05) MALAYSIA JOURAL OF MATHEMATICAL SCIECES Joural homepage: http://eispemupmedumy/joural O the Weak Localizatio Priciple of the Eigefuctio Expasios
More informationPUTNAM TRAINING INEQUALITIES
PUTNAM TRAINING INEQUALITIES (Last updated: December, 207) Remark This is a list of exercises o iequalities Miguel A Lerma Exercises If a, b, c > 0, prove that (a 2 b + b 2 c + c 2 a)(ab 2 + bc 2 + ca
More informationRademacher Complexity
EECS 598: Statistical Learig Theory, Witer 204 Topic 0 Rademacher Complexity Lecturer: Clayto Scott Scribe: Ya Deg, Kevi Moo Disclaimer: These otes have ot bee subjected to the usual scrutiy reserved for
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 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 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 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 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 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 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 informationInteresting Series Associated with Central Binomial Coefficients, Catalan Numbers and Harmonic Numbers
3 47 6 3 Joural of Iteger Sequeces Vol. 9 06 Article 6.. Iterestig Series Associated with Cetral Biomial Coefficiets Catala Numbers ad Harmoic Numbers Hogwei Che Departmet of Mathematics Christopher Newport
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 informationArchimedes - numbers for counting, otherwise lengths, areas, etc. Kepler - geometry for planetary motion
Topics i Aalysis 3460:589 Summer 007 Itroductio Ree descartes - aalysis (breaig dow) ad sythesis Sciece as models of ature : explaatory, parsimoious, predictive Most predictios require umerical values,
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 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 informationTRIGONOMETRIC POLYNOMIALS WITH MANY REAL ZEROS AND A LITTLEWOOD-TYPE PROBLEM. Peter Borwein and Tamás Erdélyi. 1. Introduction
TRIGONOMETRIC POLYNOMIALS WITH MANY REAL ZEROS AND A LITTLEWOOD-TYPE PROBLEM Peter Borwei ad Tamás Erdélyi Abstract. We examie the size of a real trigoometric polyomial of degree at most havig at least
More information2 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 7 Isoperimetric problem
Chapter 7 Isoperimetric problem Recall that the isoperimetric problem (see the itroductio its coectio with ido s proble) is oe of the most classical problem of a shape optimizatio. It ca be formulated
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 informationThe Poisson Summation Formula and an Application to Number Theory Jason Payne Math 248- Introduction Harmonic Analysis, February 18, 2010
The Poisso Summatio Formula ad a Applicatio to Number Theory Jaso Paye Math 48- Itroductio Harmoic Aalysis, February 8, This talk will closely follow []; however some material has bee adapted to a slightly
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 informationarxiv: v1 [math.co] 3 Feb 2013
Cotiued Fractios of Quadratic Numbers L ubomíra Balková Araka Hrušková arxiv:0.05v [math.co] Feb 0 February 5 0 Abstract I this paper we will first summarize kow results cocerig cotiued fractios. The we
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 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 informationMath 140A Elementary Analysis Homework Questions 3-1
Math 0A Elemetary Aalysis Homework Questios -.9 Limits Theorems for Sequeces Suppose that lim x =, lim y = 7 ad that all y are o-zero. Detarime the followig limits: (a) lim(x + y ) (b) lim y x y Let s
More informationSOME GENERALIZATIONS OF OLIVIER S THEOREM
SOME GENERALIZATIONS OF OLIVIER S THEOREM Alai Faisat, Sait-Étiee, Georges Grekos, Sait-Étiee, Ladislav Mišík Ostrava (Received Jauary 27, 2006) Abstract. Let a be a coverget series of positive real umbers.
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 informationDiscrete Orthogonal Moment Features Using Chebyshev Polynomials
Discrete Orthogoal Momet Features Usig Chebyshev Polyomials R. Mukuda, 1 S.H.Og ad P.A. Lee 3 1 Faculty of Iformatio Sciece ad Techology, Multimedia Uiversity 75450 Malacca, Malaysia. Istitute of Mathematical
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 informationThe natural exponential function
The atural expoetial fuctio Attila Máté Brookly College of the City Uiversity of New York December, 205 Cotets The atural expoetial fuctio for real x. Beroulli s iequality.....................................2
More informationThe Bilateral Laplace Transform of the Positive Even Functions and a Proof of Riemann Hypothesis
The Bilateral Laplace Trasform of the Positive Eve Fuctios ad a Proof of Riema Hypothesis Seog Wo Cha Ph.D. swcha@dgu.edu Abstract We show that some iterestig properties of the bilateral Laplace trasform
More informationCOMPUTING THE EULER S CONSTANT: A HISTORICAL OVERVIEW OF ALGORITHMS AND RESULTS
COMPUTING THE EULER S CONSTANT: A HISTORICAL OVERVIEW OF ALGORITHMS AND RESULTS GONÇALO MORAIS Abstract. We preted to give a broad overview of the algorithms used to compute the Euler s costat. Four type
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 informationBIRKHOFF ERGODIC THEOREM
BIRKHOFF ERGODIC THEOREM Abstract. We will give a proof of the poitwise ergodic theorem, which was first proved by Birkhoff. May improvemets have bee made sice Birkhoff s orgial proof. The versio we give
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 informationKU Leuven Department of Computer Science
O orthogoal polyomials related to arithmetic ad harmoic sequeces Adhemar Bultheel ad Adreas Lasarow Report TW 687, February 208 KU Leuve Departmet of Computer Sciece Celestijelaa 200A B-300 Heverlee (Belgium)
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 informationSCALING LIMITS FOR MIXED KERNELS
SCALING LIMITS FOR MIXED KERNELS DORON S. LUBINSKY A. Let µ ad ν be measures supported o, with correspodig orthoormal polyomials { p µ } ad {p ν } respectively. Defie the mixed kerel K µ,ν We establish
More informationLecture 4. We also define the set of possible values for the random walk as the set of all x R d such that P(S n = x) > 0 for some n.
Radom Walks ad Browia Motio Tel Aviv Uiversity Sprig 20 Lecture date: Mar 2, 20 Lecture 4 Istructor: Ro Peled Scribe: Lira Rotem This lecture deals primarily with recurrece for geeral radom walks. We preset
More informationTERMWISE DERIVATIVES OF COMPLEX FUNCTIONS
TERMWISE DERIVATIVES OF COMPLEX FUNCTIONS This writeup proves a result that has as oe cosequece that ay complex power series ca be differetiated term-by-term withi its disk of covergece The result has
More informationMA131 - Analysis 1. Workbook 9 Series III
MA3 - Aalysis Workbook 9 Series III Autum 004 Cotets 4.4 Series with Positive ad Negative Terms.............. 4.5 Alteratig Series.......................... 4.6 Geeral Series.............................
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 informationSection 11.8: Power Series
Sectio 11.8: Power Series 1. Power Series I this sectio, we cosider geeralizig the cocept of a series. Recall that a series is a ifiite sum of umbers a. We ca talk about whether or ot it coverges ad i
More informationComparison Study of Series Approximation. and Convergence between Chebyshev. and Legendre Series
Applied Mathematical Scieces, Vol. 7, 03, o. 6, 3-337 HIKARI Ltd, www.m-hikari.com http://d.doi.org/0.988/ams.03.3430 Compariso Study of Series Approimatio ad Covergece betwee Chebyshev ad Legedre Series
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 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 informationarxiv: v1 [math.ca] 18 Aug 2017
UNIVERSALITY AT AN ENDPOINT FOR ORTHOGONAL POLYNOMIALS WITH GERONIMUS-TYPE WEIGHTS BRIAN SIMANEK arxiv:708.0578v [math.ca] 8 Aug 207 Abstract. We provide a ew closed form expressio for the Geroimus polyomials
More informationRandom Walks on Discrete and Continuous Circles. by Jeffrey S. Rosenthal School of Mathematics, University of Minnesota, Minneapolis, MN, U.S.A.
Radom Walks o Discrete ad Cotiuous Circles by Jeffrey S. Rosethal School of Mathematics, Uiversity of Miesota, Mieapolis, MN, U.S.A. 55455 (Appeared i Joural of Applied Probability 30 (1993), 780 789.)
More informationUniversity of Colorado Denver Dept. Math. & Stat. Sciences Applied Analysis Preliminary Exam 13 January 2012, 10:00 am 2:00 pm. Good luck!
Uiversity of Colorado Dever Dept. Math. & Stat. Scieces Applied Aalysis Prelimiary Exam 13 Jauary 01, 10:00 am :00 pm Name: The proctor will let you read the followig coditios before the exam begis, ad
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 informationENGI Series Page 6-01
ENGI 3425 6 Series Page 6-01 6. Series Cotets: 6.01 Sequeces; geeral term, limits, covergece 6.02 Series; summatio otatio, covergece, divergece test 6.03 Stadard Series; telescopig series, geometric series,
More informations = and t = with C ij = A i B j F. (i) Note that cs = M and so ca i µ(a i ) I E (cs) = = c a i µ(a i ) = ci E (s). (ii) Note that s + t = M and so
3 From the otes we see that the parts of Theorem 4. that cocer us are: Let s ad t be two simple o-egative F-measurable fuctios o X, F, µ ad E, F F. The i I E cs ci E s for all c R, ii I E s + t I E s +
More informationSPECTRUM OF THE DIRECT SUM OF OPERATORS
Electroic Joural of Differetial Equatios, Vol. 202 (202), No. 20, pp. 8. ISSN: 072-669. URL: http://ejde.math.txstate.edu or http://ejde.math.ut.edu ftp ejde.math.txstate.edu SPECTRUM OF THE DIRECT SUM
More informationRelations between the continuous and the discrete Lotka power function
Relatios betwee the cotiuous ad the discrete Lotka power fuctio by L. Egghe Limburgs Uiversitair Cetrum (LUC), Uiversitaire Campus, B-3590 Diepebeek, Belgium ad Uiversiteit Atwerpe (UA), Campus Drie Eike,
More informationProblem Set 2 Solutions
CS271 Radomess & Computatio, Sprig 2018 Problem Set 2 Solutios Poit totals are i the margi; the maximum total umber of poits was 52. 1. Probabilistic method for domiatig sets 6pts Pick a radom subset S
More information5 Birkhoff s Ergodic Theorem
5 Birkhoff s Ergodic Theorem Amog the most useful of the various geeralizatios of KolmogorovâĂŹs strog law of large umbers are the ergodic theorems of Birkhoff ad Kigma, which exted the validity of the
More information