Ioan Gavrea - Academia.edu (original) (raw)

Papers by Ioan Gavrea

Nucleation and Atmospheric Aerosols, 2007

Let Pin denote the set of all algebraic polynomials of degree at most n. Let P(x) = Sigmak = 0nak... more Let Pin denote the set of all algebraic polynomials of degree at most n. Let P(x) = Sigmak = 0nakxk and ||P||dsigma = (∫R|P(x)|2dsigma(x))1/2, where dsigma(x) is a nonnegative measure on R. Milovanovic determined best constants Cnk such that |ak|<=Cnk||P||dsigma, for k = 0,1,...n. In the present work, we will propose a new way of proving the above inequality, which will lead us to finding the optimal constant C, such that ||P||∞<=C||P|dsigma, where ||.||∞ denotes the uniform norm on [0,1].

Applied Mathematics and Computation, Feb 1, 2015

We provide a sharp estimate of the Peano error representation formula for linear functionals. As ... more We provide a sharp estimate of the Peano error representation formula for linear functionals. As applications, we obtain sharp estimates for the remainder term in general quadrature formulae and expand some Ostrowski-like type inequalities to linear functionals. The new approach extends and unifies many earlier results on the subject (Dragomir and

Applied Mathematics and Computation, Aug 1, 2014

ABSTRACT We obtain a new representation of the sum of the squared Bernstein polynomials and use i... more ABSTRACT We obtain a new representation of the sum of the squared Bernstein polynomials and use it to validate a conjecture asserting that this sum is a convex function. The result is extended to some other classical approximation operators.

Journal of Mathematical Analysis and Applications, Jul 1, 2002

Is is shown that for n → +∞ the Leibnizian combination L n (fg) − f L n (g) − gL n (f) converges ... more Is is shown that for n → +∞ the Leibnizian combination L n (fg) − f L n (g) − gL n (f) converges uniformly to zero on a compact interval W if the positive operators L n belong to a certain class (including Bernstein, Gauss-Weierstrass and many others), and if the moduli of continuity of f, g satisfy ω W (f ; h)ω W (g; h) = o(h) as h → 0+. A counterexample shows that Lipschitz conditions are not appropriate to bring about a second-order version of this formula.

Journal of Mathematical Analysis and Applications, Dec 1, 2015

Abstract We prove that a result of Tachev concerning the optimal rate of convergence of the class... more Abstract We prove that a result of Tachev concerning the optimal rate of convergence of the classical Bernstein operators remains valid for the class of discretely defined positive linear operators preserving constants.

Applied Mathematics Letters, Apr 1, 2010

Let n be an integer, n > 1. Theorem. If A : C[a, b] → R is a continuous linear functional orthogo... more Let n be an integer, n > 1. Theorem. If A : C[a, b] → R is a continuous linear functional orthogonal to all polynomials of degree at most n − 1, then the the inequalities A 2 (f) ≤ (−1) k (2k − 1)! A s A t (t − s) 2k−1 + f (k) 2 2 , are satisfied for all f ∈ C n [a, b] and k = 2,. .. , n. The previous theorem generalizes results contained in the following papers:

Journal of Mathematical Analysis and Applications, Feb 1, 2018

We provide complete asymptotic expansions for some sequences of Bernstein-type and Meyer-König an... more We provide complete asymptotic expansions for some sequences of Bernstein-type and Meyer-König and Zeller-type operators preserving the monomials e 0 and e j , j > 1. In particular, this answers a conjecture related to a Voronovskaja-type theorem.

Journal of Approximation Theory, Sep 1, 2011

We have devised a new method for the study of the asymptotic behavior of the iterates of positive... more We have devised a new method for the study of the asymptotic behavior of the iterates of positive linear operators. This technique enlarges the class of operators for which the limit of the iterates can be computed. c

Journal of Mathematical Analysis and Applications, Dec 1, 2010

In this note we study the limit behavior of the iterates of a large class of positive linear oper... more In this note we study the limit behavior of the iterates of a large class of positive linear operators preserving the affine functions and, as a byproduct of our result, we obtain the limit of the iterates of Meyer-König and Zeller operators.

Journal of Mathematical Analysis and Applications, Jun 1, 2013

ABSTRACT We provide a highly convergent version of the generalized Euler sequence and we determin... more ABSTRACT We provide a highly convergent version of the generalized Euler sequence and we determine the element with the optimal rate of convergence in several classes of sequences of a prescribed form.The new approach extends and unifies many previous efforts in this direction. In addition, we answer an open problem on the asymptotic expansion of the Harmonic number sequence.

Results in Mathematics, Dec 6, 2018

We answer and generalize an open problem on the two-dimensional Bernstein polynomials on the unit... more We answer and generalize an open problem on the two-dimensional Bernstein polynomials on the unit triangle.

Journal of Approximation Theory, Apr 1, 2015

We prove that the classical Bernstein Voronovskaja-type theorem remains valid in general for all ... more We prove that the classical Bernstein Voronovskaja-type theorem remains valid in general for all sequences of positive linear approximation operators. c

Abstract and Applied Analysis, 2011

We present a general result concerning the limit of the iterates of positive linear operators act... more We present a general result concerning the limit of the iterates of positive linear operators acting on continuous functions defined on a compact set. As applications, we deduce the asymptotic behaviour of the iterates of almost all classic and new positive linear operators.

Miskolc Mathematical Notes, 2018

We establish some mean value theorems involving n-simple functionals in the sense of Popoviciu. I... more We establish some mean value theorems involving n-simple functionals in the sense of Popoviciu. In particular, we obtain the Kowalewski mean value formula.

Applied Mathematics and Computation, Nov 1, 2013

ABSTRACT The classical Le Verrier algorithm for the calculation of the characteristic polynomial ... more ABSTRACT The classical Le Verrier algorithm for the calculation of the characteristic polynomial of a matrix has led us to a solution of an open problem concerning the Euler-Mascheroni constant gamma. More exactly, we prove that there exists uniquely a sequence of rational numbers (l(n))(n >= 1) such that, for any integer q >= 1, the sequence Sigma(n)(k=1)1/k - log n - log (1 + l(1)/n + ... + l(q)/n(q)), is an n(-q-1) order approximation of gamma.

Journal of Approximation Theory, 2010

We generalize a recent result of de la Cal and Cárcamo concerning an extremal property of Bernste... more We generalize a recent result of de la Cal and Cárcamo concerning an extremal property of Bernstein operators.

Applied Mathematics Letters, Dec 1, 2011

In this note we introduce a simple and efficient technique for studying the asymptotic behavior o... more In this note we introduce a simple and efficient technique for studying the asymptotic behavior of the iterates of a large class of positive linear operators preserving constant functions.

DOAJ (DOAJ: Directory of Open Access Journals), Aug 1, 1982

Georgian Mathematical Journal, Jun 1, 2017

We answer a conjecture and an open problem concerning integral sequences of the form n f(x)f(x) ⋅... more We answer a conjecture and an open problem concerning integral sequences of the form n f(x)f(x) ⋅ ⋅ ⋅ f(x n) dx, n ≥ .

Springer optimization and its applications, 2016

We provide an analysis of the rate of convergence of positive linear approximation operators defi... more We provide an analysis of the rate of convergence of positive linear approximation operators defined on C[0, 1]. We obtain a sufficient condition for a sequence of positive linear approximation operators to possess a Mamedov-type property and give an application to the Durrmeyer approximation process.