bernstein operators for extended chebyshev systems abstract let u n c n a b be an extended chebyshev space of dimension n 1 suppose that f 0 u n is strictly positive and f 1 u n has the property that f 1 f 0 is strictly increasing we search for conditions ensuring the existence of points t 0 t n a b and positive coefficients 0 n such that for all f c a b the operator b n c a b u n defined by b n f k 0 n f t k k p n k satisfies b n f 0 f 0 and b n f 1 f 1 here it is assumed that p n k k 0 n is a bernstein basis defined by the property that each p n k has a zero of order k at a and a zero of order n k at b

blaschke oscillatory equations of the form f a z f 0 abstract we study the zero sequences of the non trivial solutions of f a z f 0 where a z is analytic in the unit disc we offer several aspects illustrating the fact that it is not so uncommon for these zero sequences to be blaschke sequences the typical results can be divided into two categories 1 we search for conditions on a z under which the zero sequences of solutions of are blaschke sequences 2 for given blaschke sequences satisfying certain conditions we construct an analytic function a z of minimal growth such that these blaschke sequences are the zero sequences of certain solutions of this discussion is a continuation of the recent paper j heittokangas solutions of f a z f 0 in the unit disc having blaschke sequences as the zeros comput methods funct theory 5 2005 49 63

entire functions that share one value with one or two of their derivatives we prove two uniqueness theorems for entire functions of finite order that share one finite value with one or two of their derivatives

growth and poles of meromorphic solutions of some difference equations abstract this paper is devoted to studying the growth property and the pole distribution of meromorphic solutions f of some complex difference equations with all coefficients being rational functions or of growth s r f we find the lower bound of the lower order or the relation between lower order and the convergence exponent of poles of meromorphic solutions of such equations

the radial oscillation of entire solutions of complex differential equations abstract this paper is devoted to studying the radial oscillation of solutions of linear differential equations f a z f b z f f z where a z b z and f z are transcendental entire functions with some added conditions on coefficients we show that there exists some close relation between the borel directions of solutions and that of f z and also obtain some estimates of growth of solutions in the angular domains

a note on the optimal stability of bases of univariate functions this note is concerned with the characterizations and uniqueness of bases of finite dimensional spaces of univariate continuous functions which are optimally stable for evaluation with respect to bases whose elements have no sign changes

on the question of whether f e zf b z f 0 can admit a solution f 0 of finite order on montre que si b z est soit i une fonction entiere transcendante d ordre 1 soit ii un polynome de degre impair alors toute solution f 0 de l equation f e z f b z f 0 est d ordre infini on obtient un resultat partiel quand b z est un polynome de degre pair

further results on entire functions that share one value with their derivatives abstract this paper deals with problems of the uniqueness of entire functions that share one value with their derivatives the results in this paper generalize a result of jank mues and volkmann and answer a question posed by h zhong and a question of yi and yang

entire functions that share one value with their linear differential polynomials this paper studies the uniqueness problem on entire function that share a finite nonzero value with their linear differential polynomials and proves some theorems which generalize some results given by jank mues and volkmann p li j p wang and h x yi

entire functions that have the same fixed points with their first derivatives let f be a nonconstant entire function and let k 2 be an integer we denote by ef z z c f z z counting multiplicities the set consisting of all the fixed points of f this paper proves that if f and f have the same fixed points namely ef z ef z and if f k z z whenever f z z then f f

finite order solutions of linear differential equations in the unit disc abstract necessary and sufficient conditions for the analytic coefficients of the complex linear differential equation f k a k 1 z f k 1 a 1 z f a 0 z f a k z are found such that all solutions satisfy f lim sup r 1 log t r f log 1 r moreover estimates for the number of linearly independent solutions of maximal growth are found in terms of the growth of the coefficients in addition sufficient conditions for the coefficients such that the zero sequence z n of any non trivial solution f of with a k 0 satisfies 1 z n 1 are found several non trivial examples are given in order to show that the established results are in a sense sharp

error analysis of efficient evaluation algorithms for tensor product surfaces backward stability of the casteljau algorithm and two more efficient algorithms for polynomial tensor product surfaces with interest in cagd is shown the conditioning of the corresponding bases are compared these algorithms are also compared with the corresponding horner algorithm and their higher accuracy is shown a running error analysis of the algorithms is also carried out providing algorithms which calculate a posteriori sharp error bounds simultaneously to the evaluation of the surface without increasing significantly the computational cost

on factorization of certain entire functions abstract let p and q be polynomials and let be an entire function suppose that q and are nonconstant we show that the function h z p z e z q z has a factorization h z g z with nonlinear meromorphic left and right factors and g if and only if p q and have such a factorization with a common right factor this confirms conjectures of f gross g d song and c c yang

dual exponential polynomials and linear differential equations abstract we study linear differential equations with exponential polynomial coefficients where exactly one coefficient is of order greater than all the others the main result shows that a nontrivial exponential polynomial solution of such an equation has a certain dual relationship with the maximum order coefficient several examples illustrate our results and exhibit possibilities that can occur

evaluation of the derivative of a polynomial in bernstein form in geometric modelling and computer aided design a polynomial is usually represented in bernstein form forward error analysis for the computation of the derivative of a polynomial in bernstein form is performed the corresponding running error is also analyzed we have compared our results with those corresponding to a polynomial represented in power basis and using legendre and chebyshev forms since the conversions between the bernstein and the power bases and between bernstein and chebyshev polynomials were already studied in the literature for the sake of completeness the conversion between bernstein and chebyshev polynomials of the first kind is analyzed in an appendix

converse growth estimates for odes with slowly growing solutions let f 1 f 2 be linearly independent solutions of f af 0 where the coefficient a is an analytic function in the open unit disc mathbb d of the complex plane mathbb c it is shown that many properties of this differential equation can be described in terms of the subharmonic auxiliary function u log f 1 f 2 for example the case when sup z in mathbb d a z 1 z 2 2 infty and f 1 f 2 is normal is characterized by the condition sup z in mathbb d nabla u z 1 z 2 infty different types of blaschke oscillatory equations are also described in terms of harmonic majorants of u even if f 1 f 2 are bounded linearly independent solutions of f af 0 it is possible that sup z in mathbb d a z 1 z 2 2 infty or f 1 f 2 is non normal these results relate to sharpness discussion of recent results in the literature and are succeeded by a detailed analysis of differential equations with bounded solutions analogues for the nevanlinna class are also considered by taking advantage of nevanlinna interpolating sequences it is shown that instead of considering solutions with prescribed zeros it is possible to construct a bounded solution of f af 0 in such a way that it solves an interpolation problem natural to bounded analytic functions while a z 2 1 z 2 3 dm z remains to be a carleson measure

optimal bases for a class of mixed spaces and their associated spline spaces we consider spaces of the form span where the functions u i i 1 4 are algebraic polynomials or trigonometric or hyperbolic functions we find intervals 0 a where we can guarantee that the spaces possess normalized totally positive bases and so shape preserving representations we construct their normalized b bases with optimal shape preserving properties we also present a unified approach to deal with the associated spline spaces a recursive procedure to obtain bases with optimal shape preserving and stability properties is presented

on the relationship between the lower order of coefficients and the growth of solutions of differential equations abstract some criteria for entire coefficients a z and b z are given in terms of the lower order forcing the solutions of f a z f b z f 0 to grow fast in the literature similar criteria have been published in terms of the usual order the case when the coefficient a z has an asymptotic growth t r a log m r a 0 1 outside of an exceptional set is also discussed previously laine wu 2000 and kim kwon 2001 have made use of this asymptotic growth in the case 1

on solutions of linear differential equations with entire coefficients abstract the first result of the paper concerns the effect of perturbation of the entire coefficients of certain linear differential equations on the oscillation of the solutions subsequent results involve the separation of the zeros of a bank laine function

on the complex oscillation of some linear differential equations we treat the linear differential equation f k a z f 0 wherek 2 is an integer anda z is a transcendental entire function of order a it is shown that any non trivial solution of the equation satisfies f a where f is the exponent of convergence of the zero sequence off under the conditionkn r 1 a t r a r efor ak 2kand an exceptional seteof finite linear measure the second order equationf ep1 z ep2 z q z f 0 wherep1 z p2 z are non constant polynomials andq z is an entire function is also studied

running relative error for the evaluation of polynomials running relative error for evaluating polynomials is analyzed and applied to different evaluation algorithms horner vs de casteljau and clenshaw algorithms numerical experiments show that the derived running relative error bounds are sharp and that the de casteljau algorithm presents better stability properties than the remaining algorithms

on the radial distribution of julia sets of entire solutions of f n a z f 0 abstract this paper is devoted to studying the dynamical properties of solutions of f n a z f 0 where n 2 is an integer and a z is a transcendental entire function of finite order we find the lower bound on the radial distribution of julia sets of e z provided that e f 1 f 2 f n and f 1 f 2 f n is a solution base of such equations

finite order solutions of second order linear differential equations on considere l equation differentielle f a z f b z f 0 ou a z et b z sont des fonctions entieres on trouve des conditions sur a z et b z qui garantissent que chaque solution f 0 de l equation a un ordre fini

dynamic evaluation of free form curves and surfaces free form curves and surfaces represented by control points together with well defined basis functions are widely used in computer aided geometric design efficient evaluation of points and derivatives of free form curves and surfaces plays an important role in interactive rendering or cnc machining in this paper we show that free form curves with properly defined basis functions are the solutions of linear differential systems by employing typical numerical methods for solving the differential systems points and derivatives of free form curves and surfaces can be computed in a dynamical way compared with traditional methods for evaluating free form curves and surfaces there are two advantages of the proposed technique first the proposed method is universal and efficient for evaluating a large class of free form curves and surfaces second the evaluation needs only arithmetic operations even when the free form curves and surfaces are defined using some transcendental functions

entire functions that share one value cm with their derivatives this paper studies the uniqueness problem on entire function that share a finite nonzero value cm with their derivatives and proves two main theorems which generalize some results given by jank mues and volkmann p li and c c yang h l zhong etc an example shows that the condition of one of our theorems is necessary

on the meromorphic solutions of non homogeneous linear differential equations with polynomial coefficients on considere un operateur differentiel lineaire d ordre k l k de la forme l k w w k a k 1 w k 1 a 0 w ou k 2 et a 0 a k 1 sont polynomiaux on etudie les fonctions f entieres ou meromorphes sans zero pour lesquelles on peut avoir une fonction meromorphe a l equation l k w f qui n a pas de zeros

on the growth of solutions of a class of higher order linear differential equations with coefficients having the same order abstract in this paper the authors investigate the growth of solutions of a class of higher order linear differential equations f k a k 1 f k 1 a 0 f 0 when most coefficients in the above equations have the same order with each other and obtain some results which improve previous results due to k h kwon k h kwon nonexistence of finite order solutions of certain second order linear differential equations kodai math j 19 1996 378 387 and z x chen z x chen the growth of solutions of the differential equation f e z f q z f 0 sci china ser a 31 2001 775 784 in chinese z x chen on the hyper order of solutions of higher order differential equations chinese ann math ser b 24 2003 501 508 in chinese z x chen on the growth of solutions of a class of higher order differential equations acta math sci ser b 24 2004 52 60 in chinese z x chen c c yang quantitative estimations on the zeros and growth of entire solutions of linear differential equations complex var 42 2000 119 133

entire functions that share a polynomial with their derivatives abstract let f be a nonconstant entire function k and q be positive integers satisfying k q and let q be a polynomial of degree q this paper studies the uniqueness problem on entire functions that share a polynomial with their derivatives and proves that if the polynomial q is shared by f and f cm and if f k z q z 0 whenever f z q z 0 then f f we give two examples to show that the hypothesis k q is necessary

on the optimal stability of the bernstein basis we show that the bernstein polynomial basis on a given interval is optimally stable in the sense that no other nonnegative basis yields systematically smaller condition numbers for the values or roots of arbitrary polynomials on that interval this result follows from a partial ordering of the set of all nonnegative bases that is induced by nonnegative basis transformations we further show by means of some low degree examples that the bernstein form is not uniquely optimal in this respect however it is the only optimally stable basis whose elements have no roots on the interior of the chosen interval these ideas are illustrated by comparing the stability properties of the power bernstein and generalized ball bases

the possible orders of solutions of linear differential equations with polynomial coefficients we find specific inforrnation about the possible orders of transeendental solutions of equations of the form f n pn l z f n l po z f0 where po z pl z apnl z are polynomials with po z g 0 several examples are given

on limit directions of julia sets of entire solutions of linear differential equations abstract this paper is devoted to studying the limit directions of julia sets of solutions of f n a n 1 z f n 1 a 0 z f 0 where n 2 is an integer and a j z j 0 1 n 1 are entire functions of finite lower order with some additional conditions on coefficients we know that every non trivial solution f z of such equations has infinite lower order and prove that the limit directions of julia sets of f z must have a definite range of measure

least supported bases and local linear independence we introduce the concept of least supported basis which is very useful for numerical purposes we prove that this concept is equivalent to the local linear independence of the basis for any given locally linearly independent basis we characterize all the bases of the space sharing the same property several examples for spline spaces are given

a collocation method to find solutions of linear complex differential equations in circular domains in this study we introduce a collocation approach for solving high order linear complex differential equations in circular domain by using collocation points defined in a circular domain and bessel functions of the first kind this method transforms the linear complex differential equations into a matrix equation the matrix equation corresponds to a system of linear equations with the unknown bessel coefficients proposed method gives the analytic solution when the exact solutions are polynomials numerical examples are given to demonstrate the validity and applicability of the technique and the comparisons are made with existing results the results obtained from the examples demonstrate the efficiency and accuracy of the present work all of the numerical computations have been computed on computer using a code written in matlab

periodic solutions for 2kth order ordinary differential equations with resonance in this paper a non variational version of a max min principle is extended and some unique existence results are obtained for the periodic boundary value problem of the higher order ordinary differential equations under a resonant condition

zero separation results for solutions of second order linear differential equations abstract the oscillation of solutions of f a f 0 is discussed by focusing on four separate situations in the complex case a is assumed to be either analytic in the unit disc d or entire while in the real case a is continuous either on 1 1 or on 0 in all situations a is expected to grow beyond bounds that ensure finite oscillation for all non trivial solutions and the separation between distinct zeros of solutions is considered in the complex case it is shown that the growth of the maximum modulus of a determines the minimal separation of zeros of all solutions and vice versa this gives rise to new concepts called zero separation exponents which measure the separation of zeros of either all solutions or of individual analytic functions in d these quantities are defined in terms of the hyperbolic distance while in the complex plane the euclidean distance is used as a by product of these findings the 1955 result of b schwarz which asserts that sup z d a z 1 z 2 2 if and only if the zero sequences of all solutions are separated in the hyperbolic sense is rediscovered the striking plane analogue established reveals that the euclidean distance between all distinct zeros of every solution is uniformly bounded away from zero if and only if a is a constant as an outgrowth of the results new information on the zero distribution of solutions in the classical polynomial coefficient case is also obtained the main results are proved by using a method of localization which naturally induces characterizations of certain subclasses of locally univalent functions in terms of the growth of their pre schwarzian and schwarzian derivatives in the real case it is shown that the separation of zeros of non trivial solutions is restricted according to the growth of a but not conversely

shape preserving properties of generalized bernstein operators on extended chebyshev spaces we study the existence and shape preserving properties of a generalized bernstein operator b n fixing a strictly positive function f 0 and a second function f 1 such that f 1 f 0 is strictly increasing within the framework of extended chebyshev spaces u n the first main result gives an inductive criterion for existence suppose there exists a bernstein operator b n c a b u n with strictly increasing nodes fixing f 0 f 1 in u n if u n subset u n 1 and u n 1 has a non negative bernstein basis then there exists a bernstein operator b n 1 c a b u n 1 with strictly increasing nodes fixing f 0 and f 1 in particular if f 0 f 1 f n is a basis of u n such that the linear span of f 0 f k is an extended chebyshev space over a b for each k 0 n then there exists a bernstein operator b n with increasing nodes fixing f 0 and f 1 the second main result says that under the above assumptions the following inequalities hold b n f geq b n 1 f geq f furthermore b nf is f0 f1 convex for all f0 f1 convex functions f in c a b

fatou sets of entire solutions of linear differential equations abstract this paper is devoted to studying the dynamical properties of solutions of f a z f 0 where a z is an entire function of finite order with some added conditions on a z we prove that all the fatou components of e z are simply connected provided that e f 1 f 2 and f 1 f 2 are two linearly independent solutions of such equations

a generalization of the airy integral for f z n f 0 it is well known that the airy integral is a solution of the airy differential equation f zf 0 and that the airy integral is a contour integral function with special properties we show that there exist analogous special contour integral solutions of the more general equation f z n f 0 where n is any positive integer related results are given

on the complex oscillation of meromorphic solutions of second order linear differential equations in the unit disc abstract the main purpose of this paper is to investigate the oscillation theory of meromorphic solutions of the second order linear differential equation f a z f 0 for the case where a is meromorphic in the unit disc d z z 1

minimax theorems and applications to the existence and uniqueness of solutions of some differential equations in this paper minimax theorems due to stepan a tersian were generalized the existence and uniqueness of solutions for several semilinear equations were proved by employing our generalized theorems and the existence and uniqueness results of nonresonance problem for these semilinear equations under the asymptotic un uniformity conditions were presented

on the optimal stability of bases of univariate functions this paper is concerned with bases of finite dimensional spaces of univariate continuous functions which are optimally stable for evaluation the only bases considered are those whose elements have no sign changes among these an optimally stable basis is characterized under the assumption that the set of points where each basis function is nonzero is an interval a uniqueness result and many examples of such optimally stable bases are also provided

a corner cutting algorithm for evaluating rational bezier surfaces and the optimal stability of the basis the usual method for evaluating rational bezier surfaces uses the projection operator and the representation provided by the bernstein basis we prove the optimal stability of the basis used in this representation we also propose an alternative method for evaluating rational surfaces through that representation we show the stability properties of this last method and prove that it has better properties than other known methods from the point of view of avoiding underflow and overflow

linear differential equations with entire coefficients of p q order in the complex plane in this paper we firstly investigate the complex high order linear differential equations in which the coefficients are entire functions of p q order and obtain some results which improve and generalize some previous results in cao 2009 3 chen and yang 2000 4 heittokangas et al 2006 12 kinnunen 1998 17 tu and yi 2008 20

complex oscillation and removable sets abstract let a be a transcendental entire function of finite order and let e be the product of linearly independent solutions of w a z w 0 we prove the existence of sequences of annuli m such that if e has relatively few zeros in the union of the m then e has relatively few zeros in the whole plane

composite bank laine functions and a question of rubel a bank laine function is an entire function e satisfying e z 1 at every zero of e we determine all bank laine functions of form e fog with f g entire further we prove that if h is a transcendental entire function of finite order then there exists a path tending to infinity on which h and all its derivatives tend to infinity thus establishing for finite order a conjecture of rubel

a non variational version of a max min principle f u w 4 2 milwl12 1 then there exists a unique uo e h such that vf ug 0 furthermoref uo min f u in 1975 ueh lazer landesman meyer l l m 3 extended these results to the following situation let x and y be two closed subspaces of h such that h x y x is finite dimensional and x and y are not necessarily orthogonal let t vf then t h h and is a c mapping its frechet derivative at u e his given by t rc d f u let ml and ml be two positive constants such that vu e h vx e x and vy e y we have

representations of solutions of periodic second order linear differential equations on considere l equation y g z y h z y 0 a coefficients entiers periodiques arbitraires g z et h z de periode 0 et des solutions y z ayant la propriete que l exposant de convergence de la suite nulle est finie

b splines and optimal stability it is proved that among all nonnegative bases of its space the b spline basis is optimally stable for evaluating spline functions

running error for the evaluation of rational bezier surfaces through a robust algorithm running error analysis of the corner cutting algorithm for rational bezier surfaces is carried out and the sharpness of the corresponding error bounds is shown

a general class of bernstein like bases this paper presents a unified approach to deal with spaces containing simultaneously algebraic and trigonometric or hyperbolic polynomials bases with optimal shape preserving and stability properties are constructed evaluation and subdivision algorithms for them are provided bases for the corresponding and mixed spline spaces are also constructed some nice properties of these bases and the generated curves are shown

linear differential equations with coefficients in weighted bergman and hardy spaces complex linear differential equations of the form with coefficients in weighted bergman or hardy spaces are studied it is shown for example that if the coefficient a j z of belongs to the weighted bergman space a 1 k j where a 0 for all j 0 k 1 then all solutions are of order of growth at most a measured according to the nevanlinna characteristic in the case when a 0 all solutions are shown to be not only of order of growth zero but of bounded characteristic conversely if all solutions are of order of growth at most a 0 then the coefficient aj z is shown to belong to a pj for all p j 0 1 k j and j 0 k 1 analogous results when the coefficients belong to certain weighted hardy spaces are obtained the non homogeneous equation associated to is also briefly discussed

growth of meromorphic solutions of linear difference equations abstract in this paper the authors continue to study the growth of meromorphic solutions of homogeneous or non homogeneous linear difference equations with entire coefficients and obtain some results which are improvement and extension of previous results in chiang and feng 2008 7 and laine and yang 2007 19 examples are also given to illustrate the sharpness of our results

existence and uniqueness results for the bending of an elastic beam equation at resonance etude d un probleme non lineaire aux valeurs limites pour la flexion d une poutre elastique en appui simple aux deux extremites dans le cas de resonance theoremes d existence et d unicite

the growth oscillation and fixed points of solutions of complex linear differential equations in the unit disc abstract we consider the complex differential equations of the form a k z f k a k 1 z f k 1 a 1 z f a 0 z f f z where a 0 0 a 1 a k and f are analytic functions in the unit disc d z c z 1 some results on the finite iterated order and the finite iterated convergence exponent of zero points in d of meromorphic analytic solutions are obtained the fixed points of solutions of differential equations are also investigated in this paper

entire functions that share a small function with their derivatives abstract let f be an entire function of finite order and a an entire function of order less than f s if f a and f a have the same zeros with the same multiplicities then f a c f a for some non zero constant c

removable sets in the oscillation theory of complex differential equations abstract letf1 f2be two linearly independent solutions of the linear differential equationf a z f 0 wherea z is transcendental entire and assume that the exponents of convergence for the zero sequences off1 f2satisfy max f1 f2 our main result proves that the zeros ofe f1f2are uniformly distributed in the sense that quite arbitrary large areas of the complex plane can be removed in such a way that if only zeros outside of these areas will be counted for the exponents of convergences their maximum still remains infinite

running error for the evaluation of rational bezier surfaces error analysis of the usual method to evaluate rational bezier surfaces is performed the corresponding running error analysis is also carried out and the sharpness of our running error bounds is shown we also modify the evaluation algorithm to include such error bounds without increasing significantly its computational cost

the weighted w2 p estimate on the solution of the gellerstedt equation in the upper half space abstract in this note we are concerned with the weighted w 2 p estimate for the gellerstedt equation which is degenerate on the boundary r 2 based on the estimates of the generalized airy functions and mihlin s multiplier theorem we establish a weighted w 2 p a priori estimate