Approximation by complex Bernstein and convolution type operators

350 Pages·2009·1.42 MB·English

by  Sorin G. Gal

APPROXIMATION BY
COMPLEX BERNSTEIN
AND CONVOLUTION TYPE
OPERATORS
SERIES ON CONCRETE AND APPLICABLE MATHEMATICS
ISSN: 1793-1142
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Department of Mathematical Sciences
The University of Memphis
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Series on Concrete and Applicable Mathematics – Vol. 8
APPROXIMATION BY
COMPLEX BERNSTEIN
AND CONVOLUTION TYPE
OPERATORS
Sorin G Gal
University of Oradea, Romania
World Scientific
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APPROXIMATION BY COMPLEX BERNSTEIN AND CONVOLUTION
TYPE OPERATORS
Series on Concrete and Applicable Mathematics — Vol. 8
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Preface
The Bernstein polynomials attached to f :[0;1] R and given by
!
n
k n
B (f)(x)= p (x)f( ); p (x)= xk(1 x)n k;x [0;1];
n n;k n;k (cid:0)
n k (cid:0) 2
k=0 (cid:18) (cid:19)
X
probablyarethe mostfamousalgebraicpolynomialsin ApproximationTheoryand
wereintroducedin1912byS.N.Bernstein[38]inordertogivethe(cid:12)rstconstructive
(and simple) proof to the Weierstrass’ approximation theorem. Many books and
papers were dedicated to their study, probably the most known being the book of
Lorentz [125]. The importance of Bernstein polynomials also consists in the fact
that their form has suggested and still continues to suggest to mathematicians the
construction of a great variety of other approximation operators, like the Schurer
polynomials, Kantorovich polynomials, Stancupolynomials, q-Bernstein polynomi-
als,Durrmeyerpolynomials,Favard-Sza(cid:19)sz-Mirakjan operators,Baskakov operators,
and the list can continue with very many others.
AnaturalquestionconcerningtheBernsteinpolynomialsofarealvariable(and
by analogy, concerning any Bernstein-type operator of a real variable x) is the
following : if in the expression of B (f)(x) one replaces x [0;1] by z in some
n
2
regions in C (containing [0;1]) where f is supposed to be analytic, (a process we
callcomplexi(cid:12)cation),thenwhatconvergencepropertieshavethecomplexBernstein
polynomials
n
k n
B (f)(z)= p (z)f( ); p (z)= zk(1 z)n k;z C ?
n n;k n;k (cid:0)
n k (cid:0) 2
k=0 (cid:18) (cid:19)
X
Inotherwords,theproblemistostudytheoverconvergencephenomenon(notin
the sense known as Wash’s overconvergencein the interpolation of functions !) for
theBernsteinpolynomials,thatistoextendtheirconvergencepropertiesandorders
oftheseconvergenciestolargersetsinthecomplexplanethantherealinterval[0;1].
The (cid:12)rst goal of the present book is to give some answers in Chapter 1 to the
above question of overconvergence, for several classes of Bernstein-type operators.
In essence, it will be shown that for all the Bernstein-type operators, the orders of
approximation from the real axis are preserved in complex domains too.
vii
viii Approximation byComplex Bernsteinand Convolution TypeOperators
We recall that concerning the complex Bernstein polynomials, Wright [199],
Kantorovich[113], Bernstein [39; 40; 41], Lorentz [125] and Tonne [190] have given
interesting answers to this question. It is worth noting that an entire Chapter
4 of 38 pages is dedicated to it in the book of Lorentz [125]. In that book in-
teresting convergence properties of B (f)(z) and of its so-called degenerate form,
n
in various domains in C, like compact disks, ellipses, loops, autonomous sets are
presented. Notice that in the above mentioned papers no quantitative estimates
of these convergence results were obtained. Also, convergence results without any
quantitative estimate were obtained for the complex Favard-Sza(cid:19)sz-Mirakjanopera-
tors by Dressel-Gergen and Purcell [65] and for the complex Jakimovski-Leviatan
operators by Wood [200]. The above qualitative results are theoretically based
on the "bridge"made by the classicalresult of Vitali (see Theorem1.0.1), between
the(well-established)approximationresultsfortheBernstein-typeoperatorsofreal
variable and those for the Bernstein-type operators of complex variable.
It is worthnoting that in the other books orlong surveysdealing with complex
approximation, like those of Sewell [162], Dzjadyk [69], Gaier [76], Suetin [186],
Andrievskii-Belyi-Dzjadyk[26], [27], orthe surveysofDyn’kin [62]andAndrievskii
[25], the topic of the present book is not considered. Also, in other books like
Lorentz [125](Chapter 4)and Gal [77] (Chapters3and 4), the topic of the present
book is attended in a tangential (and somehow di(cid:11)erent) way only.
InthePrefaceoftheirimportantbook[60]in1993concerningtheApproximation
Theory of functions of real variable, DeVore and Lorentz note that, I cite "the
ApproximationTheoryoffunctionsofcomplexvariableswouldrequirenewbooks".
The present book seeks to be one among the answers to this requirement and can
brie(cid:13)y be described as follows.
In Chapter 1 one deepens the study of the approximation properties for the
complex Bernstein polynomials B (f)(z) in compact disks and in some special
n
compact subsets of C. In addition, similar results for other Bernstein-type polyno-
mials/operatorsincluding those mentioned above are presented.
In detail, Chapter 1 can be described as follows :
{Section1.0containsthemainresultsandconceptsincomplexanalysisrequired
fortheproofsoftheresultsinthisbook. Forexample,wementionheretheVitali’s
theorem,Cauchy’sformula,Bernstein’sinequality,Faberpolynomialsassociatedto
a domain in C, Faber series, Faber coe(cid:14)cients, Faber mapping.
{ in Section 1.1 the exact orders in simultaneous approximation by B (f)(z)
n
and its derivatives, Voronovskaja’s result with quantitative upper estimate and
shape preserving properties of B (f)(z) are obtained ; Subsection 1.1.1 contains
n
the results on compact disks centered at origin while Subsection 1.1.2 contains
some approximationresults on compact sets in C for the so-calledBernstein-Faber
polynomials ;
{ Section 1.2 contains convergence results with quantitative estimates of the
iterates of B (f)(z), connected with the theoryof the semigroupsof operatorsand
n
Preface ix
the shape preserving properties of these iterates, in the sense that beginning with
anindextheypreservesomepropertiesoff inGeometricFunctionTheory,likethe
starlikeness, convexity and spirallikeness ;
{ in Section 1.3 the exact order in the generalized Voronovskaja’s theorem for
B (f)(z) is obtained ;
n
{Section1.4presentstheexactordersofapproximationbyButzer’slinearcom-
binations of complex Bernstein polynomials and of Bernstein-Faberpolynomials in
compact disks and in compact Faber sets, respectively ;
{ in Sections 1.5, 1.6, 1.7, 1.8, 1.9 and 1.10 we prove some similar properties
forthecomplexq-Bernsteinpolynomials,Bernstein-Stancupolynomials,Bernstein-
Kantorovich polynomials, Favard-Sza(cid:19)sz-Mirakjan operators, Baskakov operators
and Bala(cid:19)zs-Szabados operators, respectively. Besides the approximation results in
compactdisksforallthesecomplexBernstein-typeoperators,itisworthmentioning
heretheapproximationresultsfortheBernstein-Stancu-Faberpolynomialsincom-
pactsetsinC,aweighted-kindapproximationresultfortheFavard-Sza(cid:19)sz-Mirakjan
operator in strips and the study of two kinds of complex Baskakov operators gen-
erated by the real one ;
{Section1.11containsbibliographicalnotesandsomeopenproblems. Theopen
problems mainly consist in proposals of similar studies for other types of complex
Bernstein-type operators too.
In Chapter 2 we extend some of the results in Chapter 1 to the case of sev-
eral complex variables. In Section 2.1 the concepts and results in the complex
analysis of functions of several complex variables that we need for this chapter are
presented. Section 2.2dealswith the approximationbytwokindsofbivariatecom-
plex Bernstein polynomials, while Sections 2.3 and 2.4 treat the bivariate case of
complex Favard-Sza(cid:19)sz-Mirakjan and Baskakov operators, respectively. All the ap-
proximation results are obtained in compact polydisks. Section 2.5 contains some
bibliographical notes and open problems.
Chapter 3 deals with the approximation and geometric properties of several
types of complex convolutions. Section 3.1 contains the approximation proper-
tiesofsomecomplexlinearconvolutionpolynomials: ofdelaVall(cid:19)eePoussin,Fej(cid:19)er,
Riesz-Zygmund,JacksonandRogosinskikinds. Moreexactly,forthesecomplexlin-
earconvolutionsVoronovskaja-typeresultsandtheexactordersofapproximationin
compactdisksareproved. Section3.2studiesseveralkindsoflinearnon-polynomial
convolutions. Thus, in Subsection 3.2.1 one studies the approximation properties
(including the exact orders of approximation) in compact disks and compact sets
in C of the non-polynomial complex convolutions of Picard, Poisson-Cauchy, and
Gauss-Weierstrass. Also, their geometric properties are studied and applications
to PDE in complex setting (i.e to heat and Laplace equations of complex spatial
variable) are presented. In the Subsections 3.2.2, 3.2.3, 3.2.4 and 3.2.5 the ap-
proximation and geometric properties in compact disks of the complex q-Picard
andq-Gauss-Weierstrassconvolutions,Post-Widdercomplexconvolution,rotation-