Ðåçþìåòà íà ïóáëèêàöèèòå íà
äîö.ä-ð Ãàí÷î Òà÷åâ Òà÷åâ
20. L.Beutel,H.Gonska,D.Kacso G.Tachev, On the Second Moments of
Variation-diminishing Splines,
Duisburg: Schriftenreihe des Fachbereichs mathematik der Gerhard-MercatorUniversitaet SM-DU-530(2002),in Journal of Concrete and Applicable
Mathematics,no.1, vol.2,2004,91-117. MR-2132355; ZB-02173984
Abstract. A representation theorem for the second moment of Schoenberg's variation-diinishing splines with equidistant knots is proved, thus correcting an earlier result of Marsden. Based upon this representation, upper
and lower bounds for the moments are given in certain cases.
Ðåçþìå. Äîêàçàíà å òåîðåìà çà âòîðèÿ ìîìåíò íà Øîåíáåðã ñïëàéí
îïåðàòîðà ñ ðàâíîîòäàëå÷åíè âúçëè, êàòî ïî òîçè íà÷èí êîðèãèðàìå ïîðàííè ðåçóëòàòè íà Ìàðñäåí. Èçïîëçâàéêè òîçè ðåçóëòàò ïîëó÷àâàìå
ãîðíè è äîëíè îöåíêè çà ìîìåíòà â íÿêîè ñëó÷àè.
21. L.Beutel,H.Gonska,D.Kacso G.Tachev, On variation-diminishing Schoenberg operators:new quantitative statements, in Multivariate Approximation and Interpolation with Applications (ed. by M.Gasca), Monograas de la Academia de Ciencias de Zaragoza 20 ,2002,9-58. MR04a:41027;ZB-01999241
Abstract. We give quantitative results for variationdiminishing splines,
focusing on the case of equidistant knots. New direct inequalities are obtained, both in terms of the classical second modulus of continuity and in
terms of the second Ditzian-Totik modulus. These new results are based upon a detailed analysis of the second moments and very recent theorems for
positive linear operator approximation. The potential for simultaneous approximation is described by means of an estimate involving both the ˆrst and
the second classical modulus of continuity. The topic of global smoothness
preservation is also addressed. Furthermore, we discuss the degree of simultaneous approximation in the multivariate case, namely for Boolean sums and
tensor products of Schoenberg splines.
Ðåçþìå.  òàçè ñòàòèÿ äàâàìå êîëè÷åñòâåíè îöåíêè çà Øîåíáåðã
ñïëàéí îïåðàòîðà, êàòî ñå êîíöåíòðèðàìå íà ñëó÷àÿ ñ ðàâíîîòäàëå÷åíè
1
âúçëè. Íîâè äèðåêòíè îöåíêè ñà ïîëó÷åíè, êàêòî â òåðìèíèòå íà êëàñè÷åñêèÿ
âòîðè ìîäóë íà íåïðåêúñíàòîñò, òàêà è ñ âòîðèÿ ìîäóë íà Ditzian-Totik.
Òåçè íîâè ðåçóëòàòè ñà áàçèðàíè íà äåòàéëåí àíàëèç íà âòîðèÿ ìîìåíò
è íàñêîðî-ïîëó÷åíè òåîðåìè çà àïðîêñèìàöèè ñ ëèíåéíè ïîëîæèòåëíè
îïåðàòîðè. Ðàçãëåäàíà å âúçìîæíîñòòà çà åäíîâðåìåííà àïðîêñèìàöèÿ
íà ôóíêöèÿòà è ïðîèçâîäíèòå â òåðìèíèòå íà ïúðâèÿ è âòîðè ìîäóë íà
íåïðåêúñíàòîñò. Âúïðîñúò çà çàïàçâàíå íà ãëàäêîñòòà å ñúùî èçñëåäâàí.
Ïî-íàòàòúê ðàçãëåæäàìå ñòåïåíòà íà åäíîâðåìåííà àïðîêñèìàöèÿ â ìíîãîìåðíèÿ
ñëó÷àé, çà Áóëåâè ñóìè è òåíçîðíî ïðîèçâåäåíèå íà Øîåíáåðã ñïëàéíè.
22. H.H.Gonska,D.Kacso and G.Tachev, Direct Estimates and Bernsteintype Inequalites for Schoenberg Splines, in "RoGer 2002 - Sibiu"(Proc.
5th Romanian-German Seminar on Approximation Theory and its
Applications), Sibiu, 11.06-15.06.2002,Mathematical Analysis and
Approximation Theory ,119-129. MR-2076825;ZB-1041.41012
Abstract. We give direct estimates in terms of a certain second order
K-functional K2 , where the weight is a function closely related to the second
moment of the Schoenberg splines. The Bernstein-type inequalities will be
established for cubic splines.
Ðåçþìå. Ïîëó÷àâàìå ïðàâè îöåíêè â òåðìèíèòå íà òåãëîâè Ê-ôóíêöèîíàë
îò âòîðè ðåä, êúäåòî òåãëîòî å ôóíêöèÿ, òÿñíî ñâúðçàíà ñ âòîðèÿ ìîìåíò
íà Øîåíáåðã ñïëàéí îïåðàòîðà. Ïîëó÷åíè ñà è íåðàâåíñòâà îò òèïà íà
Áåðíùàéí çà êóáè÷íèòå ñïëàéíè.
23. L.Beutel,H.Gonska,D.Kacso and G.Tachev,Variation-diminishing splines,
revisted.2002,Proc. of the International Symposium Dedicated to
the 75th Anniversary of D.D.Stancu,Cluj -Napoca,Romania,2002,5475. MR-04g:41013;ZB-02092666
Abstract We survey and discuss several recent results on univariate
Schoenberg splines. This include new uniform and pointwise inequalities, the
degree of simultaneous approximation for rst and second order derivatives,
and the preservation of global smoothness expressed in terms of second order moduli. There is an emphasis on the case of equidistant knot sequences.
Several open problems are listed at the end of the note.
Ðåçþìå. Ðàçãëåæäàìå íÿêîè îò ïîñëåäíèòå ðåçóëòàòè çà åäíîìåðíèòå
ñïëàéíè íà Øîåíáåðã. Òîâà âêëþ÷âà íîâè ðàâíîìåðíè è ïîòî÷êîâè íåðàâåíñòâà,
2
ñòåïåíòà íà åäíîâðåìåííà àïðîêñèìàöèÿ çà ïúðâàòà è âòîðà ïðîèçâîäíè
è çàïàçâàíå íà ãëîáàëíàòà ãëàäêîñò, èçðàçåíà â òåðìèíèòå íà âòîðèÿ
ìîäóë. Íÿêîè íåðåøåíè ïðîáëåìè ñà èçáðîåíè â êðàÿ íà ñòàòèÿòà.
24. G.Tachev, On the cubic Schoenberg Splines, in Jubilee Scientic
Conf., vol.8( Soa, 2002) Ann. Inst. Archit. Genie Civil ,41,2001-2002,455465. MR-03i:00008,41A15(41A17), ZB 1187.41004
Abstract We give direct estimate in terms of second order Ditzian-Totik
modulus ωΦ2 (f, δ), where the step-weight Φ is a function closely related to the
second moment of Schoenberg splines. Two inewualities of Bernstein type will
be establishedfor cubic splines.
Ðåçþìå. Äàâàìå ïðàâè îöåíêè â òåðìèíèòå íà âòîðèÿ ìîäóë íà ÄèòöèàíÒîòèê ωΦ2 (f, δ), êúäåòî òåãëîâàòà ôóíêöèÿ Φ å òÿñíî-ñâúðçàíà ñ âòîðèÿ
ìîìåíò íà Øîåíáåðã ñïëàéí îïåðàòîðà. Äâå íåðàâåíñòâà îò òèïà íà
Áåðíùàéí ñà óñòàíîâåíè çà êóáè÷íèòå ñïëàéíè.
25. G.Tachev,Three open problems, in "RoGer 2002 - Sibiu"(Proc.
5th Romanian-German Seminar on Approximation Theory and its
Applications), Sibiu, 11.06-15.06.2002,Mathematical Analysis and
Approximation Theory ,329-330. ZB-1030.41001
Abstract. Three open problems are formulated for:
- to nd the minimal condition number of knot sequences, satisfying certain pointwise conditions;
- to evaluate the remainder term of approximation of logarithmic function
by Bernstein polynomials;
- to evaluate the constant in the lower bound for the Bernstein operator.
Ðåçþìå. Ôîðìóëèðàíè ñà òðè íåðåøåíè ïðîáëåìà, êàêòî ñëåäâà:
- äà ñå íàìåðè ìèíèìàëíîòî ÷èñëî íà îáóñëîâåíîñò çà ðåäèöè îò
âúçëè, óäîâëåòâîðÿâàùè äàäåíè ïîòî÷êîâè óñëîâèÿ;
- äà ñå îöåíè îñòàòú÷íèÿ ÷ëåí ïðè ïðèáëèæåíèå íà ëîãàðîòìè÷íàòà
ôóíêöèÿ ñ ïîëèíîìèòå íà Áåðíùàéí.
- äà ñå íàìåðÿò îöåíêè çà êîíñòàíòàòà â äîëíàòà îöåíêà çà ïðèáëèæåíèå
ñ îïåðàòîðà íà Áåðíùàéí â ðàâíîìåðíà íîðìà.
3
26.H.Gonska,I.Gavrea, R.Paltanea and G.Tachev,General Estimates with
Ditzian-Totik moduli of second order, in East J. Approx. Th. ,vol.9,no.2,2003,175194. MR-04e:41006, ZB-05147510
Abstract. For linear positive operators, mapping the space C[a, b] of
continuous functions into itself and reproducing linear functions, a general
estimate in terms of second order Ditzian-Totik modulus with the classical
weight is given. This inequality includes a free parameter h and exhibits
an explicit and small value of the constant in front of the modulus.It thus
improves and simplies earlier results by Gavrea, Gonska and Tachev. In
particulat, for the case of the classical Bernstein operators a more precise
version of a now classical estimate by Ditzian, Totik and Zhou is given.
Ðåçþìå. Çà ëèíåéíèòå ïîëîæèòåëíè îïåðàòîðè, èçîáðàçÿâàùè ïðîñòðàíñòâîòî
îò íåïðåêúñíàòè ôóíêöèè C[a, b] â ñåáå ñè è çàïàçâàùè ëèíåéíèòå ôóíêöèè
å ïîëó÷åíà îáùà îöåíêà â òåðìèíèòå íà âòîðèÿ ìîäóë íà Äèòöèàí-Òîòèê
ñ êëàñè÷åñêà òåãëîâà ôóíêöèÿ. Òîâà íåðàâåíñòâî âêëþ÷âà ïàðàìåòúð h è
ïðåäñòàâÿ åæïëèöèòíî ìàëêè ñòîéíîñòè íà êîíñòàíòàòà ïðåä ìîäóëà. Ïî
òîçè íà÷èí ïîäîáðÿâàìå è îïðîñòÿâàìå ïðåäèøíè ðåçóëòàòè íà Ãàâðåà,
Ãîíñêà è Òà÷åâ.  ÷àñòíîñò, çà ñëó÷àÿ íà êëàñè÷åñêèÿ îïåðàòîð íà Áåðíùàéí
ñà ïîëó÷åíè ìíîãî ïî-ïðåöèçíè îöåíêè îò êëàñè÷åñêèòå îöåíêè íà Ditzian,
Totik and Zhou.
27.H.H.Gonska,D.Kacso and G.Tachev,Inverse Estimates for the cubic
Schoenberg Splines, 2005, in "RoGer 2004 - Sibiu"(Proc. 6th RomanianGerman Seminar on Approximation Theory and its Applications),
Cluj-Napoca, 11.06-15.06.2004,Mathematical Analysis and Approximation Theory ,19-24.
Abstract. For Schoenberg splines fo degrees k = 1, 2, 3 we establish inverse results in terms of the second order classical and Ditzian-Totik moduli.
As a consequence, results relaring the order of approximation by Schoenberg
splines to the smoothness of the function are derived.
Ðåçþìå. Çà Øîåíáåðã ñïëàéíèòå îò ñòåïåí k = 1, 2, 3 óñòàíîâÿâàìå
îáðàòíè îöåíêè â òåðìèíèòå íà âòîðèÿ êëàñè÷åñêè ìîäóë è âòîðèÿ ìîäóë
íà Äèòöèàí-Òîòèê. Êàòî ñëåäñòâèå ñà ïîëó÷åíè ðåçóëòàòè çà ñòåïåíòà íà
àïðîêñèìàöèÿ ñúñ ñïëàéíèòå íà Øîåíáåðã â çàâèñèìîñò îò ãëàäêîñòòà íà
ôóíêöèÿòà.
4
28. G.Tachev,Approximation by rational spline functions ,(2006) in Calcolo,vol.43 ,(4),279-286.ZB-05151033MR-07k:41032.IF=0.621
Abstract. We discuss the linear precision property of NURBS functions.
The degree of approximation of continuous functions is studied.
Ðåçþìå. Èçñëåäâà ñå ñâîéñòâîòî "linear precision"(çàïàçâàíå íà ëèíåéíèòå
ôóíêöèè) ïðè ïðèáëèæåíèå ñ NURBS-ôóíêöèèòå. Èçñëåäâàíà å ñòåïåíòà
íà àïðîêñèìàöèÿ íà íåïðåêúñíàòèòå ôóíêöèè.
29. G.Tachev,Inequalities for linear positive operators,(2007), in International Journal of Applied Mathematics and Statistics-special issue
on FIDA,, vol.12,no.D07,112-123.ZB-1137.41339MR-09a:41018
Abstract This survey paper grew out of a series of results, achieved
at Duisburg in collaboration with mathematicians from Germany and Romania during my visits in the period of last seven years,1999-2005. Direct
theorems for approximation of continuous real-valued functions by positive
linear operators in terms of the second order modulus of smoothness are
presented.Special emphasis is on the magnitude of the absolute constants
appearing in the right-hand side of the inequalities. The degree of approximation for Bernstein operator, Schoenberg variation-diminishing spline operator and NURBS (non-uniform rational Bspline operator) is studied. Each
of these three linear positive operators is a generalization of the preceding
one and has an important applications in broad areas of mathematics such
as real and functional analysis, approximation theory, numerical methods,
computer-aided geometric design (CAGD), the theory of probabilities and
mathematical statistics,geometry.
Ðåçþìå. Òàçè îáçîðíà ñòàòèÿ å íàïèñàíà ïî ïîêàíà è å âêëþ÷åíà
â èçäàíèåòî íà FIDA-vol.12,2007, ïîñâåòåíî íà 300 ã. îò ðîæäåíèåòî íà
Leonard Euler. Òÿ îáõâàùà ñåðèÿ îò ðåçóëòàòè, ïîëó÷åíè îò ñúâìåñòíàòà
ìè ðàáîòà ñ ìàòåìàòèöè îò Ãåðìàíèÿ è Ðóìúíèÿ ïî âðåìå íà âèçèòèòå
ìè â Äóèñáóðã ïðåç ïåðèîäà 1999-2005.
30. G.Tachev, Weighted approximation of second derivatives for Schoenberg splines, to appear in Proc. Int. Conf. UACEG 2009: Science and
Practice, (2009).
5
abstract We prove weighted uniform approximation on [0, 1] of second
derivative of f ∈ C 2 [0, 1] by the second derivative of cubic Schoenberg splines.
The weight function Φ is closely related to the second moment of Schoenberg
splines.
Ðåçþìå. Äîêàçâàìå ðàâíîìåðíà òåãëîâà àïðîêñèìàöèÿ íà âòîðàòà
ïðîèçâîäíà íà ôóíêöèÿ f ∈ C 2 [0, 1] ÷ðåç âòîðàòà ïðîèçâîäíà íà ñïëàéíèòå
íà Øîåíáåðã. Òåãëîâàòà ôóíêöèÿ å ñâúðçàíà ñ âòîðèÿ ìîìåíò íà ñïëàéíèòå
íà Øîåíáåðã.
32.G.Tachev, Voronovskaja's Theorem Revisited, in Journal of Mathematical Analysis and Applications,343 (2008),399-404, ZB-1140.41002
MR-09c:41048.IF=1.046
Abstract We represent a new quantitative variant of Voronovskaja's theorem for Bernstein operator. This estimate improves the recent quantitative
versions of Voronovskaja's theorem for certain Bernstein-type operators, obtained by H. Gonska, P. Pitul and I. Rasa in 2006.
Ðåçþìå. Äîêàçâàìå íîâ âàðèàíò íà Òåîðåìàòà íà Âîðîíîâñêà çà
îïåðàòîðà íà Áåðíùàéí. Òàçè îöåíêà ïîäîáðÿâà ïîñëåäíèòå êîëè÷åñòâåíè
âåðñèè íà Òåîðåìàòà íà Âîðîíîâñêà çà îïåðàòîðè îò òèïà íà Áåðíùàéí,
ïîëó÷åíè îò H. Gonska, P. Pitul and I. Rasa ïðåç 2006.
33.G.Tachev , (2008), The Distance between Bezier Curve and its Control Polygon,to appear in Collection of papers dedicated to the 60-th
Anniv. of M.M.Konstantinov.
Abstract. We consider integer and rational parametric Bezier curves and
study the distance between the curve and its control polygon. To measure the
distance we use rst and second order moduli of smoothness of vectorvalued
function. Some direct approximation theorems are presented.
Ðåçþìå. Èçñëåäâàìå ïîëèíîìèàëíè è ðàöèîíàëíè ïàðàìåòðè÷íè êðèâè
íà Áåçèå è ðàçñòîÿíèåòî ìåæäó êðèâàòà è íåéíèÿ êîíòðîëåí ïîëèãîí. Çà
äà èçìåðèì ðàçñòîÿíèåòî èçïîëçâàìå ïúðâèÿ è âòîðè ìîäóë íà ãëàäêîñò
íà âåêòîðíî-çíà÷íà ôóíêöèÿ. Ïîëó÷åíè ñ àäèðåêòíè îöåíêè çà àïðîêñèìàöèÿ.
6
34. G. Tachev, A lower bound for the second moment of Schoenberg operator, in General Mathematics, 16 (2008), no.4, 165-170, MR-2471282
ZB-05626027.
Abstract. In this paper we represent a new lower bound for the second moment for Schoenberg variation-diminishing spline operator. We apply
this estimate for f ∈ C 2 [0, 1] and generalize the results obtained earlier by
Gonska, Pitul and Rasa.
Ðåçþìå.  òàçè ñòàòèÿ ïîëó÷àâàìå íîâà äîëíà ãðàíèöà çà âòîðèÿ
ìîìåíò íà Øîåíáåðã îïåðàòîðà. Ïðèëàãàìå òàçè îöåíêà çà f ∈ C 2 [0, 1] è
îáîáùàâàìå ðåçóëòàòèòå ïîëó÷åíè ïî-ðàíî îò Gonska, Pitul and Rasa.
35. G. Tachev, Approximation, numerical dierentiation and integration
based on Taylor polynomial, in J. Inequal. Pure and Appl. Math. (JIPAM), 10(1),(2009), electronic,ZB-1168.65325,MR-2491928.SJP=0.031
(SCImago Journal Country Rank)
Abstract. We represent new estimates of errors of quadrature formula,
formula of numerical dierentiation and approximation using Taylor polynomial. To measure the errors we apply representation of the remainder in
Taylor formula by least concave majorant of the modulus of continuity of
the n-th derivative of an n-times dierentiable function. Our quantitative
estimates are special applications of a more general inequality for Pn-simple
functionals.
Ðåçþìå. Ïîëó÷àâàìå íîâè îöåíêè çà ãðåøêàòà íà êâàäðàòóðíè ôîðìóëè,
ôîðìóëè çà ÷èñëåíî äèôåðåíöèðàíå è çà ïðèáëèæåíèå ñ ïîëèíîìèòå íà
Òåéëúð. Çà èçìåðâàíå íà ãðåøêàòà ïðèëàãàìå ïðåäñòàâÿíå íà îñòàòú÷íèÿ
÷ëåí îò ôîðìóëàòà íà Òåéëúð ÷ðåç íàé-ìàëêàòà âäëúáíàòà ìàæîðàíòà íà
ìîäóëà íà íåïðåêúñíàòîñò íà í-òà ïðîèçâîäíà íà í-ïúòè äèôåðåíöèðóåìà
ôóíêöèÿ. Êîëè÷åñòâåíèòå îöåíêè â òàçè ñòàòèÿ ñà ÷àñòåí ñëó÷àé íà ïîîáùî íåðàâåíñòâî çà Pn ïðîñòè ôóíêöèîíàëè, âúâåäåíè îò Éîàí Ãàâðåà.
36. H.Gonska and G.Tachev, (2009), A Quantitative Variant of Voronovskaja's Theorem, in Results in Mathematics,53(2009),287-294, MR-2524730
ZB-1181.41026.IF=0.513
Abstract. A general quantitative Voronovskaja theorem for Bernstein
operators is given which bridges the gap between such estimates in terms of
7
the least concave majorant of the rst order modulus of continuity
p and the
rst order Ditzian.Totik modulus with classical weight ϕ(x) = x(1 − x).
Ðåçþìå. Ïîëó÷åíà å îáùà Òåîðåìà îò òèïà íà Âîðîíîâñêà â êîëè÷åñòâåí
âàðèàíò, êîÿòî îáõâàùà îöåíêè ñ íàé-ìàëêàòà âäëúáíàòà àæîðàíòà íà
ïúðâèÿ ìîäóë íà íåïðåêúñíàòîñò p
è îöåíêè ñ ïúðâèÿ ìîäóë íà ÄèòöèàíÒîòèê ñ êëàñè÷åñêî òåãëî ϕ(x) = x(1 − x).
37. G.Tachev, On the second moment of rational Bernstein functions,
in Journal of Computational Analysis and Applications,JoCAAA,
12(2010),no.2, 471-479. ZB 1192.41003, MR-2650454.IF=0.697
Abstract. We study the dependance of the second moment of the Bernstein rational functions on the weight numbers. For the second degree rational
Bernstein function we prove that the minimal value of the second moment
uniform on [0, 1] is attained when all weight numbers are equal. At the end
we represent a quantitative variant of Voronovskaja's Theorem.
Ðåçþìå.Èçñëåäâà ñå çàâèñèìîñòòà íà âòîðèÿ ìîìåíò íà Áåðíùàéí
ðàöèîíàëíèòå ôóíêöèè îò òåãëîâèòå ÷èñëà. Çà ðàöèîíàëíàòà Áåðíùàéí
ôóíêöèÿ îò âòîðà ñòåïåí äîêàçâàìå, ÷å íàé-ìàëêàòà ñòîéíîñò íà âòîðèÿ
ìîìåíò ðàâíîìåðíî â [0, 1] ñå ïîëó÷àâà êîãàòî âñè÷êè òåãëîâè ÷èñëà
ñà ðàâíè. Íàêðàÿ ïðåäñòàâÿìå êîëè÷åñòâåí âàðèàíò íà Òåîðåìàòà íà
Âîðîíîâñêà.
38. G.Tachev, Approximation of a continuous curve by its BernsteinBezier operators, Mediterranean J. of Math., 8(2011), no.3, 381-393. ZB
1231.65036, MR-2824587.IF=0.463
Abstract. In this paper we consider integer and rational parametric
B ezier curves and study the distance between the curve and its control
polygon. To measure the distance we use rst and second order moduli of
smoothness of vector-valued function. We consider also NURBS curves with
equidistant knots. Some direct approximation theorems will be presented.
Ðåçþìå.  òàçè ñòàòèÿ èçñëåäâàìå ïîëèíîìèàëíè è ðàöèîíàëíè ïàðàìåòðè÷íè
êðèâè íà Áåçèå è ðàçñòîÿíèåòî ìåæäó êðèâàòà è íåéíèÿ êîíòðîëåí ïîëèãîí.
Ïîëó÷åíè ñà äèðåêòíè òåîðåìè çà àïðîêñèìàöèè.
8
39. H.Gonska and G.Tachev, Gr
uss-type inequality for positive linear operators with second order Ditzian-Totik moduli, Mat. Vesnik, 63(2011),
íî. 4, 247-252. MR-12g:41027.SJR=0.031-(SCImago Journal Country
Rank)
Abstract. We prove two Gruss-type inequalities for positive linear operator approximation, i.e., inequalities explaining the non-multiplicativity of
such mappings. Instead of the least concave majorant of the rst order modulus of continuity, we employ second order moduli of smoothness and show
in the case of the classical Bernstein operators that in certain cases this leads
to better results than those obtained earlier.
Ðåçþìå. Äîêàçâàìå äâå íåðàâåíñòâà îò òèïà íà Ãðþñ çà ïðèáëèæåíèå
ñ ëèíåéíè ïîëîæèòåëíè îïåðàòîðè. Íîâîòî å , ÷å èçïîëçâàìå âòîðèÿ
ìîäóë íà ãëàäêîñò è ïîëó÷àâàìå ïî-äîáðè îöåíêè çà ñëó÷àÿ íà îïåðàòîðà
íà Áåðíùàéí.
-
40. G.Tachev, Voronovskaja Theorem for Schoenberg operator, in Mathematical Inequalities and Applications,MIA,15 (2012),no.1, 49-59. ZB
1236.41010,MR-2919430.IF=0.558
Abstract. In this paper we represent new quantitative variants of Voronovska
jaÃfs Theorem for Schoenberg variation-diminishing spline operator. We estimate the rate of uniform convergence for f ∈ C 2 [0, 1] and generalize the
results obtained earlier by Goodman, Lee, Sharma, Gonska etc.
Ðåçþìå.  òàçè ñòàòèÿ ïðåäñòàâÿìå íîâè êîëè÷åñòâåíè âàðèàíòè íà
Òåîðåìàòà íà Âîðîíîâñêà çà îïåðàòîðà íà Øîåíáåðã. Îöåíÿâàìå ñòåïåíòà
íà ðàâíîìåðíà ñõîäèìîñò çà f ∈ C 2 [0, 1] è îáîáùàâàìå ðåçóëòàòèòå ,
ïîëó÷åíè ïî-ðàíî îò Goodman, Lee, Sharma, Gonska.
41. G.Tachev, New estimates in Voronovskaja's theorem, in Numerical
Algorithms, 59(2012), 119-129. ZB-06006049, MR-2886440.IF=1.042
Abstract.In the present article we establish pointwise variant of E. V.
Voronovskaja's 1932 result, concerning the degree of approximation of Bernstein operator, applied to functions f ∈ C 3 [0, 1]
9
Ðåçþìå.  íàñòîÿùàòà ñòàòèÿ äîêàçâàìå ïîòî÷êîâà îöåíêà - ïîòî÷êîâ
âàðèàíò íà Òåîðåìàòà íà Âîðîíîâñêà îò 1932 ã., çàñÿãàù ñòåïåíòà íà
àïðîêñèìàöèÿ ñ îïåðàòîðà íà Áåðíùàéí, ïðèëîæåí êúì ôóíêöèè f ∈
C 3 [0, 1]
42. G.Tachev, The complete asymptotic expansion for Bernstein operators, in Journal of Math. Anal. and Appl.,JMAA,385 (2012),1179-1183.
ZB 1232.41016, MR-2834919.IF=1.001
Abstract.In this paper we study the asymptotic behavior of the classical Bernstein operators, applied to q-times continuously dierentiable functions. Our main results extend the results of S.N. Bernstein and R.G. Mamedov for all q-odd natural numbers and thus generalize the theorem of E.V.
Voronovskaja. The exact degree of approximation is also proved.
Ðåçþìå.  òàçè ñòàòèÿ èçñëåäâàìå àñèìïòîòè÷íîòî ïîâåäåíèå íà
êëàñè÷åñêèÿ îïåðàòîð íà Áåðíùàéí, ïðèëîæåí êúì q -ïúòè íåïðåêúñíàòîäèôåðåíöèðóåìà ôóíêöèÿ. Îñíîâíàòà òåîðåìà ðàçøèðÿâà ðåçóëòàòèòå
íà Ìàìåäîâ è Áåðíùàéí çà âñÿêî q - íå÷åòíî ÷èñëî è òàêà îáîáùàâàìå
è Òåîðåìàòà íà Âîðîíîâñêà. Äîêàçàíà å è îïòèìàëíîñò íà ñòåïåíòà íà
àïðîêñèìàöèÿ.
43. G.Tachev, On multiplicativity of Bernstein operator, in Computers
and Mathematics with Applications, CMA,62 (2011), no.8, 3236-3240.
ZB-1232.41005, MR-2837756.IF=1.747
Abstract.For continuous functions f and g, we prove that the Bernstein
operator Bn is multiplicative for all n ≥ 1 and all x ∈ [0, 1] if and only if at
least one of the functions f and g is a constant function. Some other variants
of multiplicativity are also considered.
Ðåçþìå.Çà íåïðåêúñíàòè ôóíêöèè f, g äîêàçâàìå, ÷å îïåðàòîðà íà
Áåðíùàéí Bn å ìóëòèïëèêàòèâåí çà âñÿêî n ≥ 1 è âñÿêî x ∈ [0, 1] òîãàâà
è ñàìî òîãàâà, êîãàòî ïîíå åäíà îò òåçè äâå ôóíêöèè å êîíñòàíòà. Íÿêîè
äðóãè âàðèàíòè íà ìóëòèïëèêàòèâíîñò ñúùî ñà ðàçãëåäàíè.
10
44. G.Tachev, Pointwise Approximation by Bernstein Polynomials, in
Bulletin of the Australian Mathematical Society, BAMS,85(2012),
no.2, 353-358. ZB-06047084,MR-2924764.IF=0.545
Abstract.We improve the degree of pointwise approximation of continuous functions f (x) by Bernstein operators, when x is close to the endpoints
of [0; 1]. We apply the new estimate to establish upper and lower pointwise
estimates for the test function g(x) = x ln(x) + (1 − x) ln(1 − x). At the
end we prove a general statement for pointwise approximation by Bernstein
operators.
Ðåçþìå. Ïîäîáðÿâàìå èçâåñòíàòà ïîòî÷êîâà îöåíêà íà Äèòöèàí çà
àïðîêñèìàöèÿ íà íåïðåêúñíàòà ôóíêöèÿ f (x) ñ ïîëèíîìèòå íà Áåðíùàéí,
êîãàòî x å áëèçî äî êðàèùàòà íà èíòåðâàëà [0, 1]. Ïðèëàãàìå òàçè íîâà
îöåíêà çà äà ïîëó÷èì ãîðíà è äîëíà ãðàíèöà çà ïîòî÷êîâà àïðîêñèìàöèÿ
íà òåñò-ôóíêöèÿòà g(x) = x ln(x) + (1 − x) ln(1 − x). Íàêðàÿ äîêàçâàìå
îáùà ïîòî÷êîâà îöåíêà çà ïðèáëèæåíèå ñ îïåðàòîðà íà Áåðíùàéí.
45. G.Tachev, The rate of approximation by rational Bernstein functions
in terms of second order moduli of continuity, in Numerical Functional
Analysis and Applications, 33(2) (2012), 206-215. ZB-06029827, MR2876778.IF=0.711
Abstract.The aim of this note is to study the impact of the weight numbers in two concrete cases on the rate of approximation by rational Bernstein
functions. The approximation is measured in terms of second order moduli
of continuity. We consider also the case of rational Bernstein curves and the
role of the weights to modify the shape of the curves.
Ðåçþìå.Öåëòà íà òàçè ñòàòèÿ å äà ñå èçñëåäâà âëèÿíèåòî íà òåãëîâèòå
÷èñëà â äâà êîíêðåòíè ñëó÷àÿ âúðõó ñêîðîñòòà íà àïðîêñèìàöèÿ ñ ðàöèîíàëíèòå
Áåðíùàéí ôóíêöèè. Àïðîêñèìàöèÿòà ñå èçìåðâà â òåðìèíèòå íà âòîðèÿ
ìîäóë íà íåïðåêúñíàòîñò. Ðàçãëåæäà ñå ñúùî è ñëó÷àÿ íà ðàöèîíàëíè
êðèâè íà Áåðíùàéí è ðîëÿòà íà òåãëàòà çà ãåîåìòðè÷íî ìîäåëèðàíå íà
âèäà íà êðèâàòà.
-
11
46. G.Tachev, From Bernstein polynomial to Lagrange interpolant, in
Proc. of 2-nd Int. Conf. MDIS-Sibiu,Romania 2011,ISBN 978-60612-0243-0, Lucian Blaga University Press, Ed. Dana Simian, 192-199 .
Abstract.For a given continuous function f(x) on [0, 1] we construct
sequence of algebraic polynomials based on Bernstein approximation. We
prove that the limit of this sequence is the Lagrange interpolation polynomial
of degree n. Application to the representation of polynomial curves will be
given.
Ðåçþìå. Çà äàäåíà íåïðåêúñíàòà ôóíêöèÿ f (x) âúðõó [0, 1] êîíñòðóèðàìå
ðåäèöà îò àëãåáðè÷íè ïîëèíîìè, áàçèðàíà íà ïîëèíîìèòå íà Áåðíùàéí.
Äîêàçâàìå, ÷å ãðàíèöàòà íà òàçè ðåäèöà å èíòåðïîëàöèîííèÿ ïîëèíîì
íà Ëàãðàíæ îò ñòåïåí n. Êàòî ïðèëîæåíèå ðàçãëåæäàìå è ïðåäñòàâÿíå
íà ïîëèíîìèàëíè êðèâè.
47. Sorin Gal and Gancho Tachev, On the Constant in The Lower Estimate for the Bernstein Operator, to appear in Mathematica Balkanica,
26(2012).
Abstract.For functions belonging to the classes C 2 [0, 1] and C 3 [0, 1], we
establish the lower estimate with an explicit constant in approximation by
Bernstein polynomials in terms of the second order Ditzian-Totik modulus
of smoothness. Several applications to some concrete examples of functions
are presented.
Ðåçþìå. Çà ôóíêöèè îò ïðîñòðàíñòâàòà C 2 [0, 1] and C 3 [0, 1] óñòàíîâÿâàìå
äîëíà ãðàíèöà è êîíêðåòíè êîíñòàíòè ïðè àïðîêñèìàöèÿ ñ ïîëèíîìèòå
íà Áåðíùàéí â òåðìèíèòå íà âòîðèÿ ìîäóë íà Äèòöèàí-Òîòèê. Íÿêîëêî
ïðèëîæåíèÿ çà êîíêðåòíè åëåìåíòàðíè ôóíêöèè ñà äàäåíè â ïîñëåäíàòà
÷àñò.
-
48. G.Tachev, On the conjecture of Cao, Gonska and Kacso, in Stud.
Univ. Babes-Bolyai Math., 57(2012), no.1, 83-88. .MR-2922183
Abstract.We consider the question if lower estimates in terms of the second order Ditzian-Totik modulus are possible, when we measure the pointwise
approximation of continuous function by Bernstein operator. In this case we
12
conrm the conjecture made by Cao, Gonska and Kacso. To prove this we
rst establish sharp upper and lower bounds for pointwise approximation of
the function g(x) = x ln(x)+(1−x) ln(1−x), x ∈ [0, 1] by Bernstein operator.
Ðåçþìå.Ðàçãëåæäà ñå âúïðîñà, äàëè å âúçìîæíà äîëíà ãðàíèöà â
òåðìèíèòå íà âòîðèÿ ìîäóë íà Äèòöèàí-Òîòèê, êîãàòî èçìåðâàìå ïîòî÷êîâàòà
àïðîêñèìàöèÿ ñ ïîëèíîìèòå íà Áåðíùàéí.  òîçè êîíêðåòåí ñëó÷àé ïîòâúðæäàâàìå
õèïîòåçàòà íà Cao, Gonska and Kacso.
49. Margareta Heilmann and Gancho Tachev, Commutativity, Direct and
Strong Converse Results for Phillips Operators, in East J. Approx. Th.,
17(2011),no. 3, 299-317.ZB-06077250,MR-2953082.
Abstract.We study the so-called Phillips operators which can be considered as genuine Szasz-Mirakjan-Durrmeyer operators. As main results we
prove the commutativity of the operators as well as their commutativity
with an appropriate dierential operator and establish a strong converse inequality of type A for the approximation of real valued continuous bounded
functions f on [0; ∞). Together with the corresponding direct theorem we derive an equivalence result for the error of approximation and an appropriate
K-functional and modulus of smoothness.
Ðåçþìå. Èçñëåäâàò ñå àïðîêñèìàòèâíèòå ñâîéñòâà íà îïåðàòîðà íà
Ôèëèïñ, êîéòî ìîæå äà ñå ðàçãëåæäà îùå è êàòî genuine Szasz-MirakjanDurrmeyer îïåðàòîð. Êàòî ãëàâåí ðåçóëòàò äîêàçâàìå êîìóòàòèâíîñò íà
òîçè îïåðàòîð, êàêòî è êîìóòàòèâíîñò ñ ïîäõîäÿù äèôåðåíöèàëåí îïåðàòîð
è äîêàçâàìå ñèëíî îáðàòíî íåðàâåíñòâî îò òèï À çà àïðîêñèìàöèÿ íà
ðåàëíî-çíà÷íè îãðàíè÷åíè è íåïðåêúñíàòè ôóíêöèè â èíòåðâàëà [0; ∞).
Çàåäíî ñúñ ñúîòâåòíàòà ïðàâà òåîðåìà óñòàíîâÿâàìå åêâèâàëåíòíîñò íà
ãðåøêàòà íà àïðîêñèìàöèÿòà ñ ïîäõîäÿù Ê-ôóíêöèîíàë è ìîäóë íà ãëàäêîñò.
50. H.Gonska, J.Prestin and G.Tachev, New estimates on Holder approximation by Bernstein operators, accepted in Appl. Math. Letters,26(2013),
no.1,48-53.IF=1.371,ZB-06106993,MR-2971397
Abstract.In this work we discuss the rate of simultaneous approximation
of H
older continuous functions by Bernstein operators, measured by Holder
norms with dierent exponents. We extend the known results on this topic.
13
Ðåçþìå. ñòàòèÿòà ðàçãëåæäàìå ñòåïåíòà íà ïðèáëèæåíèå íà ôóíêöèÿ,
íåïðåêúñíàòà ïî Õüîëäåð è íåéíèòå ïðîèçâîäíè ñ îïåðàòîðà íà Áåðíùàéí,
èçìåðåíà â Õüîëäåðîâè íîðìè ñ ðàçëè÷íè åêñïîíåíòè â äâåòå ñòðàíè. Ïî
òîçè íà÷èí ðàçøèðÿâàìå èçâåñòíèòå äîñåãà ðåçóëòàòè â òàçè îáëàñò.
51. G.Tachev, Global smoothness preservation with second order moduli
of smoothness, accepted in Math. Slovaca, 2012. IF=0.269
Abstract.We establish the global smoothness preservation of a function
f by the sequence of linear positive operators. Our estimate is in terms of the
second order Ditzian-Totik modulus of smoothness. Application is given to
the Bernstein operator.
Ðåçþìå. Äîêàçâàì çàïàçâàíå íà ãëîáàëíàòà ãëàäêîñò íà ôóíêöèÿ
f ïðè ïðèáëèæàâàíåòî è ñ ðåäèöà îò ëèíåéíè ïîëîæèòåëíè îïåðàòîðè.
Îöåíêàòà íè å â òåðìèíèòå íà âòîðèÿ ìîäóë íà ãëàäêîñò íà ÄèòöèàíÒîòèê. Ïîëó÷åíèÿò ðåçóëòàò å ïðèëîæåí çà ïîëèíîìèòå íà Áåðíùàéí.
52. Teodora Zapryanova and Gancho Tachev, Generalized Inverse Theorem for Schoenberg Operator, in Journal of Modern Mathematics FrontierJMMF, 1(2012), no.2, 11-16.
Abstract.For Schoenberg splines of degree k ≥ 1 -xed and n → ∞ ,
we establish inverse result in terms of the second order classical moduli. As
a consequence, we generalize the earlier inverse estimates of Berens-Lorenz
(n=1) and of Beutel, Gonska etc. for (k = 1, 2, 3).
Ðåçþìå. Çà îïåðàòîðà íà Øîåíáåðã îò ñòåïåí k ≥ 1 -ôèêñèðàíî è
n → ∞ ,óñòàíîâÿâàìå îáðàòíà òåîðåìà â òåðìèíèòå íà âòîðèÿ ìîäóë íà
íåïðåêúñíàòîñò. Kaòî ñëåäñòâèå îáîáùàâàìå èçâåñòíèòå ïî-ðàíî îáðàòíè
òåîðåìè íà Áåðåíñ-Ëîðåíòö (n=1) è íà Áîéòåë, Ãîíñêà î äð. çà (k=1,2,3).
53. H.Gonska, J.Prestin, G.Tachev and D.X.Zhou, Simultaneous Approximation with Bernstein polinomials in H
older norms, in Mathematische
Nachrichten,286(2013),no.4, 349-359, IF=0.682.
14
Abstract.In this paper we discuss approximation of continuous functions f on [0, 1] in Holder norms including simultaneous approximation of
derivatives of f.
Ðåçþìå. â òàçè ñòàòèÿ èçó÷àâàìå ïðèáëèæåíèåòî íà íåïðåêúñíàòà
ôóíêöèÿ âúðõó [0, 1] â Õüîëäåðîâè íîðìè, âêëþ÷âàéêè åäíîâðåìåííîòî
ïðèáëèæåíèå íà ïðîèçâîäíèòå íà ôóíêöèÿòà.
54. Heiner Gonska and Gancho Tachev, On the composition and decomposition of positive linear operators IV: Favard-Bernstein operators revisited,
in General Mathematics, 20(2012), no.5, 37-46.
Abstract.Following a 1939 article of Favard we consider the composition
of classical Bernstein operators and piecewise linear interpolation at mutually
distinct knots in [0, 1], not necessarily equidistant. We prove direct theorems
in terms of the classical and the Ditzian-Totik modulus of second order.
Ðåçþìå. Ñëåäâàéêè èçâåñòíàòà ñòàòèÿ íà Ôàâàðä îò 1939 ã. ðàçãëåæäàìå
êîìïîçèöèÿ íà êëàñè÷åñêèÿ îïåðàòîð íà Áåðíùàéí è ÷àñòè÷íî-ëèíåéíèÿ
èíòåðïîëàíò ñ âúçëè â èíòåðâàëà [0, 1] -ïðîèçâîëíè. Äîêàçâàìå ïðàâè
òåîðåìè â òåðìèíèòå íà êëàñè÷åñêèÿ âòîðè ìîäóë íà ãëàäêîñò è âòîðèÿ
ìîäóë íà Äèòöèàí-Òîòèê.
55. Margareta Heilmann and Gancho Tachev, Linear Combinations of
Genuine Szasz-Mirakjan - Durrmeyer Operators, in Springer Proceedings
in Mathematics and Statistics with Volume title Advances in Applied
Mathematics and Approximation Theory-Contributions from AMAT
2012 Conference, Turkey, ed by G. Anastassiou and O. Duman,(2012), 5-th
Chapter, 85-106.
Abstract.We study approximation properties of linear combinations of
the genuine Sz asz-Mirakjan-Durrmeyer operators which are also konown as
Phillips operators. We prove a full quantitative Vornonovskaja-type theorem
generalizing and improving earlier results by Agrawal, Gupta and May. A
Voronovskaja-type result for simultaneous approximation is also established.
Furthermore global direct theorems for the approximation and weighted simultaneous approximation in term of the Ditzian-Totik modulus of smoothness are proved.
15
Ðåçþìå. òàçè ñòàòèÿ èçó÷àâàìå àïðîêñèìàòèâíèòå ñâîéñòâà íà ëèíåéíè
êîìáèíàöèè îò îïåðàòîðè íà Ôèëèïñ-èëè òàêà íàðå÷åíèòå genuine Sz aszMirakjan-Durrmeyer îïåðàòîðè. Äîêàçâàìå Òåîðåìà îò òèïà íà Âîðîíîâñêà
è òàêà îáîáùàâàìå ðåäèöà ðåçóëòàòè íà Agrawal, Gupta and May. Ñëó÷àÿò
íà simultaneous approximation å ñúùî ðàçãëåäàí.
56. Teodora Zapryanova and Gancho Tachev, Approximation by the iterates of Bernstein operator, in AIP Conf. Proc., 1497, (2012), 184-189.SJP=0.025.
Abstract.We study the degree of pointwise approximation of the iterated Bernstein operators to its limiting operator. We obtain a quantitative
estimates related to the conjecture of Gonska and Rasa.
Ðåçþìå. Èçñëåäâàìå ñòåïåíòà íà ïîòî÷êîâà àïðîêñèìàöèÿ ñ èòåðàöèèòå
íà îïåðàòîðà íà Áåðíùàéí. Ïîëó÷åíàòà êîëè÷åñòâåíà îöåíêà å ñâúðçàíà
ñ åäíà õèïîòåçà íà Ãîíñêà è Ðàøà.
-
57. Gancho T. Tachev, Rened estimates for the equivalence between
Ditzian-Totik moduli of smoothness and K -functionals, in "Theory and
Applications of Mathematics Computer Sciences"TAMSC, 2(2012),
no.2,48-54.
Abstract.The aim of this note is to study the magnitude of the constants in the equivalence between the rst and second order Ditzian-Totik
moduli of smoothness and related K-functionals. Applications to some classic
approximation operators are given.
Ðåçþìå.â òàçè ñòàòèÿ, íàïèñàíà ïî ïîêàíà, èçó÷àâàìå ìàãíèòóäà íà
êîíñòàíòèòå â åêâèâàëåíòíîñòòà íà ïúðâèÿ è âòîðèÿ ìîäóë íà ÄèòöèàíÒîòèê ñúñ ñúîòâåòíèòå Ê-ôóíêöèîíàëè. Äàäåíè ñà ïðèëîæåíèÿ çà êëàñè÷åñêè
îïåðàòîðè.
16