Rate of convergence for certain families of summation integral type operators
|
|
- Margaret Walker
- 5 years ago
- Views:
Transcription
1 J Math Anal Appl wwwelseviercom/locate/jmaa Rate of convergence for certain families of summation integral type operators Vijay Gupta a,,mkgupta b a School of Applied Sciences, Netaji Subhas Institute of Technology, Sector 3 Dwarka, New Delhi 1145, India b Department of Mathematics, Ch Charan Singh University, Meerut 254, India Received 14 August 23 Available online 8 July 24 Submitted by HM Srivastava Abstract In this paper, we study the rate of convergence for the iterative combinations of a certain family of linear positive operators in terms of higher-order integral modulus of smoothness We have used the technique of linear approimating method, namely Steklov mean, to prove the main result In the end, we propose some other sequences of linear positive operators and obtain the recurrence formulae for the central moments, Voronovskaja type asymptotic formulae and error estimations 24 Elsevier Inc All rights reserved Keywords: Linear positive operators; Iterative combinations; Steklov mean; Modulus of smoothness 1 Introduction Very recently, Srivastava and Gupta [8] defined a certain family of linear positive operators, which include some well-known operators as special cases, they estimated the rate of convergence for functions of bounded variation For f L p [,, p 1, the mied summation integral type operators see, eg, [2,5] discussed in this paper are defined as * Corresponding author addresses: vijay@nsitacin V Gupta, mkgupta22@hotmailcom MK Gupta X/$ see front matter 24 Elsevier Inc All rights reserved doi:1116/jjmaa24429
2 where V n f, = V Gupta, MK Gupta / J Math Anal Appl = n K n, t = n K n, tf t dt b n,v s n,v 1 tf t dt 1 n f, 1 b n,v s n,v 1 t 1 n δt, δt being Dirac delta function, and n v 1 b n,v = v 1 n v nt ntv, s v n,v t = e v! It has been easily verified that the operators V n are linear positive operators It turns out that the order of approimation by these operators is at best On 1, n,howsoever smooth the function may be Thus to improve the order of approimation, we may consider some combinations of the operators 1 In this contet, the linear combinations of the operators V n were recently considered in [2] for improving the order of approimation Yet another approach for improving the order of approimation is the iterative combinations due to Micchelli [7], who improved the order of approimation of Bernstein polynomials; the results of [7] were later sharpened in [3] One cannot apply the iterative combinations easily for all linear positive operators; for eample, in the sequence of operators see [1,6,8] t is not mapped eactly to, so it is much more difficult to handle the iterative combinations for the operators discussed in these papers But for the operators 1, we have V n t, =, which is the interesting property of these operators V n, thus one can consider the iterative combinations of 1 to improve the rate of convergence Here we consider the iterative combinations V n,k f, of the operators V n,whichforf L p [, are defined as V n,k f, = [ I I V n k] ft, k = 1 r1 k V r r n ft,, 2 where V r n denotes the rth iterate superposition of the operator V n It is obvious that Bernstein polynomials summation type operators are as such not L p approimation methods Nevertheless several linear positive operators of summation type have been appropriately modified to become L p -approimation methods The operators studied in the present paper make it possible to approimate pth 1 p< power Lebesgue integrable functions on [, This, along with the application of iterative combinations V n,k of the operators V n, motivated us to study in this direction Several other approimation properties on some summation integral type operators were studied in [1,5,6], etc In the present paper, we consider iterative combination due to Micchelli for the operators V n defined by 2 and obtain an error estimate in terms of higher-order integral modulus r=1
3 61 V Gupta, MK Gupta / J Math Anal Appl of smoothness in L p -norm In the end, we propose some other mied summation integral type operators and give their central moments in the form of recurrence relations 2 Auiliary results In this section we mention some basic lemmas Lemm Let the function µ n,m, m N the set of non-negative integers be defined by µ n,m = n b n,v s n,v 1 tt m dt m 1 n, < Then 2 µ n, = 1, µ n,1 =, µ n,2 = n Furthermore, the following recurrence relation holds: nµ n,m1 = 1 µ 1 n,m m2 µ n,m 1 mµ n,m, m 1 It follows from the above recurrence relation that µ n,m = O n [m1/2] n, < ; m N Remark 1 For each m N the mth order moment µ {q} n,m for the operators Vn q is defined by µ {q} n,m = Vn q t m, 3 For q = 1, µ 1 n,m µ n,m Lemm There holds the following relation: m m j µ n,m {q1} m 1 = j i! Di µ {q} n,m j µ n,ij, j= i= D d d Lemma 3 We have µ {q} n,m = O n [m1/2] n, < ; m N Proof The proof of the above lemma easily follows by using the principle of mathematical induction and Lemm By direct application of Lemmas 2 and 3, we have V n,k t i, = O n k n, 4
4 V Gupta, MK Gupta / J Math Anal Appl where V n,k is the iterative combination defined by 2 Throughout the present paper, we consider < < <a 3 <b 3 <b 2 <b 1 < and I r =[a r,b r ], r = 113 Lemma 4 Let f L p [,, p>1,alsoletf 2k eists on I 1 with f 2k 1 ACI 1 and f 2k L p I 1, then for n sufficiently large, we have Vn,k f, f Lp I 2 C 1n k{ f 2k Lp I 2 f L p [, } 5 Also, if f L 1 [,, f 2k 1 L 1 I 1 with f 2k 2 ACI 1 and f 2k 1 BVI 1, then for n sufficiently large, we have V n,k f, f L1 I 2 C 2n k{ f 2k 1 BVI1 f 2k 1 L1 I 2 f L1 [, }, 6 where the constants C 1 and C 2 are independent of f and n Proof First let p>1 By the hypothesis, for all t [, and I 2,wehave 2k 1 V n,k f, f= i=1 f i V n,k t i, i! 1 2k 1! V n,k φt t t w 2k 1 f 2k w dw, V n,k Ft, 1 φt, = E 1 E 2 E 3, where φt denotes the characteristic function of I 1 and 2k 1 t i Ft,= ft f i i! i= Applying 4 and the interpolation property due to Goldberg and Meir [4], we obtain 2k 1 E 1 Lp I 2 C 3 n k f i Lp I 2 C 4n k{ f Lp I 2 f 2k Lp I 2 } i=1 To estimate E 2,letH f be the Hardy Littlewood maimal function [9] of f 2k on I 1 Using Holder s inequality and Lemm, we have E 2 Vn φt t t w 2k 1 f 2k w dw, { V n φt t 2kq, } 1/q { V n φt Hf t p, } 1/p b1 C 5 n k K n, t hf t p dt 1/p
5 612 V Gupta, MK Gupta / J Math Anal Appl Therefore, we have E 2 Lp I 2 C 6 n k f 2k Lp I 1 For t [, \[,b 1 ], I 2, there eists δ> such that t δ Thus E3 V n Ft, 1 φt, δ 2k V n Ft, t 2k, δ [V 2k n ft t 2k, 2k 1 = E31 E 32 Applying Holder s inequality and Lemm, we have δ 2k{ V n ft p, } 1/p{ Vn t 2kq, } 1/q E 31 C 7 n k{ V n ft p } 1/p, Finally, by Fubini s theorem, we obtain E 31 p L p I 2 = b2 i= E p 31 d C8 n pk f p L p [,, also by the interpolation property [4], we have E 32 Lp I 2 C 9n k{ f Lp I 2 f 2k Lp I 2 } Thus E 3 Lp I 2 C 1 n k{ f Lp [, f 2k Lp I 2 } f i V n t 2ki, ] i! Combining E 1, E 2 and E 3, we get 5 Net suppose p = 1 By the hypothesis, for almost all I 2 and for all t [,,we have 2k 1 V n,k f, f= i=1 f i V n,k t i, i! 1 2k 1! V n,k φt t t w 2k 1 f 2k 1 w dw, V n,k Ft, 1 φt, = H 1 H 2 H 3 Applying Lemm and the interpolation property due to Goldberg and Meir [4], we have H 1 L1 I 2 C 11 n k{ f L1 I 2 f 2k 1 L1 I 2 } Net, for the estimation of H 2 we proceed as follows:
6 V Gupta, MK Gupta / J Math Anal Appl H 2 V n φt b 2 b 1 t t w 2k 1 f 2k 1 w dw, K n, t t 2k 1 t L1 I 2 df 2k 1 w dt d For each n there eists a nonnegative integer r = rn satisfying rn 1/2 < mab 1, b 2 r 1n 1/2 H 2 r l= b 2 { l1n 1/2 ln 1/2 φtk n, t t l1n 1/2 r l= b 2 { ln 1/2 2k 1 φw df 2k 1 w } dt d l1n 1/2 φtk n, t t l1n 1/2 φt 2k 1 df 2k 1 w } dt d Suppose φ w,c,d denotes the characteristic function of the interval [ cn 1/2, dn 1/2 ],wherec,d are positive integers Then we have { r H2 l 4 n 2 l=1 b1 b 2 l1n 1/2 φ w,,l 1 { r l 4 n 2 l=1 b1 ln 1/2 φtk n, t t b 2 ln 1/2 φ w, l 1, df 2k 1 w } dt 2k3 l1n 1/2 φtk n, t t df 2k 1 w } dt 2k3
7 614 V Gupta, MK Gupta / J Math Anal Appl b 2 { a1 n 1/2 2k 1 φtk n, t t n 1/2 b1 φ w, 1, 1 df 2k 1 w } dt d Applying Lemm and using Fubini s theorem, we obtain { r b1 w H2 C 12n 2k1/2 l 4 df 2k 1 w l=1 b 1 wl1n 1/2 w w l1n 1/2 d df d 2k 1 w b 1 wn 1/2 w n 1/2 d df 2k 1 w } C 13 n k f 2k 1 w BVI1 Thus H 2 L1 I 2 C 14 n k f 2k 1 BVI1 Finally we estimate H 3 ;forallt [, \[,b 1 ] and all [,b 2 ], we choose δ> such that t δ Thus H L1 3 I 2 Vn Ft, 1 φt, L1 I 2 b 2 2k 1 r= K n, t ft 1 φt dt d 1 r! b 2 K n, t f r t t r 1 φt dt d Applying Lemm and the interpolation property due to Goldberg and Meir [4], the second term in the right-hand side of the above epression is bounded by C 15 n k { f L1 [, f 2k 1 L1 I 2 } The first term is estimated as follows For t sufficiently large there eist positive constants C 16 and C 17 satisfying the condition t 2k /t 2k 1>C 17,forallt C 16, t [,b 2 ] Therefore, using Fubini s theorem and Lemm, we have b 2 K n, t ft 1 φt dt d C 16 b 2 b 2 C 16 K n, t ft 1 φt dt d
8 V Gupta, MK Gupta / J Math Anal Appl C 18 n k C 16 ft dt 1 C 19 n k f L1 [, C 17 C 16 b 2 t 2k K n, t ft ddt t 2k 1 Collecting the estimates of H 1, H 2 and H 3, the required estimate 6 follows 3 Error estimation We first define the linear approimating method viz Steklov mean which is the main tool to prove the error estimation Let f L p [a,b],1 p<, then for sufficiently small η> the Steklov mean f η,m of mth order corresponding to f is defined by f η,m = η m η/2 η/2 η/2 η/2 { ft 1 m 1 m u ft} dt 1 dt 2 dt m, where u = m i=1 t i, t [a,b], and m η ft is the mth forward differences with step length η It is verified that f η,m has derivatives with respect to order m, f η,m n 1 AC[,b 1 ] and mth derivatives of f η,m eists almost everywhere and belonging to L p [,b 1 ]Moreover, i f r η,m Lp [,b 2 ] K 1 η r ω r f,η,p,[,b 1 ], r = 1, 2,,m; ii f f η,m Lp [,b 2 ] K 2 ω m f,η,p,[,b 1 ]; iii f η,m Lp [,b 2 ] K 3 f Lp [,b 1 ]; iv f m η,m Lp [,b 2 ] K 4 η m f Lp [,b 1 ], where K i s i = 1, 2, 3, 4 are certain constants independent of f and η Theorem 1 Let f L p [,, p 1 Then for n sufficiently large, we have Vn,k f, f Lp I 2 C { 2 ω2k f,n 1/2,p,I 1 n k } f Lp [,, where C 2 is a constant independent of f and n Proof By linearity property, we have Vn,k f, f Lp I 2 Vn,k f f η,2k, Lp I 2 Vn,k f η,2k, f η,2k Lp I 2 f η,2k f Lp I 2 = First we estimate 1,letχt be the characteristic function of the interval I 3 Thus
9 616 V Gupta, MK Gupta / J Math Anal Appl V n f fη,2k t, = V n χtf fη,2k t, V n 1 χt f fη,2k t, Now applying Fubini s theorem, we have b 2 Vn χt f fη,2k t, p d b 3 b 2 K n, t f f η,2k p ddt a 3 f f η,2k p L p I 3 Net using Lemm and Holder s inequality, for allp 1, we have V n 1 χt f fη,2k t, Lp I 2 C 21n k f f η,2k Lp [, Thus 1 C 22 { ω2k f,η,p,i 1 n k f Lp [, } Also it is observed that f 2k 1 η,2k BVI3 = f 2k η,2k L 1 I 3 Applying Lemma 4 and the property of Steklov mean s, we get 2 C 23 n k{ 2k f Lp I 3 f } η,2k Lp {, η,2k C 24 n k{ η 2k ω 2k f,η,p,i 1 f Lp [, } Finally, by the property of Steklov mean s, we get 3 f η,2k f Lp I 2 C 25 ω 2k f,η,p,i 1 Choosing η = n 1/2, and combining the estimates of 1, 2 and 3, the theorem follows 4 Some eamples Let s n,v, b n,v and p n,v be Szasz, Baskakov and Beta basis functions, respectively, which are defined as n nv n v 1 s n,v = e, b n,v = v 1 n v, v! v p n,v = 1 Bn,v 1 v 1 n v 1 To approimate Lebesgue integrable functions on the interval [,, we propose some other new summation integral type operators
10 V Gupta, MK Gupta / J Math Anal Appl Szasz Baskakov type operators: S n f, = n 1 s n,v b n,v 1 tf t dt e n f If S n,m S n t m,, then they satisfy the recurrence relation: n m 2S n,m1 = [ S n,m 1 m 2S n,m 1 ] [ m 2m 1 ] S n,m ; n>m 2 Beta Szasz type operators: P n f, = p n,v s n,v 1 tf t dt 1 n 1 f If P n,m P n t m,, we have the recurrence relation: np n,m1 = 1 [ P 1 n,m mp n,m 1 ] m P n,m mp n,m 1 Szasz Beta type operators: B n f, = s n,v p n,v 1 tf t dt e n f If B n,m B n t m,, we have the recurrence relation: n m 1B n,m1 = [ B n,m 1 m 2B n,m 1 ] [ m 1 2m ] B n,m ; n>m 1 From the above recurrence relations for different operators, we obtain the following Voronovskaja type asymptotic formulae and error estimations Theorem 2 Let f be bounded and integrable on [, and has second derivative at a point [,,then i ii iii iv lim n[ V n f, f ] = n 2 f 2 ; 2 lim n[ S n f, f ] = 2f 1 n lim n n[ P n f, f ] = f 1 lim n[ B n f, f ] = f 1 n 2 f 2 ; 2 2 f 2 ; 2 2 f 2 2
11 618 V Gupta, MK Gupta / J Math Anal Appl Theorem 3 Let f C[, Then for [,a], <a<, we have i V n f, f C[,a] [ 1 aa 2 ] ω f,n 1/2 ; ii Sn f, f [ C[,a] 1 n[naa 2 ] 6a2 ] ω f,n 1/2 ; n 2n 3 iii Pn f, f [ ] a1 an 2 na C[,a] 1 ω f,n 1/2 ; n iv B n f, f [ ] C[,a] 1 n[a2 n 2 2na] ω f,n 1/2, n 1n 2 where ωf, is the modulus of continuity of f and is the sup norm on the interval [,a] Remark 2 It is observed that the mied summation integral type operators are easier to construct but each of these operators have different properties in approimation theory It may be remarked here that the above mentioned operators S n, P n and B n are linear positive operators, but we cannot apply the iterative combinations for these operators the reason is mentioned in the introduction Therefore, for these operators we cannot obtain the result analogous to our Theorem 1 Also as another eample we cannot obtain the rate of convergence for bounded variation functions easily for the above mentioned operators analogous to the results of [6,8] The main problem is that one cannot find a relation between summation and integration having different basis functions These problems are still unresolved and these may be treated as open problems for the readers Acknowledgment The authors are grateful to the reviewers for the critical review, leading to a better presentation of the paper References [1] PN Agrawal, KJ Thamer, Approimation of unbounded functions by a new sequence of linear positive operators, J Math Anal Appl [2] PN Agrawal, AJ Mohammad, On L p approimation by a linear combination of a new sequence of linear positive operators, Turkish J Math [3] PN Agrawal, HS Kasana, On the iterative combinations of Bernstein polynomials, Demonstratio Math [4] S Goldberg, V Meir, Minimum moduli of ordinary differential operators, Proc London Math Soc [5] MK Gupta, V Vasishtha, On the iterative combinations of a sequence of linear positive operators, Math Comput Modelling [6] V Gupta, Rate of convergence by a new sequence of linear positive operators, Comput Math Appl [7] CA Micchelli, The saturation class and iterates of the Bernstein polynomials, J Appro Theory [8] HM Srivastava, V Gupta, A certain family of summation integral type operators, Math Comput Modelling [9] AF Timan, Theory of Approimation of Functions of a Real Variable, Hindustan, Delhi, 1966
Journal of Inequalities in Pure and Applied Mathematics
Journal of Inequalities in Pure and Applied Mathematics ON SIMULTANEOUS APPROXIMATION FOR CERTAIN BASKAKOV DURRMEYER TYPE OPERATORS VIJAY GUPTA, MUHAMMAD ASLAM NOOR AND MAN SINGH BENIWAL School of Applied
More informationSIMULTANEOUS APPROXIMATION BY A NEW SEQUENCE OF SZÃSZ BETA TYPE OPERATORS
REVISTA DE LA UNIÓN MATEMÁTICA ARGENTINA Volumen 5, Número 1, 29, Páginas 31 4 SIMULTANEOUS APPROIMATION BY A NEW SEQUENCE OF SZÃSZ BETA TYPE OPERATORS ALI J. MOHAMMAD AND AMAL K. HASSAN Abstract. In this
More informationJournal of Inequalities in Pure and Applied Mathematics
Journal of Inequalities in Pure and Applied Mathematics ON A HYBRID FAMILY OF SUMMATION INTEGRAL TYPE OPERATORS VIJAY GUPTA AND ESRA ERKUŞ School of Applied Sciences Netaji Subhas Institute of Technology
More informationV. Gupta, A. Aral and M. Ozhavzali. APPROXIMATION BY q-szász-mirakyan-baskakov OPERATORS
F A S C I C U L I M A T H E M A T I C I Nr 48 212 V. Gupta, A. Aral and M. Ozhavzali APPROXIMATION BY -SZÁSZ-MIRAKYAN-BASKAKOV OPERATORS Abstract. In the present paper we propose the analogue of well known
More informationDIRECT AND INVERSE THEOREMS FOR SZÂSZ-LUPAS TYPE OPERATORS IN SIMULTANEOUS APPROXIMATION. Naokant Deo. 1. Introduction
MATEMATIQKI VESNIK 58 26), 19 29 UDK 517.984 originalni nauqni rad research paper DIRECT AND INVERSE THEOREMS FOR SZÂSZ-LUPAS TYPE OPERATORS IN SIMULTANEOUS APPROXIMATION Naokant Deo Abstract. In this
More informationON THE (p, q) STANCU GENERALIZATION OF A GENUINE BASKAKOV- DURRMEYER TYPE OPERATORS
International Journal of Analysis and Applications ISSN 91-869 Volume 15 Number 17 18-145 DOI: 1894/91-869-15-17-18 ON THE p q STANCU GENERALIZATION OF A GENUINE BASKAKOV- DURRMEYER TYPE OPERATORS İSMET
More informationA GENERALIZATION OF POST-WIDDER OPERATORS
ANALELE ŞTIINŢIFICE ALE UNIVERSITĂŢII AL.I. CUZA DIN IAŞI S.N.) MATEMATICĂ Tomul LXII 16 f.1 A GENERALIZATION OF POST-WIDDER OPERATORS BASED ON -INTEGERS BY DIDEM AYDIN ALI ARAL and GÜLEN BAŞCANBAZ-TUNCA
More informationarxiv: v2 [math.ca] 23 Feb 2016
On p, q Baskakov-Durrmeyer-Stancu Operators Vishnu Narayan Mishra a,b,1, Shikha Pandey a a Department of Applied Mathematics & Humanities, Sardar Vallabhbhai National Institute of Technology, Ichchhanath
More informationMOMENTS OF A q BASKAKOV BETA OPERATORS IN CASE 0 < q < Introduction
Journal of Classical Analysis Volume 2, Number 1 213, 9 22 doi:1.7153/jca-2-2 MOMENTS OF A BASKAKOV BETA OPERATORS IN CASE < < 1 A. R. GAIROLA, P.N.AGRAWAL, G.DOBHAL AND K. K. SINGH Abstract. In this paper
More informationInternational Journal of Pure and Applied Mathematics Volume 60 No ,
International Journal of Pure and Applied Mathematics Volume 60 No. 3 200, 259-267 ON CERTAIN CLASS OF SZÁSZ-MIRAKYAN OPERATORS IN EXPONENTIAL WEIGHT SPACES Lucyna Rempulska, Szymon Graczyk 2,2 Institute
More informationSome Approximation Properties of Szasz-Mirakyan-Bernstein Operators
EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol 7, No 4, 04, 49-48 ISSN 307-5543 wwwejpamcom Some Approximation Properties of Szasz-Mirakyan-Bernstein Operators Tuncay Tunç, Ersin Şimşek, Department
More informationReceived 8 June 2003 Submitted by Z.-J. Ruan
J. Math. Anal. Appl. 289 2004) 266 278 www.elsevier.com/locate/jmaa The equivalence between the convergences of Ishikawa and Mann iterations for an asymptotically nonexpansive in the intermediate sense
More informationL p -convergence of Bernstein Kantorovich-type operators. by Michele Campiti (Bari) and Giorgio Metafune (Lecce)
ANNALES POLONICI MATHEMATICI LXIII.3 (996) L p -convergence of Bernstein Kantorovich-type operators by Michele Campiti (Bari) and Giorgio Metafune (Lecce) Abstract. We study a Kantorovich-type modification
More informationEquivalence constants for certain matrix norms II
Linear Algebra and its Applications 420 (2007) 388 399 www.elsevier.com/locate/laa Equivalence constants for certain matrix norms II Bao Qi Feng a,, Andrew Tonge b a Department of Mathematical Sciences,
More informationViscosity approximation methods for nonexpansive nonself-mappings
J. Math. Anal. Appl. 321 (2006) 316 326 www.elsevier.com/locate/jmaa Viscosity approximation methods for nonexpansive nonself-mappings Yisheng Song, Rudong Chen Department of Mathematics, Tianjin Polytechnic
More informationThe Hausdorff measure of a class of Sierpinski carpets
J. Math. Anal. Appl. 305 (005) 11 19 www.elsevier.com/locate/jmaa The Hausdorff measure of a class of Sierpinski carpets Yahan Xiong, Ji Zhou Department of Mathematics, Sichuan Normal University, Chengdu
More informationMATH39001 Generating functions. 1 Ordinary power series generating functions
MATH3900 Generating functions The reference for this part of the course is generatingfunctionology by Herbert Wilf. The 2nd edition is downloadable free from http://www.math.upenn. edu/~wilf/downldgf.html,
More informationTaylor Series and Asymptotic Expansions
Taylor Series and Asymptotic Epansions The importance of power series as a convenient representation, as an approimation tool, as a tool for solving differential equations and so on, is pretty obvious.
More informationPiecewise Smooth Solutions to the Burgers-Hilbert Equation
Piecewise Smooth Solutions to the Burgers-Hilbert Equation Alberto Bressan and Tianyou Zhang Department of Mathematics, Penn State University, University Park, Pa 68, USA e-mails: bressan@mathpsuedu, zhang
More informationThe Hardy Operator and Boyd Indices
The Hardy Operator and Boyd Indices Department of Mathematics, University of Mis- STEPHEN J MONTGOMERY-SMITH souri, Columbia, Missouri 65211 ABSTRACT We give necessary and sufficient conditions for the
More informationJournal of Mathematical Analysis and Applications
J. Math. Anal. Appl. 35 29) 276 282 Contents lists available at ScienceDirect Journal of Mathematical Analysis and Applications www.elsevier.com/locate/jmaa A Turán-type inequality for the gamma function
More informationEuler-Maclaurin summation formula
Physics 4 Spring 6 Euler-Maclaurin summation formula Lecture notes by M. G. Rozman Last modified: March 9, 6 Euler-Maclaurin summation formula gives an estimation of the sum N in f i) in terms of the integral
More informationASYMPTOTIC BEHAVIOR OF SOLUTIONS OF DISCRETE VOLTERRA EQUATIONS. Janusz Migda and Małgorzata Migda
Opuscula Math. 36, no. 2 (2016), 265 278 http://dx.doi.org/10.7494/opmath.2016.36.2.265 Opuscula Mathematica ASYMPTOTIC BEHAVIOR OF SOLUTIONS OF DISCRETE VOLTERRA EQUATIONS Janusz Migda and Małgorzata
More informationHW #1 SOLUTIONS. g(x) sin 1 x
HW #1 SOLUTIONS 1. Let f : [a, b] R and let v : I R, where I is an interval containing f([a, b]). Suppose that f is continuous at [a, b]. Suppose that lim s f() v(s) = v(f()). Using the ɛ δ definition
More informationNested Inequalities Among Divergence Measures
Appl Math Inf Sci 7, No, 49-7 0 49 Applied Mathematics & Information Sciences An International Journal c 0 NSP Natural Sciences Publishing Cor Nested Inequalities Among Divergence Measures Inder J Taneja
More informationBounds of Hausdorff measure of the Sierpinski gasket
J Math Anal Appl 0 007 06 04 wwwelseviercom/locate/jmaa Bounds of Hausdorff measure of the Sierpinski gasket Baoguo Jia School of Mathematics and Scientific Computer, Zhongshan University, Guangzhou 5075,
More informationMean convergence theorems and weak laws of large numbers for weighted sums of random variables under a condition of weighted integrability
J. Math. Anal. Appl. 305 2005) 644 658 www.elsevier.com/locate/jmaa Mean convergence theorems and weak laws of large numbers for weighted sums of random variables under a condition of weighted integrability
More informationBiorthogonal Spline Type Wavelets
PERGAMON Computers and Mathematics with Applications 0 (00 1 0 www.elsevier.com/locate/camwa Biorthogonal Spline Type Wavelets Tian-Xiao He Department of Mathematics and Computer Science Illinois Wesleyan
More informationMills ratio: Monotonicity patterns and functional inequalities
J. Math. Anal. Appl. 340 (008) 136 1370 www.elsevier.com/locate/jmaa Mills ratio: Monotonicity patterns and functional inequalities Árpád Baricz Babeş-Bolyai University, Faculty of Economics, RO-400591
More informationPersistence and global stability in discrete models of Lotka Volterra type
J. Math. Anal. Appl. 330 2007 24 33 www.elsevier.com/locate/jmaa Persistence global stability in discrete models of Lotka Volterra type Yoshiaki Muroya 1 Department of Mathematical Sciences, Waseda University,
More informationON THE MOMENTS OF ITERATED TAIL
ON THE MOMENTS OF ITERATED TAIL RADU PĂLTĂNEA and GHEORGHIŢĂ ZBĂGANU The classical distribution in ruins theory has the property that the sequence of the first moment of the iterated tails is convergent
More informationA note on fixed point sets in CAT(0) spaces
J. Math. Anal. Appl. 320 (2006) 983 987 www.elsevier.com/locate/jmaa Note A note on fied point sets in CAT(0) spaces P. Chaoha,1, A. Phon-on Department of Mathematics, Faculty of Science, Chulalongkorn
More informationNote on Absolute Matrix Summability Factors
Int. J. Contemp. Math. Sciences, Vol. 5, 2010, no. 35, 1737-1745 Note on Absolute Matrix Summability Factors Pragati Sinha, Hirdesh Kumar 1 and Vipin Kumar Gupta Department of Mathematics S. M. P.G.) College
More informationViscosity Iterative Approximating the Common Fixed Points of Non-expansive Semigroups in Banach Spaces
Viscosity Iterative Approximating the Common Fixed Points of Non-expansive Semigroups in Banach Spaces YUAN-HENG WANG Zhejiang Normal University Department of Mathematics Yingbing Road 688, 321004 Jinhua
More informationOn the Convergence of the Summation Formulas Constructed by Using a Symbolic Operator Approach
On the Convergence of the Summation Formulas Constructed by Using a Symbolic Operator Approach Tian-Xiao He 1, Leetsch C. Hsu 2, and Peter J.-S. Shiue 3 1 Department of Mathematics and Computer Science
More informationExistence of global solutions of some ordinary differential equations
J. Math. Anal. Appl. 340 (2008) 739 745 www.elsevier.com/locate/jmaa Existence of global solutions of some ordinary differential equations U. Elias Department of Mathematics, Technion IIT, Haifa 32000,
More informationON THE DEGREE OF APPROXIMATION BY POSITIVE LINEAR OPERATORS USING THE B SUMMABILITY METHOD.* A.S. RANADIVE and S.P. SINGH. for n +I ::; m ::; n +p
Revista Colombiana de Matematicas Vol. XXV (1991) pgs. 1-10 ON THE DEGREE OF APPROXIMATION BY POSITIVE LINEAR OPERATORS USING THE B SUMMABILITY METHOD.* by A.S. RANADIVE and S.P. SINGH ABSTRACT. The aim
More informationW. Lenski and B. Szal ON POINTWISE APPROXIMATION OF FUNCTIONS BY SOME MATRIX MEANS OF CONJUGATE FOURIER SERIES
F A S C I C U L I M A T H E M A T I C I Nr 55 5 DOI:.55/fascmath-5-7 W. Lenski and B. Szal ON POINTWISE APPROXIMATION OF FUNCTIONS BY SOME MATRIX MEANS OF CONJUGATE FOURIER SERIES Abstract. The results
More informationBasicity of System of Exponents with Complex Coefficients in Generalized Lebesgue Spaces
Gen. Math. Notes, Vol. 3, No., September 25, pp.2-2 ISSN 229-784; Copyright c ICSRS Publication, 25 www.i-csrs.org Available free online at http://www.geman.in Basicity of System of Exponents with Complex
More informationExistence and multiple solutions for a second-order difference boundary value problem via critical point theory
J. Math. Anal. Appl. 36 (7) 511 5 www.elsevier.com/locate/jmaa Existence and multiple solutions for a second-order difference boundary value problem via critical point theory Haihua Liang a,b,, Peixuan
More informationProblem Set 9 Solutions
8.4 Problem Set 9 Solutions Total: 4 points Problem : Integrate (a) (b) d. ( 4 + 4)( 4 + 5) d 4. Solution (4 points) (a) We use the method of partial fractions to write A B (C + D) = + +. ( ) ( 4 + 5)
More informationImpulsive stabilization of two kinds of second-order linear delay differential equations
J. Math. Anal. Appl. 91 (004) 70 81 www.elsevier.com/locate/jmaa Impulsive stabilization of two kinds of second-order linear delay differential equations Xiang Li a, and Peixuan Weng b,1 a Department of
More informationJournal of Combinatorial Theory, Series B
Journal of Combinatorial Theory, Series B 100 2010) 161 170 Contents lists available at ScienceDirect Journal of Combinatorial Theory, Series B www.elsevier.com/locate/jctb The average order of a subtree
More informationCHAPTER III THE PROOF OF INEQUALITIES
CHAPTER III THE PROOF OF INEQUALITIES In this Chapter, the main purpose is to prove four theorems about Hardy-Littlewood- Pólya Inequality and then gives some examples of their application. We will begin
More informationTwo dimensional exterior mixed problem for semilinear damped wave equations
J. Math. Anal. Appl. 31 (25) 366 377 www.elsevier.com/locate/jmaa Two dimensional exterior mixed problem for semilinear damped wave equations Ryo Ikehata 1 Department of Mathematics, Graduate School of
More informationPartial Differential Equations, 2nd Edition, L.C.Evans Chapter 5 Sobolev Spaces
Partial Differential Equations, nd Edition, L.C.Evans Chapter 5 Sobolev Spaces Shih-Hsin Chen, Yung-Hsiang Huang 7.8.3 Abstract In these exercises always denote an open set of with smooth boundary. As
More informationSOME RESULTS FOR MAX-PRODUCT OPERATORS VIA POWER SERIES METHOD. 1. Introduction
Acta Math. Univ. Comenianae Vol. LXXXVII, 2 (208), pp. 9 98 9 SOME RESULTS FOR MAX-PRODUCT OPERATORS VIA POWER SERIES METHOD T. YURDAKADIM and E. TAŞ Abstract. In this paper, we obtain an approximation
More informationTHE STRUCTURE OF A RING OF FORMAL SERIES AMS Subject Classification : 13J05, 13J10.
THE STRUCTURE OF A RING OF FORMAL SERIES GHIOCEL GROZA 1, AZEEM HAIDER 2 AND S. M. ALI KHAN 3 If K is a field, by means of a sequence S of elements of K is defined a K-algebra K S [[X]] of formal series
More informationFourier transform of tempered distributions
Fourier transform of tempered distributions 1 Test functions and distributions As we have seen before, many functions are not classical in the sense that they cannot be evaluated at any point. For eample,
More informationGeometry of Banach spaces and sharp versions of Jackson and Marchaud inequalities
Geometry of Banach spaces and sharp versions of Jackson and Marchaud inequalities Andriy Prymak joint work with Zeev Ditzian January 2012 Andriy Prymak (University of Manitoba) Geometry of Banach spaces
More informationLinear Algebra and its Applications
Linear Algebra its Applications 432 21) 394 41 Contents lists available at ScienceDirect Linear Algebra its Applications journal homepage: wwwelseviercom/locate/laa On the Perron exponents of discrete
More informationAn Introduction to the Gamma Function
UNIVERSITY OF WARWICK Second Year Essay An Introduction to the Gamma Function Tutor: Dr. Stefan Adams April 4, 29 Contents 1 Introduction 2 2 Conve Functions 3 3 The Gamma Function 7 4 The Bohr-Möllerup
More informationCharacterizations on Heavy-tailed Distributions by Means of Hazard Rate
Acta Mathematicae Applicatae Sinica, English Series Vol. 19, No. 1 (23) 135 142 Characterizations on Heavy-tailed Distributions by Means of Hazard Rate Chun Su 1, Qi-he Tang 2 1 Department of Statistics
More informationConstrained Leja points and the numerical solution of the constrained energy problem
Journal of Computational and Applied Mathematics 131 (2001) 427 444 www.elsevier.nl/locate/cam Constrained Leja points and the numerical solution of the constrained energy problem Dan I. Coroian, Peter
More informationPrimes in arithmetic progressions to large moduli
CHAPTER 3 Primes in arithmetic progressions to large moduli In this section we prove the celebrated theorem of Bombieri and Vinogradov Theorem 3. (Bombieri-Vinogradov). For any A, thereeistsb = B(A) such
More informationEXISTENCE OF SOLUTIONS TO A BOUNDARY-VALUE PROBLEM FOR AN INFINITE SYSTEM OF DIFFERENTIAL EQUATIONS
Electronic Journal of Differential Equations, Vol. 217 (217, No. 262, pp. 1 12. ISSN: 172-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu EXISTENCE OF SOLUTIONS TO A BOUNDARY-VALUE
More informationClarkson Inequalities With Several Operators
isid/ms/2003/23 August 14, 2003 http://www.isid.ac.in/ statmath/eprints Clarkson Inequalities With Several Operators Rajendra Bhatia Fuad Kittaneh Indian Statistical Institute, Delhi Centre 7, SJSS Marg,
More informationMath 565: Introduction to Harmonic Analysis - Spring 2008 Homework # 2, Daewon Chung
Math 565: Introduction to Harmonic Analysis - Spring 8 Homework #, Daewon Chung. Show that, and that Hχ [a, b] : lim H χ [a, b] a b. H χ [a, b] : sup > H χ [a, b] a b. [Solution] Let us pick < min a, b
More informationThe Fibonacci sequence modulo π, chaos and some rational recursive equations
J. Math. Anal. Appl. 310 (2005) 506 517 www.elsevier.com/locate/jmaa The Fibonacci sequence modulo π chaos and some rational recursive equations Mohamed Ben H. Rhouma Department of Mathematics and Statistics
More information= 1 2 x (x 1) + 1 {x} (1 {x}). [t] dt = 1 x (x 1) + O (1), [t] dt = 1 2 x2 + O (x), (where the error is not now zero when x is an integer.
Problem Sheet,. i) Draw the graphs for [] and {}. ii) Show that for α R, α+ α [t] dt = α and α+ α {t} dt =. Hint Split these integrals at the integer which must lie in any interval of length, such as [α,
More informationProperties of the Scattering Transform on the Real Line
Journal of Mathematical Analysis and Applications 58, 3 43 (001 doi:10.1006/jmaa.000.7375, available online at http://www.idealibrary.com on Properties of the Scattering Transform on the Real Line Michael
More informationMATH MEASURE THEORY AND FOURIER ANALYSIS. Contents
MATH 3969 - MEASURE THEORY AND FOURIER ANALYSIS ANDREW TULLOCH Contents 1. Measure Theory 2 1.1. Properties of Measures 3 1.2. Constructing σ-algebras and measures 3 1.3. Properties of the Lebesgue measure
More informationWeak and strong convergence theorems of modified SP-iterations for generalized asymptotically quasi-nonexpansive mappings
Mathematica Moravica Vol. 20:1 (2016), 125 144 Weak and strong convergence theorems of modified SP-iterations for generalized asymptotically quasi-nonexpansive mappings G.S. Saluja Abstract. The aim of
More informationInterpolating the arithmetic geometric mean inequality and its operator version
Linear Algebra and its Applications 413 (006) 355 363 www.elsevier.com/locate/laa Interpolating the arithmetic geometric mean inequality and its operator version Rajendra Bhatia Indian Statistical Institute,
More informationON QUADRAPELL NUMBERS AND QUADRAPELL POLYNOMIALS
Hacettepe Journal of Mathematics and Statistics Volume 8() (009), 65 75 ON QUADRAPELL NUMBERS AND QUADRAPELL POLYNOMIALS Dursun Tascı Received 09:0 :009 : Accepted 04 :05 :009 Abstract In this paper we
More informationON A CLASS OF LINEAR POSITIVE BIVARIATE OPERATORS OF KING TYPE
STUDIA UNIV. BABEŞ BOLYAI, MATHEMATICA, Volume LI, Number 4, December 2006 ON A CLASS OF LINEAR POSITIVE BIVARIATE OPERATORS OF KING TYPE OCTAVIAN AGRATINI Dedicated to Professor Gheorghe Coman at his
More informationMATH 117 LECTURE NOTES
MATH 117 LECTURE NOTES XIN ZHOU Abstract. This is the set of lecture notes for Math 117 during Fall quarter of 2017 at UC Santa Barbara. The lectures follow closely the textbook [1]. Contents 1. The set
More informationSpectral radii of graphs with given chromatic number
Applied Mathematics Letters 0 (007 158 16 wwwelseviercom/locate/aml Spectral radii of graphs with given chromatic number Lihua Feng, Qiao Li, Xiao-Dong Zhang Department of Mathematics, Shanghai Jiao Tong
More informationPositive solutions of three-point boundary value problems for systems of nonlinear second order ordinary differential equations
J. Math. Anal. Appl. 32 26) 578 59 www.elsevier.com/locate/jmaa Positive solutions of three-point boundary value problems for systems of nonlinear second order ordinary differential equations Youming Zhou,
More informationJournal of Mathematical Analysis and Applications
J. Math. Anal. Appl. 356 009 79 737 Contents lists available at ScienceDirect Journal of Mathematical Analysis and Applications www.elsevier.com/locate/jmaa Takagi functions and approximate midconvexity
More informationStudy of some equivalence classes of primes
Notes on Number Theory and Discrete Mathematics Print ISSN 3-532, Online ISSN 2367-8275 Vol 23, 27, No 2, 2 29 Study of some equivalence classes of primes Sadani Idir Department of Mathematics University
More informationThis article was published in an Elsevier journal. The attached copy is furnished to the author for non-commercial research and education use, including for instruction at the author s institution, sharing
More informationFRACTIONAL BOUNDARY VALUE PROBLEMS ON THE HALF LINE. Assia Frioui, Assia Guezane-Lakoud, and Rabah Khaldi
Opuscula Math. 37, no. 2 27), 265 28 http://dx.doi.org/.7494/opmath.27.37.2.265 Opuscula Mathematica FRACTIONAL BOUNDARY VALUE PROBLEMS ON THE HALF LINE Assia Frioui, Assia Guezane-Lakoud, and Rabah Khaldi
More informationI and I convergent function sequences
Mathematical Communications 10(2005), 71-80 71 I and I convergent function sequences F. Gezer and S. Karakuş Abstract. In this paper, we introduce the concepts of I pointwise convergence, I uniform convergence,
More informationON THE COMPLETE CONVERGENCE FOR WEIGHTED SUMS OF DEPENDENT RANDOM VARIABLES UNDER CONDITION OF WEIGHTED INTEGRABILITY
J. Korean Math. Soc. 45 (2008), No. 4, pp. 1101 1111 ON THE COMPLETE CONVERGENCE FOR WEIGHTED SUMS OF DEPENDENT RANDOM VARIABLES UNDER CONDITION OF WEIGHTED INTEGRABILITY Jong-Il Baek, Mi-Hwa Ko, and Tae-Sung
More informationTranspose of the Weighted Mean Matrix on Weighted Sequence Spaces
Transose of the Weighted Mean Matri on Weighted Sequence Saces Rahmatollah Lashkariour Deartment of Mathematics, Faculty of Sciences, Sistan and Baluchestan University, Zahedan, Iran Lashkari@hamoon.usb.ac.ir,
More informationJournal of Mathematical Analysis and Applications
J. Math. Anal. Appl. 381 (2011) 506 512 Contents lists available at ScienceDirect Journal of Mathematical Analysis and Applications www.elsevier.com/locate/jmaa Inverse indefinite Sturm Liouville problems
More informationDifferential subordination related to conic sections
J. Math. Anal. Appl. 317 006 650 658 www.elsevier.com/locate/jmaa Differential subordination related to conic sections Stanisława Kanas Department of Mathematics, Rzeszów University of Technology, W. Pola,
More informationON A CERTAIN GENERALIZATION OF THE KRASNOSEL SKII THEOREM
Journal of Applied Analysis Vol. 9, No. 1 23, pp. 139 147 ON A CERTAIN GENERALIZATION OF THE KRASNOSEL SKII THEOREM M. GALEWSKI Received July 3, 21 and, in revised form, March 26, 22 Abstract. We provide
More informationEXTENDED LAGUERRE INEQUALITIES AND A CRITERION FOR REAL ZEROS
EXTENDED LAGUERRE INEQUALITIES AND A CRITERION FOR REAL ZEROS DAVID A. CARDON Abstract. Let fz) = e bz2 f z) where b 0 and f z) is a real entire function of genus 0 or. We give a necessary and sufficient
More information2014:05 Incremental Greedy Algorithm and its Applications in Numerical Integration. V. Temlyakov
INTERDISCIPLINARY MATHEMATICS INSTITUTE 2014:05 Incremental Greedy Algorithm and its Applications in Numerical Integration V. Temlyakov IMI PREPRINT SERIES COLLEGE OF ARTS AND SCIENCES UNIVERSITY OF SOUTH
More informationJUHA KINNUNEN. Harmonic Analysis
JUHA KINNUNEN Harmonic Analysis Department of Mathematics and Systems Analysis, Aalto University 27 Contents Calderón-Zygmund decomposition. Dyadic subcubes of a cube.........................2 Dyadic cubes
More informationOn the mean values of an analytic function
ANNALES POLONICI MATHEMATICI LVII.2 (1992) On the mean values of an analytic function by G. S. Srivastava and Sunita Rani (Roorkee) Abstract. Let f(z), z = re iθ, be analytic in the finite disc z < R.
More informationQuantization dimension of probability measures supported on Cantor-like sets
J. Math. Anal. Appl. 338 (2008) 742 750 Note www.elsevier.com/locate/jmaa Quantization dimension of probability measures supported on Cantor-like sets Sanguo Zhu Department of Mathematics, Huazhong University
More informationRate of approximation for Bézier variant of Bleimann-Butzer and Hahn operators
General Mathematics Vol 13, No 1 25), 41 54 Rate of approximation for Bézier variant of Bleimann-Butzer and Hahn operators Vijay Gupta and Alexandru Lupaş Dedicated to Professor Emil C Popa on his 6th
More informationUniform convergence of N-dimensional Walsh Fourier series
STUDIA MATHEMATICA 68 2005 Uniform convergence of N-dimensional Walsh Fourier series by U. Goginava Tbilisi Abstract. We establish conditions on the partial moduli of continuity which guarantee uniform
More informationDirect 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
More informationBounds for the Eventual Positivity of Difference Functions of Partitions into Prime Powers
3 47 6 3 Journal of Integer Seuences, Vol. (7), rticle 7..3 ounds for the Eventual Positivity of Difference Functions of Partitions into Prime Powers Roger Woodford Department of Mathematics University
More informationHerz (cf. [H], and also [BS]) proved that the reverse inequality is also true, that is,
REARRANGEMENT OF HARDY-LITTLEWOOD MAXIMAL FUNCTIONS IN LORENTZ SPACES. Jesús Bastero*, Mario Milman and Francisco J. Ruiz** Abstract. For the classical Hardy-Littlewood maximal function M f, a well known
More informationOn rate of convergence in distribution of asymptotically normal statistics based on samples of random size
Annales Mathematicae et Informaticae 39 212 pp. 17 28 Proceedings of the Conference on Stochastic Models and their Applications Faculty of Informatics, University of Debrecen, Debrecen, Hungary, August
More informationHardy-Littlewood maximal operator in weighted Lorentz spaces
Hardy-Littlewood maximal operator in weighted Lorentz spaces Elona Agora IAM-CONICET Based on joint works with: J. Antezana, M. J. Carro and J. Soria Function Spaces, Differential Operators and Nonlinear
More informationJournal of Number Theory
Journal of Number Theory 130 (2010) 1737 1749 Contents lists available at ScienceDirect Journal of Number Theory www.elsevier.com/locate/jnt A binary linear recurrence sequence of composite numbers Artūras
More informationWeighted Composition Operators on Sobolev - Lorentz Spaces
Int. J. Contemp. Math. Sciences, Vol. 6, 2011, no. 22, 1071-1078 Weighted Composition Operators on Sobolev - Lorentz Spaces S. C. Arora Department of Mathematics University of Delhi, Delhi - 110007, India
More informationSome Approximation Results For (p, q)-lupaş-schurer Operators
Filomat 3:1 018, 17 9 https://doi.org/10.98/fil180117k Published by Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.rs/filomat Some Approimation Results
More informationA Symbolic Operator Approach to Several Summation Formulas for Power Series
A Symbolic Operator Approach to Several Summation Formulas for Power Series T. X. He, L. C. Hsu 2, P. J.-S. Shiue 3, and D. C. Torney 4 Department of Mathematics and Computer Science Illinois Wesleyan
More informationFUNDAMENTALS OF REAL ANALYSIS by. IV.1. Differentiation of Monotonic Functions
FUNDAMNTALS OF RAL ANALYSIS by Doğan Çömez IV. DIFFRNTIATION AND SIGND MASURS IV.1. Differentiation of Monotonic Functions Question: Can we calculate f easily? More eplicitly, can we hope for a result
More informationON BASICITY OF A SYSTEM OF EXPONENTS WITH DEGENERATING COEFFICIENTS
TWMS J. Pure Appl. Math. V.1, N.2, 2010, pp. 257-264 ON BASICITY OF A SYSTEM OF EXPONENTS WITH DEGENERATING COEFFICIENTS S.G. VELIYEV 1, A.R.SAFAROVA 2 Abstract. A system of exponents of power character
More informationg(x) = P (y) Proof. This is true for n = 0. Assume by the inductive hypothesis that g (n) (0) = 0 for some n. Compute g (n) (h) g (n) (0)
Mollifiers and Smooth Functions We say a function f from C is C (or simply smooth) if all its derivatives to every order exist at every point of. For f : C, we say f is C if all partial derivatives to
More informationOn the validity of the Euler Lagrange equation
J. Math. Anal. Appl. 304 (2005) 356 369 www.elsevier.com/locate/jmaa On the validity of the Euler Lagrange equation A. Ferriero, E.M. Marchini Dipartimento di Matematica e Applicazioni, Università degli
More informationUNIFORM DENSITIES OF REGULAR SEQUENCES IN THE UNIT DISK. Peter L. Duren, Alexander P. Schuster and Kristian Seip
UNIFORM DENSITIES OF REGULAR SEQUENCES IN THE UNIT DISK Peter L. Duren, Alexander P. Schuster and Kristian Seip Abstract. The upper and lower uniform densities of some regular sequences are computed. These
More information