Miskolc Mathematical Notes HU e-issn Uniform approximation by means of some piecewise linear functions. Zoltán Finta
|
|
|
- Sherman Logan
- 5 years ago
- Views:
Transcription
1 Miskolc Mathematical Notes HU e-issn Vol. 3 (00, No, pp. 0- DOI: 0.854/MMN Uiform approimatio by meas of some piecewise liear fuctios Zoltá Fita
2 Mathematical Notes, Miskolc, Vol. 3., No.., (00, pp. 0 UNIFORM APPROXIMATION BY MEANS OF SOME PIECEWISE LINEAR FUNCTIONS Zoltá Fita Babeş - Bolyai Uiversity, Departmet of Mathematics, M. Kogăliceau St., 3400 Cluj-Napoca, Romaia fzolta@math.ubbcluj.ro [Received: September, 00] Abstract. I the preset paper we costruct ew piecewise liear fuctios usig a special partitio of the iterval [, ]. This costructio leads to the defiitio of some ew liear operators ad we shall obtai global estimates for the remaider i approimatig cotiuous fuctios by these operators usig the secod order modulus of smoothess of Ditzia - Totik. Mathematical Subject Classificatio: 4A0, 4A35 Keywords: piecewise liear fuctio, Ditzia - Totik modulus of smoothess. Itroductio Let (δ k k be a sequece such that δ k δ k (k,,..., ad / δ < δ <... < δ <δ 0 <c 0 /. Furthermore, let ( k k : < <...< < be a partitio of the iterval [, ] with the properties: : (i c δ k k k c δ k (k,..., : (ii for ay u [ k, k ] (k,..., we have k k c u. Here ad throughout c 0,c,c ad c deote absolute costats ad the value of c may vary with each occurrece, eve o the same lie. Theeisteceof(δ k k ad ( k k is guarateed by the costructive proof of DeVore ad Yu give i [, p. 36 ad p. 39]. This proof establishes a poitwise estimate of the Tima - Teljakovski type for mootoe polyomial approimatio [, p. 34, Theorem ]. The idea of DeVore ad Yu was successfully applied by Leviata [3, p. 3, Theorem ] to give a global estimate o mootoe approimatio. Both proofs are based o a two - stage approimatio. At first the fuctio f C[, ] is approimated by a piecewise liear fuctio S f which iterpolates f at the poits k (k,,...,. By Newto s formula we have (S f( f( ( k ( k [ k,, k ; f ],
3 0 Z. Fita k k (k,...,, where the square brackes deote the divided differece of f at k,, k. Usig the proof of [3, p. 7, Theorem 7 ] we ca deduce the followig result: kf S fk cωϕ(f,, (. where kkisthesup-ormo[, ], ϕ(, [, ] ad ωϕ(f,t f(k is the Ditzia - Totik modulus of smoothess [], where sup 0<ht k hϕ( hϕ( f( f( hϕ( f(f( hϕ(, if ± hϕ( [, ] ad hϕ( f( 0, otherwise. I this ote we are iterested i uiform approimatio by piecewise liear fuctios differet from S f.. Approimatio by piecewise liear fuctios Usig a represetatio theorem give by Popoviciu for the operator S [5], we cosider the followig piecewise liear fuctio: (U f( f( ( X k k k [ k, k, k ; t ] t f( k γ ϕ( k f( k f( k γ ϕ( k ª f(, ( where f C[, ] ad γ such that c δ c γ. Here we deote by [ k, k, k ; t ] t thefactthatthedivideddifferece is applied to the variable t. The operator U : C[, ] C[, ] is liear which preserves the liear fuctios. The fuctio U f iterpolates f at the edpoits of the iterval [, ]. Moreover, if f C[, ] is a positive, cove fuctio, the U f is also positive. Ideed, for k,..., we have f( k γ ϕ( k f( k f( k γ ϕ( k f( k 0. Hece, by defiitio of U f we obtai (U f( (S f( 0, [, ] because S : C[, ] C[, ] is a positive liear operator. Our first result is the followig: Theorem. If f C[, ], thekf U fk cω ϕ(f,. Proof. By (i we get γ ϕ( γ c δ. Hece γ ϕ(. Agai, by (i ad by the properties of (δ k k we have γ ϕ( k γ c δ c δ k k k (k,...,. The
4 Uiform approimatio by meas of some piecewise liear fuctios 03 k k γ ϕ( k k (k,...,. Withthesamehypotheses we obtai γ ϕ( k γ c δ c δ k k k (k,...,. Hece k k γ ϕ( k k (k,...,. This meas that U f is well - defied. Now, i view of Popoviciu s represetatio theorem [5] we have (S f( f( ( X k k k [ k, k, k ; t ] t f( k f(. ( Hece (U f( (S f( f( γ ϕ( f( f( γ ϕ( ª (. for ; (U f( (S f( k f( k γ ϕ( k k k f( k f( k γ ϕ( k ª k f( k γ ϕ( k k k f( k f( k γ ϕ( k ª (. for k k (k,..., ; (U f( (S f( f( γ ϕ( f( f( γ ϕ( ª (.3 for. O the other had, by defiitio of the Ditzia-Totik modulus of smoothess we obtai from (, (3, (4 ad γ the estimates (U f( (S f( γ ϕ( f( k γ ϕ( f(k ω ϕ(f,,
5 04 Z. Fita where ; (U f( (S f( where k k ; k k γ ϕ( k f( k k k k γ ϕ( k f( k k k γ ϕ( f(k ω ϕ(f,, (U f( (S f( γ ϕ( f( k γ ϕ( f(k ω ϕ(f,, where. Hece ku f S fk ωϕ(f,. Usig ( we obtai the assertio of the theorem. Our et piecewise liear fuctio is the followig: (V f( k k k k k k c δ k c δ k Z k c δ k k f(u du Z k k c δ k f(u du, if k << k (k,..., ad (V f( f( k, if k (k,...,. The V : C[, ] L [, ] is a liear, positive operator such that V f iterpolates the fuctio f at the poits k (k,...,. The mai differece betwee V ad S is that V f is ot ecessarily cotiuous at k (k,...,. Moreover, V e 0 e 0 ad V e e O( (here e 0 ( ad e ( for [, ]. Ideed, the first statemet is obvious ad for the secod oe we have for k << k (k,..., : (V e ( k k k c δ k k k k c δ k c δ k µ k k k Hece, by properties of (δ k k we obtai k c δ k c δk k c δ k c δ k k. k k (V e ( e ( c δ k c 0c,.
6 Uiform approimatio by meas of some piecewise liear fuctios 05 Furthermore, we deote the set of all algebraic polyomials of degree at most by Π ad the best uiform approimatio o [, ] by E (f if{kf pk : p Π }. Our et results are the followig: Theorem. Let f C[, ] ad V ( N, be defied as above. The kf V fk c (ω ϕ(f, Z ω ϕ(f,t t 3 dt E 0 (f. Corollary. Let f C[, ], N,. The kf V fk c ( Z ω ϕ(f,t t 3 dt kfk. To prove our statemets we eed a lemma: Lemma. For g C [, ], we have kg V gk c kg 0 k kϕ g 00 k ª. Proof. If k << k (k,...,, the (V g( g( k k k ( c δ k Z k c δ k k [g(u g( k ] du g( k g( k k k ( c δ k Z k k c δ k [g(u g( k ] du g( k g(. Simple computatios show that Z β α [g(u g(α] du Z β α (β u g 0 (u du ad Z β α [g(β g(u] du Z β α (u α g 0 (u du.
7 06 Z. Fita Hece ( c δ u g (u du (V g( g( k k k ( Z k c δ k Z ( k c δ k u g 0 (u du g 0 (u du c δ k k k k k k c δ k Z k Z k 0 g 0 (u du k k k c δ k k k k c δ k ( Z k c δ k Z ( k c δ k u g 0 (u du c δ k g 0 (u du k k ( Z k k ( k c δ k u g 0 (u du k k c δ k k c δ k Z k ¾ c δ k g 0 (u du k k k c δ k ( Z Z kc δ k ( k ug 0 (u du ( k c δ k ug 0 (u du k ( Z k ( k c δ k ug 0 (u du k k c δ k k c δ k Z k ¾ ( k ug 0 (u du. By partial itegratio we obtai (V g( g( k k k c δ k Z ( k g 0 ( ( k u g 00 (u du k ( k c δ k g 0 ( ( k c δ k u g 00 (u du Z k c δ k
8 Uiform approimatio by meas of some piecewise liear fuctios 07 Z k c δ k k k k c δ k Z ( k c δ k u g 00 (u du ( k c δ k g 0 ( ( k c δ k u g 00 (u du k c δ k ( k g 0 ( Z k ¾ ( k u g 00 (u du. (.4 O the other had, we get for k u k by (i ad (ii ( k ( k c δ k c δ k ( k (c δ k c δ k ( k c δ k c δ k ( k c δ k 3 c δ k ( k k 3 c δ k c u c cδ k ; (.5 we have for k u < k, by (ii ( k u ( k k c u (.6 ad for k < u k c δ k k or k k c δ k u < k we have i view of (i ad (ii ( k c δ k u ( k u (c δ k 4( k k c u. (.7 Usig the same argumets we obtai ( k c δ k ( k cc δ k (.8 for k u k ; ( k c δ k u c u (.9 for k k c δ k u < k or k < u k c δ k k ad ( k u c u (.0 for k < u k. The (5 ( ad (ii imply
9 08 Z. Fita (V g( g( k k k c δ k Z ( k ( k c δ k g 0 ( (u k g 00 (u du k Z k c δ k ( k c δ k u g 00 (u du k k k c δ k ( k c δ k ( k g 0 ( Z ( k c δ k u g 00 (u du k c δ k Z k ¾ ( k u g 00 (u du k c cδ k k k c δ k kg0 k Z c u k g 00 (u du Z k c δ k c u g 00 (u du k c cδ k k k c δ k kg0 k Z c u k c δ k g 00 (u du Z k c ¾ u g 00 (u du. (. But Z k Z k c δ k Z k Z k c δ k Z k c δ k k if k < k c δ k or Z k Z k c δ k Z k Z k c δ k Z k Z k Z k if k c δ k < k ad Z k c δ k Z k Z k c δ k Z k Z k k c δ k
10 Uiform approimatio by meas of some piecewise liear fuctios 09 if k c δ k < k or Z k c δ k Z k Z k c δ k Z k Z k Z k Z k if k < k c δ k. So we obtai i view of ( that (V g( g( k k k k k k c c δ k c δ k c cδ k kg0 k c kϕ g 00 k ma (c δ k ; k c cδ k kg0 k c kϕ g 00 k kg 0 k kϕ g 00 kma ¾ ¾ ma(c δ k ; k µ ; ¾ k k. c δ k By (i we have k k c δ k. Therefore (V g( g( c kg 0 k kϕ g 00 k ª. (. If << the (V g( g( Z c δ Z c δ c δ [g(u g( ] du g( g( c δ [g(u g( ] du g( g(
11 0 Z. Fita Z c δ Z c δ [g(u g( ] du c δ Z [g(u g( ] du c δ c δ Z " c δ Z # u g 0 (v dv Z c δ c δ Z u g 0 (v dv g 0 (u du Z Z du g 0 (u du Z du g 0 (u du g 0 (u du. Hece (V g( g( c δ Z " c δ Z # u g 0 (v dv c δ Z c δ Z c δ Z u c δ g 0 (v dv Z du g 0 (u du Z du δ c δ Z (u du ( kg 0 k c ( u du ( kg 0 k. g 0 (u du
12 Uiform approimatio by meas of some piecewise liear fuctios I view of (i we have u c δ, c δ u ad, respectively. So (V g( g( Z c δ c δ ( du ( kg 0 k c δ Z ( du ( c δ kg 0 k ( kg 0 k. Agai, (i implies c δ. This meas that (V g( g( c δ kg 0 k c kg0 k. (.3 Aalogously (V g( g( c kg0 k (.4 for <<. I coclusio (3, (4 ad (5 imply kg V gk c kg 0 k kϕ g 00 k ª, which completes the proof. ProofofTheorem. We have (V f( k k k c δ k k k k c δ k Z k c δ k Z k k k c δ k f(u du f(u du kfk for k << k (k,...,. Thus kv fk kfk. (.5 O the other had, let us deote by p Π the best th degree polyomial approimatio to f. The we kow for f C[, ] (see [, p. 79, Theorem 7..] that E (f kf p k cωϕ(f,. (.6 Moreover, we have the followig Berstei type iequality [, p. 84, Theorem 7.3.] for the best approimatio polyomial kϕ p 00 k ωϕ(f,. (.7
13 Z. Fita The, usig (6, Lemma, (7, (8 ad the proof of [4, pp , Theorem 3.] we obtai the coclusio of our theorem. Proof of Corollary. I view of [4, p. 86, Remark 3.4], the proof is a direct cosequece of the fact that we ca drop the firsttermotherighthadsideofthe estimate give i Theorem because of Z ω ϕ(f,t t 3 dt ω ϕ(f, Z dt t 3 c ω ϕ(f,, >. Ackowledgemet. This work was doe with the fiacial support of Domus Hugarica Scietiarum et Artium. REFERENCES [] DeVore,R.A.ad Yu, X. M.: Poitwise estimates for mootoe polyomial approimatio, Costr.Appro.,, (985, [] Ditzia, Z. ad Totik, V.: Moduli of Smoothess, Spriger - Verlag, Berli Heidelberg New York Lodo, 987. [3] Leviata, D.: Mootoe ad Comootoe Polyomial Approimatio, Revisited, J. Appro. Theory, 53, (988, -6. [4] Mache, D. H. ad Zhou, D. X.: Best direct ad coverse results for Lagrage - type operators, Appro. Theory Appl., (, (995, [5] Popoviciu, T.: Course of Mathematical Aalysis, Part III, Babeş-Bolyai Uiversity, Cluj, 974.
q-durrmeyer operators based on Pólya distribution
Available olie at wwwtjsacom J Noliear Sci Appl 9 206 497 504 Research Article -Durrmeyer operators based o Pólya distributio Vijay Gupta a Themistocles M Rassias b Hoey Sharma c a Departmet of Mathematics
Local Approximation Properties for certain King type Operators
Filomat 27:1 (2013, 173 181 DOI 102298/FIL1301173O Published by Faculty of Scieces ad athematics, Uiversity of Niš, Serbia Available at: http://wwwpmfiacrs/filomat Local Approimatio Properties for certai
On 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
BETWEEN QUASICONVEX AND CONVEX SET-VALUED MAPPINGS. 1. Introduction. Throughout the paper we denote by X a linear space and by Y a topological linear
BETWEEN QUASICONVEX AND CONVEX SET-VALUED MAPPINGS Abstract. The aim of this paper is to give sufficiet coditios for a quasicovex setvalued mappig to be covex. I particular, we recover several kow characterizatios
New estimates in Voronovskaja s theorem. Gancho Tachev. Numerical Algorithms ISSN Numer Algor DOI / s
New estimates i Voroovskaja s theorem Gacho Tachev Numerical Algorithms ISSN 7-398 DOI.7/ s75--9479-23 Your article is protected by copyright ad all rights are held exclusively by Spriger Sciece+Busiess
Fall 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
Direct Estimates for Lupaş-Durrmeyer Operators
Filomat 3:1 16, 191 199 DOI 1.98/FIL161191A Published by Faculty of Scieces ad Mathematics, Uiversity of Niš, Serbia Available at: http://www.pmf.i.ac.rs/filomat Direct Estimates for Lupaş-Durrmeyer Operators
Gamma Distribution and Gamma Approximation
Gamma Distributio ad Gamma Approimatio Xiaomig Zeg a Fuhua (Frak Cheg b a Xiame Uiversity, Xiame 365, Chia mzeg@jigia.mu.edu.c b Uiversity of Ketucky, Leigto, Ketucky 456-46, USA cheg@cs.uky.edu Abstract
ON SOME PROPERTIES OF THE PICARD OPERATORS. Lucyna Rempulska and Karolina Tomczak
ACHIVUM MATHEMATICUM BNO Tomus 45 9, 5 35 ON SOME POPETIES OF THE PICAD OPEATOS Lucya empulska ad Karolia Tomczak Abstract. We cosider the Picard operators P ad P ;r i expoetial weighted spaces. We give
Characterizations Of (p, α)-convex Sequences
Applied Mathematics E-Notes, 172017, 77-84 c ISSN 1607-2510 Available free at mirror sites of http://www.math.thu.edu.tw/ ame/ Characterizatios Of p, α-covex Sequeces Xhevat Zahir Krasiqi Received 2 July
Weighted Approximation by Videnskii and Lupas Operators
Weighted Approximatio by Videsii ad Lupas Operators Aif Barbaros Dime İstabul Uiversity Departmet of Egieerig Sciece April 5, 013 Aif Barbaros Dime İstabul Uiversity Departmet Weightedof Approximatio Egieerig
Sequences 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
A Bernstein-Stancu type operator which preserves e 2
A. Şt. Uiv. Ovidius Costaţa Vol. 7), 009, 45 5 A Berstei-Stacu type operator which preserves e Igrid OANCEA Abstract I this paper we costruct a Berstei-Stacu type operator followig a J.P.Kig model. Itroductio
THE REPRESENTATION OF THE REMAINDER IN CLASSICAL BERNSTEIN APPROXIMATION FORMULA
Global Joural of Advaced Research o Classical ad Moder Geometries ISSN: 2284-5569, Vol.6, 2017, Issue 2, pp.119-125 THE REPRESENTATION OF THE REMAINDER IN CLASSICAL BERNSTEIN APPROXIMATION FORMULA DAN
(p, q)-baskakov-kantorovich Operators
Appl Math If Sci, No 4, 55-556 6 55 Applied Mathematics & Iformatio Scieces A Iteratioal Joural http://ddoiorg/8576/amis/433 p, q-basaov-katorovich Operators Vijay Gupta Departmet of Mathematics, Netaji
The 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,
(p, q)-type BETA FUNCTIONS OF SECOND KIND
Adv. Oper. Theory 6, o., 34 46 http://doi.org/.34/aot.69. ISSN: 538-5X electroic http://aot-math.org p, q-type BETA FUNCTIONS OF SECOND KIND ALI ARAL ad VIJAY GUPTA Commuicated by A. Kamisa Abstract. I
A 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
ON BLEIMANN, BUTZER AND HAHN TYPE GENERALIZATION OF BALÁZS OPERATORS
STUDIA UNIV. BABEŞ BOLYAI, MATHEMATICA, Volume XLVII, Number 4, December 2002 ON BLEIMANN, BUTZER AND HAHN TYPE GENERALIZATION OF BALÁZS OPERATORS OGÜN DOĞRU Dedicated to Professor D.D. Stacu o his 75
. Prelimiaries ad otatios Let f C q [ 1 1] be give, where q 0. The for a xed r such that q < r q + 1 we dee a polyomial H r f of degree at most + r 1
![. Prelimiaries ad otatios Let f C q [ 1 1] be give, where q 0. The for a xed r such that q < r q + 1 we dee a polyomial H r f of degree at most + r 1](/thumbs/91/104957031.jpg ". Prelimiaries ad otatios Let f C q [ 1 1] be give, where q 0. The for a xed r such that q < r q + 1 we dee a polyomial H r f of degree at most + r 1")
Poitwise Gopegauz Estimates for Iterpolatio T. Kilgore ad J. Presti Abstract We derive some ew poitwise estimates for the error i simultaeous approximatio of a fuctio f C q [ 1 1] ad its derivatives by
Product 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
MATH301 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
TR/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
MATH4822E 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
Chapter 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
Lecture 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
Real and Complex Analysis, 3rd Edition, W.Rudin
Real ad Complex Aalysis, 3rd ditio, W.Rudi Chapter 6 Complex Measures Yug-Hsiag Huag 206/08/22. Let ν be a complex measure o (X, M ). If M, defie { } µ () = sup ν( j ) : N,, 2, disjoit, = j { } ν () =
Approximation by Superpositions of a Sigmoidal Function
Zeitschrift für Aalysis ud ihre Aweduge Joural for Aalysis ad its Applicatios Volume 22 (2003, No. 2, 463 470 Approximatio by Superpositios of a Sigmoidal Fuctio G. Lewicki ad G. Mario Abstract. We geeralize
About the use of a result of Professor Alexandru Lupaş to obtain some properties in the theory of the number e 1
Geeral Mathematics Vol. 5, No. 2007), 75 80 About the use of a result of Professor Alexadru Lupaş to obtai some properties i the theory of the umber e Adrei Verescu Dedicated to Professor Alexadru Lupaş
Archimedes - 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,
NEW 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
A NOTE ON BOUNDARY BLOW-UP PROBLEM OF u = u p
A NOTE ON BOUNDARY BLOW-UP PROBLEM OF u = u p SEICK KIM Abstract. Assume that Ω is a bouded domai i R with 2. We study positive solutios to the problem, u = u p i Ω, u(x) as x Ω, where p > 1. Such solutios
APPROXIMATION PROPERTIES OF STANCU TYPE MEYER- KÖNIG AND ZELLER OPERATORS
Hacettepe Joural of Mathematics ad Statistics Volume 42 (2 (2013, 139 148 APPROXIMATION PROPERTIES OF STANCU TYPE MEYER- KÖNIG AND ZELLER OPERATORS Mediha Örkcü Received 02 : 03 : 2011 : Accepted 26 :
On an Operator Preserving Inequalities between Polynomials
Applied Mathematics 3 557-563 http://dxdoiorg/436/am3685 ublished Olie Jue (http://wwwscirorg/joural/am) O a Operator reservig Iequalities betwee olyomials Nisar Ahmad Rather Mushtaq Ahmad Shah Mohd Ibrahim
Chapter 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
Convergence 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
The 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
Chapter 3. Strong convergence. 3.1 Definition of almost sure convergence
Chapter 3 Strog covergece As poited out i the Chapter 2, there are multiple ways to defie the otio of covergece of a sequece of radom variables. That chapter defied covergece i probability, covergece i
DANIELL 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
On 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,
Several properties of new ellipsoids
Appl. Math. Mech. -Egl. Ed. 008 9(7):967 973 DOI 10.1007/s10483-008-0716-y c Shaghai Uiversity ad Spriger-Verlag 008 Applied Mathematics ad Mechaics (Eglish Editio) Several properties of ew ellipsoids
Chapter 10: Power Series
Chapter : Power Series 57 Chapter Overview: Power Series The reaso series are part of a Calculus course is that there are fuctios which caot be itegrated. All power series, though, ca be itegrated because
It 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
Chapter 2 The Solution of Numerical Algebraic and Transcendental Equations
Chapter The Solutio of Numerical Algebraic ad Trascedetal Equatios Itroductio I this chapter we shall discuss some umerical methods for solvig algebraic ad trascedetal equatios. The equatio f( is said
MAT1026 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
Power Series: A power series about the center, x = 0, is a function of x of the form
You are familiar with polyomial fuctios, polyomial that has ifiitely may terms. 2 p ( ) a0 a a 2 a. A power series is just a Power Series: A power series about the ceter, = 0, is a fuctio of of the form
Oscillation and Property B for Third Order Difference Equations with Advanced Arguments
Iter atioal Joural of Pure ad Applied Mathematics Volume 3 No. 0 207, 352 360 ISSN: 3-8080 (prited versio); ISSN: 34-3395 (o-lie versio) url: http://www.ijpam.eu ijpam.eu Oscillatio ad Property B for Third
SOME 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
1 Approximating Integrals using Taylor Polynomials
Seughee Ye Ma 8: Week 7 Nov Week 7 Summary This week, we will lear how we ca approximate itegrals usig Taylor series ad umerical methods. Topics Page Approximatig Itegrals usig Taylor Polyomials. Defiitios................................................
1.3 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(
Advanced 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
lim 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
Mathematical 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..
HOMEWORK #10 SOLUTIONS
Math 33 - Aalysis I Sprig 29 HOMEWORK # SOLUTIONS () Prove that the fuctio f(x) = x 3 is (Riema) itegrable o [, ] ad show that x 3 dx = 4. (Without usig formulae for itegratio that you leart i previous
APPROXIMATION BY BERNSTEIN-CHLODOWSKY POLYNOMIALS
Hacettepe Joural of Mathematics ad Statistics Volume 32 (2003), 1 5 APPROXIMATION BY BERNSTEIN-CHLODOWSKY POLYNOMIALS E. İbili Received 27/06/2002 : Accepted 17/03/2003 Abstract The weighted approximatio
Topics. 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
Chapter 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
REAL ANALYSIS II: PROBLEM SET 1 - SOLUTIONS
REAL ANALYSIS II: PROBLEM SET 1 - SOLUTIONS 18th Feb, 016 Defiitio (Lipschitz fuctio). A fuctio f : R R is said to be Lipschitz if there exists a positive real umber c such that for ay x, y i the domai
Exponential Functions and Taylor Series
MATH 4530: Aalysis Oe Expoetial Fuctios ad Taylor Series James K. Peterso Departmet of Biological Scieces ad Departmet of Mathematical Scieces Clemso Uiversity March 29, 2017 MATH 4530: Aalysis Oe Outlie
Journal of Multivariate Analysis. Superefficient estimation of the marginals by exploiting knowledge on the copula
Joural of Multivariate Aalysis 102 (2011) 1315 1319 Cotets lists available at ScieceDirect Joural of Multivariate Aalysis joural homepage: www.elsevier.com/locate/jmva Superefficiet estimatio of the margials
University of Manitoba, Mathletics 2009
Uiversity of Maitoba, Mathletics 009 Sessio 5: Iequalities Facts ad defiitios AM-GM iequality: For a, a,, a 0, a + a + + a (a a a ) /, with equality iff all a i s are equal Cauchy s iequality: For reals
Random 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.)
MASSACHUSETTS 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
Shivley s Polynomials of Two Variables
It. Joural of Math. Aalysis, Vol. 6, 01, o. 36, 1757-176 Shivley s Polyomials of Two Variables R. K. Jaa, I. A. Salehbhai ad A. K. Shukla Departmet of Mathematics Sardar Vallabhbhai Natioal Istitute of
Approximation properties of (p, q)-bernstein type operators
Acta Uiv. Saietiae, Mathematica, 8, 2 2016 222 232 DOI: 10.1515/ausm-2016-0014 Aroximatio roerties of, -Berstei tye oerators Zoltá Fita Deartmet of Mathematics, Babeş-Bolyai Uiversity, Romaia email: fzolta@math.ubbcluj.ro
6. Uniform distribution mod 1
6. Uiform distributio mod 1 6.1 Uiform distributio ad Weyl s criterio Let x be a seuece of real umbers. We may decompose x as the sum of its iteger part [x ] = sup{m Z m x } (i.e. the largest iteger which
An elementary proof that almost all real numbers are normal
Acta Uiv. Sapietiae, Mathematica, 2, (200 99 0 A elemetary proof that almost all real umbers are ormal Ferdiád Filip Departmet of Mathematics, Faculty of Educatio, J. Selye Uiversity, Rolícej šoly 59,
TRIGONOMETRIC 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
McGill University Math 354: Honors Analysis 3 Fall 2012 Solutions to selected problems
McGill Uiversity Math 354: Hoors Aalysis 3 Fall 212 Assigmet 3 Solutios to selected problems Problem 1. Lipschitz fuctios. Let Lip K be the set of all fuctios cotiuous fuctios o [, 1] satisfyig a Lipschitz
Some Tauberian theorems for weighted means of bounded double sequences
A. Ştiiţ. Uiv. Al. I. Cuza Iaşi. Mat. N.S. Tomul LXIII, 207, f. Some Tauberia theorems for weighted meas of bouded double sequeces Cemal Bele Received: 22.XII.202 / Revised: 24.VII.203/ Accepted: 3.VII.203
BIRKHOFF 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
A 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
A multivariate rational interpolation with no poles in R m
NTMSCI 3, No., 9-8 (05) 9 New Treds i Mathematical Scieces http://www.tmsci.com A multivariate ratioal iterpolatio with o poles i R m Osma Rasit Isik, Zekeriya Guey ad Mehmet Sezer Departmet of Mathematics,
A Simplified Binet Formula for k-generalized Fibonacci Numbers
A Simplified Biet Formula for k-geeralized Fiboacci Numbers Gregory P. B. Dresde Departmet of Mathematics Washigto ad Lee Uiversity Lexigto, VA 440 dresdeg@wlu.edu Zhaohui Du Shaghai, Chia zhao.hui.du@gmail.com
Math 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
Linear chord diagrams with long chords
Liear chord diagrams with log chords Everett Sulliva Departmet of Mathematics Dartmouth College Haover New Hampshire, U.S.A. everett..sulliva@dartmouth.edu Submitted: Feb 7, 2017; Accepted: Oct 7, 2017;
Math 2784 (or 2794W) University of Connecticut
ORDERS OF GROWTH PAT SMITH Math 2784 (or 2794W) Uiversity of Coecticut Date: Mar. 2, 22. ORDERS OF GROWTH. Itroductio Gaiig a ituitive feel for the relative growth of fuctios is importat if you really
Assignment 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
INVERSE THEOREMS OF APPROXIMATION THEORY IN L p,α (R + )
Electroic Joural of Mathematical Aalysis ad Applicatios, Vol. 3(2) July 2015, pp. 92-99. ISSN: 2090-729(olie) http://fcag-egypt.com/jourals/ejmaa/ INVERSE THEOREMS OF APPROXIMATION THEORY IN L p,α (R +
International Journal of Mathematical Archive-3(4), 2012, Page: Available online through ISSN
Iteratioal Joural of Mathematical Archive-3(4,, Page: 544-553 Available olie through www.ima.ifo ISSN 9 546 INEQUALITIES CONCERNING THE B-OPERATORS N. A. Rather, S. H. Ahager ad M. A. Shah* P. G. Departmet
f 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
IRRATIONALITY MEASURES, IRRATIONALITY BASES, AND A THEOREM OF JARNÍK 1. INTRODUCTION
IRRATIONALITY MEASURES IRRATIONALITY BASES AND A THEOREM OF JARNÍK JONATHAN SONDOW ABSTRACT. We recall that the irratioality expoet µα ( ) of a irratioal umber α is defied usig the irratioality measure
Integrable 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
New Results for the Fibonacci Sequence Using Binet s Formula
Iteratioal Mathematical Forum, Vol. 3, 208, o. 6, 29-266 HIKARI Ltd, www.m-hikari.com https://doi.org/0.2988/imf.208.832 New Results for the Fiboacci Sequece Usig Biet s Formula Reza Farhadia Departmet
The picture in figure 1.1 helps us to see that the area represents the distance traveled. Figure 1: Area represents distance travelled
1 Lecture : Area Area ad distace traveled Approximatig area by rectagles Summatio The area uder a parabola 1.1 Area ad distace Suppose we have the followig iformatio about the velocity of a particle, how
Council 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
If 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?
![If 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?](/thumbs/89/99848926.jpg "If 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
Continuous Functions
Cotiuous Fuctios Q What does it mea for a fuctio to be cotiuous at a poit? Aswer- I mathematics, we have a defiitio that cosists of three cocepts that are liked i a special way Cosider the followig defiitio
Ma 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
Real Analysis Fall 2004 Take Home Test 1 SOLUTIONS. < ε. Hence lim
Real Aalysis Fall 004 Take Home Test SOLUTIONS. Use the defiitio of a limit to show that (a) lim si = 0 (b) Proof. Let ε > 0 be give. Defie N >, where N is a positive iteger. The for ε > N, si 0 < si
MIDTERM 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
ANSWERS 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
LECTURES 5 AND 6: CONVERGENCE TESTS
LECTURES 5 AND 6: CONVERGENCE TESTS I the previous lecture we saw a very basic test for covergece - the compariso test. Here we develop slightly more sophisticated machiery for determiig the covergece
On the behavior at infinity of an integrable function
O the behavior at ifiity of a itegrable fuctio Emmauel Lesige To cite this versio: Emmauel Lesige. O the behavior at ifiity of a itegrable fuctio. The America Mathematical Mothly, 200, 7 (2), pp.75-8.
Inverse theorem for the iterates of modified Bernstein type polynomials
Stud. Uiv. Babeş-Bolyai Math. 5924, No. 3, 33 35 Iverse theorem for the iterates of modified Berstei type polyomials T.A.K. Siha, P.N. Agrawal ad K.K. Sigh Abstract. Gupta ad Maheshwari [2] itroduced a
I v r e s n a d di t r th o e r m e s o f r monoton e app o r i x m t a ion b a t 1. I r n o n a
Iverse ad direct orems or mootoe aroxima tio ukeia Abdulla Al- B ermai B abylo Uiversity ciece College For Wome C omuter ciece Deartmet A bstract We rove that i s icreasig uctio o i or e ac h re a icreasig
Approximation in Holder norms with higher order. Dedicated to the professors of mathematics
Rostock. Math. Kolloq. 51, 33{50 (1997) Subject Classicatio (AMS) 41A05, 41A10, 65D05 Jurge Presti; Alia Stosiek Approximatio i Holder orms with higher order diereces Dedicated to the professors of mathematics
Math 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
Find quadratic function which pass through the following points (0,1),(1,1),(2, 3)... 11
Adrew Powuk - http://www.powuk.com- Math 49 (Numerical Aalysis) Iterpolatio... 4. Polyomial iterpolatio (system of equatio)... 4.. Lier iterpolatio... 5... Fid a lie which pass through (,) (,)... 8...
Rademacher 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