Table Of ContentDevelopments in Mathematics
Vijay Gupta
Gancho Tachev
Approximation
with Positive
Linear Operators
and Linear
Combinations
Developments in Mathematics
Volume 50
Serieseditors
KrishnaswamiAlladi,Gainesville,USA
HershelM.Farkas,Jerusalem,Israel
Moreinformationaboutthisseriesathttp://www.springer.com/series/5834
Vijay Gupta • Gancho Tachev
Approximation with Positive
Linear Operators and Linear
Combinations
123
VijayGupta GanchoTachev
DepartmentofMathematics DepartmentofMathematics
NetajiSubhasInstituteofTechnology UniversityofArchitectureCivilEngineering
NewDelhi,India andGeodesy
Sofia,Bulgaria
ISSN1389-2177 ISSN2197-795X (electronic)
DevelopmentsinMathematics
ISBN978-3-319-58794-3 ISBN978-3-319-58795-0 (eBook)
DOI10.1007/978-3-319-58795-0
LibraryofCongressControlNumber:2017940878
MathematicsSubjectClassification:41A25,41A30,30E05,30E10
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Preface
Approximation of functions by positive linear operators is an important branch of
theapproximationtheory.Toincreasetheorderofapproximationausefultoolisthe
methodoflinearcombinationsofpositivelinearoperators(p.l.o.).Themostknown
example of p.l.o. is the famous Bernstein operators introduced by S. Bernstein
[26,27],whichforf 2CŒ0;1(cid:2)isgivenby
n !
X n
B .f;x/D p .x/f.k=n/; p .x/D xk.1(cid:2)x/n(cid:2)k:
n n;k n;k
k
kD0
In1932ElenaVoronovskaja[193]—adoctoralstudentofS.Bernstein—provedthat
iff isboundedonŒ0;1(cid:2),differentiableinsomeneighbourhoodofxandhassecond
derivativef00forsomex2Œ0;1(cid:2)then
x.1(cid:2)x/
lim nŒB .f;x/(cid:2)f.x/(cid:2)D f00.x/:
n
n!1 2
If f 2 C2Œ0;1(cid:2) the convergence is uniform. This result shows that the rate of
convergence B .f;x/ (cid:2) f.x/ ! 0 as n ! 1 is of order not better than 1=n
n
if f00.x/ ¤ 0: In this sense the theorem of E. Voronovskaja is a first example
of saturation, i.e. the optimal rate of convergence is 1=n: To increase the rate of
convergence, it was P. L. Butzer [34] who introduced in 1953 the following linear
combinations:
.2r(cid:2)1/B .f;x/D2rB .f;x/(cid:2)B .f;x/;B .f;x/(cid:3)B .f;x/:
n;r 2n;r(cid:2)1 n;r(cid:2)1 n;0 n
Butzershowedthat,forsmoothfunctionsf;B .f;x/(cid:2)f.x/tendsto0fasterthan
n;r
B .f;x/(cid:2)f.x/: More general combinations are considered by Rathore [163] and
n
v
vi Preface
C. P. May in their PhD thesis and in the year 1976 in [145]. The kth order linear
combinationsL .f;k;x/oftheoperatorsL .f;x/;discussedby[145],aregivenas
n djn
k
X
L .f;k;x/D C.j;k/L .f;x/;
n djn
jD0
where
k
Y dj
C.j;k/D ;k¤0IC.0;0/D1
d (cid:2)d
j i
iD0
i¤j
andd ;d ;:::::;d arekC1distinctarbitraryandfixedpositiveintegers.
0 1 k
Z. Ditzian, the scientific advisor to C. P. May, in the famous book Moduli
of Smoothness [50] written jointly with V. Totik in 1987 generalized the known
methodsoflinearcombinations(seeChapter9inthebook).Thelinearcombinations
L forr2NoftheoperatorsL aregivenby
n;r ni
r(cid:2)1
X
L .f;x/D ˛.n/L .f;x/
n;r i ni
iD0
wherethenumbersn andcoefficients˛.n/satisfythefollowingfourconditions:
i i
(a) anDn <n <(cid:4)(cid:4)(cid:4)n (cid:5)An;
0 1 r(cid:2)1
(b) Pr(cid:2)1j˛.n/j<C;
iD0 i
(c) Pr(cid:2)1˛.n/D1;
iD0 i
(d) Pr(cid:2)1˛.n/n(cid:2)(cid:3) D0;(cid:3)D1;2;(cid:4)(cid:4)(cid:4)r(cid:2)1:
iD0 i i
The last two conditions represent a linear system for the coefficient ˛.n/ with
i
uniquesolution
r(cid:2)1
Y ni
˛.n/D :
i
n (cid:2)n
i k
kD0;k¤i
NotethatL DL (foraD1).Inourbookwefollowthismoregeneralframework
n;0 n
oflinearcombinations.InChapter9in[50],DitzianandTotikamongothersproved
thefollowingequivalenceresult(seeTheorem9.3.2)
jjL f (cid:2)fjj DO.n(cid:2)˛=2/,!2r.f;h/ DO.h˛/;
n;r B ' B
0 < ˛ < 2r; where the space B and weight function ' are defined as follows: for
p
L D B —Bernstein operator, B D CŒ0;1(cid:2);'.x/ D x.1(cid:2)x/I for L D S —
n n p n n
Szász–Mirakjan operator B D CŒ0;1/;'.x/ D xI for L D V —Baskakov
n n
Preface vii
p
operator B D CŒ0;1/;'.x/ D x.1Cx/ and lastly for their Kantorovich
variants the weight functions remain the same, where B D L Œ0;1(cid:2) for BO and
p n
B D L .0;1/;1 (cid:5) p < 1 for SO;VO and L are the linear combinations
p n n n;r
of these classical p.l.o.—Bernstein, Szász–Mirakjan and Baskakov operators and
their Kantorovich [126] modifications. The case of Post–Widder operators was
also considered. Since then in the last three decades, hundreds of papers have
appearedandconsideringdifferentproblemsconnectedwiththemethodsoflinear
combinations of p.l.o. We only mention the dissertation of M. Heilmann [115]
published in 1992 which may be considered as a second systematical study of
linearcombinationsattachedtoDurrmeyermodificationsofthreeclassicaloperators
mentionedabove.Itishardlypossibletomentionallresultsonthistopic.Together
withknownresultsinthepastweincludealsothenewresultsobtainedveryrecently
in our joint papers and also results obtained by many other mathematicians in the
past 5–10 years. Some of the results are formulated and the reader may find the
proofsinthereferencesgivenattheendofthebook.
Thebookconsistsofeightchapters.Inthefirsttwochapters,wegivetheknown
results about the closed expressions (when it is possible) of the moments and the
central moments of the operators L , two expressions which are crucial tools for
n
further investigation of approximation by linear combinations. Direct and inverse
estimates for a broad class of p.l.o. are considered in the next chapters. The cases
offiniteandunbounded intervalsofthereal-valuedandcomplex-valued functions
areconsidered.Welistalsotheresultsforapproximationbylinearcombinationsin
a pointwise form, obtained very recently. The known strong converse inequalities
of type A in the terminology of Ditzian–Ivanov [51] for linear combinations of
Bernstein and Bernstein–Kantorovich operators are also included. We represent
alsovariousVoronovskaja-typeestimatesforsomelinearcombinations.Someopen
problemsarealsooutlined,concerningtheapproximationbylinearcombinationsof
p.l.o.
Quantitative estimates for the sequences of p.l.o. play an important role not
merely inapproximating thefunctions, but also infinding the error of approxima-
tion.Oneofthemostimportantconvergenceresultsinthetheoryofapproximation
is the Voronovskaja-type theorem, which describes the rate of pointwise conver-
gence. The quantitative version of the Voronovskaja theorem for any p.l.o. acting
oncompactintervalswasobtainedin[80].AlsoAcar–Aral–Rasain[7]established
quantitativeresultsforweightedmodulusofcontinuityintherecentyears.Paˇltaˇnea
in[156,157]introducedtheweightedmodulusofcontinuity.Herewediscusssome
oftheresultsappearedintherecentyearsonsuchproblems.Alsointhelast3years
somepapersonnewhybridoperatorsappeared;wealsodiscusssomeofthem.
In the recent years R. Paˇltaˇnea in [155] proposed the generalization of Phillips
operators based on certain parameter (cid:3) > 0, which has a link to the well-known
Szász–Mirakjanoperatorsinlimitingcase.Afterthatalsomanysuchoperatorshave
been appropriately modified so that they depend on certain parameters and in the
limitingcasetheyreducetothewell-knownoperatorsavailableintheliterature.We
alsodiscusssomeofthepapersinthisdirection.
viii Preface
Itisourgoalinthisbooktodescribethemostinterestingfeaturesconnectedwith
approximationbylinearcombinationsofp.l.o.Wehopeourbookmaynotonlybe
considered as a systematic overview but also be served as a basis for future study
anddevelopmentofthismethod.
NewDelhi,India VijayGupta
Sofia,Bulgaria GanchoTachev
Some Words
The first author works as a professor in the Department of Mathematics, NSIT,
New Delhi. His area of research is Approximation Theory, especially on linear
positive operators and application of q-calculus in approximation theory. He has
collaboratedjointlywithmanyresearchersglobally.Thefirstauthorisinspiredby
the work of many researchers specially of Prof. Zeev Ditzian, Prof. Ulrich Abel,
Prof.MargaretaHeilmann,Prof.MirceaIvan,Prof.HeinerGonska,Prof.Gradimir
V. Milovanovic´, Prof. Octavian Agratini, Prof. Radu Paˇltaˇnea, Prof. Sorin Gal,
Prof.PurshottamNarainAgrawal,Prof.Th.M.RassiasandProf.AliAral.Hegota
chancetomeetsomeofthempersonallyduringhisvisitstoJaenUniversity,Spain;
IndianInstituteofTechnologyRoorkee,India;LucianBlagaUniversity,Romania,
andKirikkaleUniversity,Turkey.
ThesecondauthorworksintheDepartmentofMathematicsattheUniversityof
Architecture, Civil Engineering and Geodesy, Sofia, Bulgaria. His attention to the
theoryofpositivelinearoperators,especiallyonlinearcombinations,wasbrought
tohimduringhisvisitstotheUniversitiesofDuisburg-EssenandWuppertalinthe
period2000–2012andcollaborationwithProf.HeinerGonskaandProf.Margareta
Heilmann. The participation at Romanian-German Seminars on Approximation
Theoryinthelast15yearshasgiventheopportunitytoestablishscientificcontacts
and collaboration with Prof. Radu Paˇltaˇnea, Prof. Ioan Gavrea, Prof. Sorin Gal,
Prof. Daniela Kacso and others. In the last 4 years as a result of the joint work of
boththeauthorsseveralpapershaveappeared,dealingwithapproximationbylinear
combinationsanddirectestimatesofp.l.o.
The authors believe that the book will motivate other mathematicians to
obtain new exciting results on this topic. They are thankful to their collaborative
researchers, friends and also students for valuable suggestions. The authors thank
the Springer-Verlag team for publishing the book timely. They are grateful to
the reviewers and Ms. Razia Amzad for valuable suggestions leading to overall
improvementsofthemanuscript.
ix
