Browsing by Author "Özban, Ahmet Yaşar"

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Citation Count: 0
Approximation of Discontinuous Functions by q-Bernstein Polynomials
(Springer international Publishing Ag, 2016) Ostrovska, Sofia; Ozban, Ahmet Yasar; Mathematics
This chapter presents an overview of the results related to the q-Bernstein polynomials with q > 1 attached to discontinuous functions on [0, 1]. It is emphasized that the singularities of such functions located on the set Jq : = {0} boolean OR {q-l}(l=0, infinity), q > 1 are definitive for the investigation of the convergence properties of their q-Bernstein polynomials.
Citation Count: 1
Approximation theory and numerical analysis
(Hindawi Publishing Corporation, 2014) Ostrovska,S.; Berdysheva,E.; Nowak,G.; Özban,A.Y.; Mathematics
[No abstract available]
Durağan yinelemeli yöntemler için yeni ön koşullayıcılar
(2017) Atya, Naıma Ibrahım; Özban, Ahmet Yaşar; Mathematics
Doğrusal denklem sistemlerinin çözümü için kullanılan yinelemeli yöntemlerin ya\-kınsaklığı doğrusal sistem matrisinin spektrumunun özelliklerine bağlıdır. Bu nedenle, yakınsaklığı hızlandırmak için, verilen doğrusal sistem, ön koşullayıcılar olarak bilinen doğrusal dönüşümlerle eşdeğer bir sisteme dönüştürülür. Bu tezde, sütuna göre kesin köşegensel baskın (SKKB) $L-$matrisli ve SKKB pozitif matrisli doğrusal denklem sistemlerinin Jacobi ve Gauss-Seidel (GS) yöntemleriyle çözümü için iki yeni ön koşullayıcı tanımlanmaktadır. Yeni ön koşullayıcılar sistem matrisinin tek bir satırına, sınırlı sayıdaki satırlarına, ki kısmi ön koşullama olarak adlandırılır, veya bütün satırlarına, ki tam ön koşullama olarak adlandırılır, uygulanabilir. İlk olarak, SKKB $L-$matrisler ve SKKB pozitif matrisler için ön koşullanmış matrislerin özellikleri belirlenmektedir. Daha sonra ön koşullandırılmış sistemler için Jacobi ve GS yöntemlerinin yakınsaklık analizleri yapılmaktadır. SKKB $L-$matrisli sistemler için, ön koşullandırılmış sistemlerin Jacobi ve GS yinele\-me matrislerinin spektral yarıçaplarının ön koşullandırılmamış sistemlere karşılık gelenlerden daha küçük olduğu gösterilmektedir. SKKB pozitif matrisli sistemler için, ön koşullandırılmış sistemlerin Jacobi yineleme matrislerinin spektral yarıçapları\-nın ön koşullandırılmamış sistemlere karşılık gelenlerden daha küçük olduğunu ispatlamamıza karşın, GS yineleme matrisleri için böyle bir sonuç mevcut değildir. Sayısal sonuçlar, yeni ön koşullayıcıların, SKKB $L-$matrisli sistemler için spektral yarıçap ve yineleme sayısı bakımından literatürde mevcut olanlarla tümüyle yarışabilir durumda olduğunu gös\-termektedir. Buna karşılık, SKKB pozitif matrisli sistemlere ilişkin sayısal sonuçlar, yeni ön koşullayıcıların diğer bazı önkoşullayıcılar\-la yarışabilir nitelikte olmasına karşın, mevcut ön koşullayıcıların çoğuna karşı tercih edilebilir olmadığını ifade etmektedir. Son olarak, yeni ön koşullayıcıların sütuna göre köşegensel baskın (SKB) $L-$matrisler ve SKB-olmayan $L-$matrisler ve hatta, SKKB-olmayan pozitif matrisler için etkinlikleri, daha fazla araştırmayı haketmektedir.
Citation Count: 2
HOW DO SINGULARITIES OF FUNCTIONS AFFECT THE CONVERGENCE OF q-BERNSTEIN POLYNOMIALS?
(Element, 2015) Ostrovska, Sofiya; Ozban, Ahmet Yasar; Turan, Mehmet; Mathematics
In this article, the approximation of functions with a singularity at alpha is an element of (0, 1) by the q-Bernstein polynomials for q > 1 has been studied. Unlike the situation when alpha is an element of (0, 1) \ {q(-j)} j is an element of N, in the case when alpha = q(-m), m is an element of N, the type of singularity has a decisive effect on the set where a function can be approximated. In the latter event, depending on the types of singularities, three classes of functions have been examined, and it has been found that the possibility of approximation varies considerably for these classes.
Citation Count: 3
The q-Bernstein polynomials of the Cauchy kernel with a pole on [0,1] in the case q > 1
(Elsevier Science inc, 2013) Ostrovska, Sofiya; Ozban, Ahmet Yasar; Mathematics
The problem to describe the Bernstein polynomials of unbounded functions goes back to Lorentz. The aim of this paper is to investigate the convergence properties of the q-Bernstein polynomials B-n,B-q(f; x) of the Cauchy kernel 1/x-alpha with a pole alpha is an element of [0, 1] for q > 1. The previously obtained results allow one to describe these properties when a pole is different from q(-m) for some m is an element of {0, 1, 2, ...}. In this context, the focus of the paper is on the behavior of polynomials B-n,B-q(f; x) for the functions of the form f(m)(x) = 1/(x - q(-m)), x not equal q(-m) and f(m)(q(-m)) = a, a is an element of R. Here, the problem is examined both theoretically and numerically in detail. (C) 2013 Elsevier Inc. All rights reserved.
Citation Count: 0
Non-asymptotic norm estimates for the q-bernstein operators
(Springer New York LLC, 2013) Ostrovska,S.; Özban,A.Y.; Mathematics
The aim of this paper is to present new non-asymptotic norm estimates in C[0,1] for the q-Bernstein operators Bn,q in the case q > 1. While for 0 < q ≤ 1, {double pipe}Bn,q{double pipe} = 1 for all n ∈ ℕ, in the case q > 1, the norm {double pipe}Bn,q{double pipe} grows rather rapidly as n → + ∞ and q → + ∞. Both theoretical and numerical comparisons of the new estimates with the previously available ones are carried out. The conditions are determined under which the new estimates are better than the known ones. © Springer Science+Business Media New York 2013.
Citation Count: 6
The norm estimates of the q-Bernstein operators for varying q > 1
(Pergamon-elsevier Science Ltd, 2011) Ostrovska, Sofiya; Ozban, Ahmet Yasar; Mathematics
The aim of this paper is to present norm estimates in C [0, 1] for the q-Bernstein basic polynomials and the q-Bernstein operators B-n,B-q in the case q > 1. While for 0 < q <= 1, vertical bar vertical bar B-n,B-q vertical bar vertical bar = 1 for all n is an element of N. in the case q > 1, the norm vertical bar vertical bar B-n,B-q vertical bar vertical bar increases rather rapidly as q -> +infinity. In this study, it is proved that vertical bar vertical bar B-n,B-q vertical bar vertical bar similar to C(n)q(n(n-1)/2), q -> +infinity with C-n = 2/n (1- 1/n)(n-1). Moreover, it is shown that vertical bar vertical bar B-n,B-q vertical bar vertical bar similar to 2q(n(n-1)/2) /ne as n -> infinity, q -> +infinity. The results of the paper are illustrated by numerical examples. (C) 2011 Elsevier Ltd. All rights reserved.
Citation Count: 26
On idempotency and tripotency of linear combinations of two commuting tripotent matrices
(Elsevier Science inc, 2009) Ozdemir, Halim; Sarduvan, Murat; Ozban, Ahmet Yasar; Guler, Nesrin; Mathematics
Let T-1 and T-2 be two nonzero commuting n x n tripotent matrices and c(1), c(2) two nonzero complex numbers. Necessary and sufficient conditions for the tripotency and the idempotency of c(1)T(1) + c(2)T(2) are obtained. The problems considered here have also statistical importance when c(1), c(2) are real scalars and T-1, T-2 are real symmetric matrices. (C) 2008 Elsevier Inc. All rights reserved.
Citation Count: 32
On idempotency of linear combinations of idempotent matrices
(Elsevier Science inc, 2004) Özdemir, H; Özban, AY; Mathematics
P-1, P-2 and P-3 being any three different nonzero mutually commutative n x n idempotent matrices, and C-1 C-2 and C-3 being nonzero scalars, the problem of characterizing some situations, where a linear combination of the form P = c(1)P(1) + c(2)P(2) or P = c(1)P(1) + c(2)P(2) + c(3)P(3), is also an idempotent matrix has been considered. Moreover two interesting results about the idempotency of linear combinations of 2 x 2 idempotent matrices and 3 x 3 idempotent matrices have been given. A statistical interpretation of the idempotency problem considered in this study has also been pointed out. (C) 2003 Elsevier Inc. All rights reserved.
Citation Count: 7
On the q-Bernstein polynomials of rational functions with real poles
(Academic Press inc Elsevier Science, 2014) Ostrovska, Sofiya; Ozban, Ahmet Yasar; Mathematics
The paper aims to investigate the convergence of the q-Bernstein polynomials B-n,B-q(f; x) attached to rational functions in the case q > 1. The problem reduces to that for the partial fractions (x - alpha)(-J), j is an element of N. The already available results deal with cases, where either the pole a is simple or alpha not equal q(-m), m is an element of N-0. Consequently, the present work is focused on the polynomials Bn,q(f; x) for the functions of the form f (x) = (x - q(-m))(-j) with j >= 2. For such functions, it is proved that the interval of convergence of {B-n,B-q(f; x)} depends not only on the location, but also on the multiplicity of the pole - a phenomenon which has not been considered previously. (C) 2013 Elsevier Inc. All rights reserved.
Citation Count: 7
On the q-Bernstein Polynomials of Unbounded Functions with q > 1
(Hindawi Ltd, 2013) Ostrovska, Sofiya; Ozban, Ahmet Yasar; Mathematics
The aim of this paper is to present new results related to the q-Bernstein polynomials B-n,B-q (f;x) of unbounded functions in the case q > 1 and to illustrate those results using numerical examples. As a model, the behavior of polynomials B-n,B-q (f;x) is examined both theoretically and numerically in detail for functions on [0, 1] satisfying f(x) similar to Kx(-alpha) as x -> 0(+), where alpha > 0 and K not equal 0 are real numbers.
Citation Count: 0
On the Convergence of the q-Bernstein Polynomials for Power Functions
(Springer Basel Ag, 2021) Ostrovska, Sofiya; Ozban, Ahmet Yasar; Mathematics
The aim of this paper is to present new results related to the convergence of the sequence of the complex q-Bernstein polynomials {B-n,B-q(f(alpha); z)}, where 0 < q not equal 1 and f(alpha) = x(alpha), alpha >= 0, is a power function on [0, 1]. This study makes it possible to describe all feasible sets of convergence K for such polynomials. Specifically, if either 0 < q < 1 or alpha is an element of N-0, then K = C, otherwise K = {0} boolean OR {q(-j)}(j=0)(infinity). In the latter case, this identifies the sequence K = {0} boolean OR {q(-j)}(j=0)(infinity) as the 'minimal' set of convergence for polynomials B-n,B-q(f; z), f is an element of C[0, 1] in the case q > 1. In addition, the asymptotic behavior of the polynomials {B-n,B-q(f(alpha); z)}, with q > 1 has been investigated and the obtained results are illustrated by numerical examples.
Citation Count: 3
On the Sets of Convergence for Sequences of the q-Bernstein Polynomials with q > 1
(Hindawi Publishing Corporation, 2012) Ostrovska, Sofiya; Ozban, Ahmet Yasar; Mathematics
The aim of this paper is to present new results related to the convergence of the sequence of the q-Bernstein polynomials {B-n,B-q(f x)} in the case q > 1, where f is a continuous function on [0,1]. It is shown that the polynomials converge to f uniformly on the time scale J(q) = {q(-j)}(j-0)(infinity) boolean OR {0}, and that this result is sharp in the sense that the sequence {B-n,B-q(f;x)}(n-1)(infinity) may be divergent for all x is an element of R \ J(q). Further the impossibility of the uniform approximation for the Weierstrass-type functions is established. Throughout the paper the results are illustrated by numerical examples.
Citation Count: 0
Polynomial logistic distribution associated with a cubic polynomial
(Taylor & Francis inc, 2017) Aksoy, Umit; Ostrovska, Sofiya; Ozban, Ahmet Yasar; Mathematics
Let P(x) be a polynomial monotone increasing on ( - , +). The probability distribution possessing the distribution function is called the polynomial logistic distribution with associated polynomial P. This has recently been introduced by Koutras etal., who have also demonstrated its importance for modeling financial data. In this article, the properties of the polynomial logistic distribution with an associated polynomial of degree 3 have been investigated in detail. An example of polynomial logistic distribution describing daily exchange rate fluctuations for the US dollar versus the Turkish lira is provided.
Citation Count: 0
UNCORRELATEDNESS SETS OF DISCRETE RANDOM VARIABLES VIA VANDERMONDE-TYPE DETERMINANTS
(Walter de Gruyter Gmbh, 2019) Turan, Mehmet; Ostrovska, Sofiya; Ozban, Ahmet Yasar; Mathematics
Given random variables X and Y having finite moments of all orders, their uncorrelatedness set is defined as the set of all pairs (j, k) is an element of N-2; for which X-j and Y-kappa are uncorrelated. It is known that, broadly put, any subset of N-2 can serve as an uncorrelatedness set. This claim is no longer valid for random variables with prescribed distributions, in which case the need arises so as to identify the possible uncorrelatedness sets. This paper studies the uncorrelatedness sets for positive random variables uniformly distributed on three points. Some general features of these sets are derived. Two related Vandermonde-type determinants are examined and applied to describe uncorrelatedness sets in some special cases. (C) 2019 Mathematical Institute Slovak Academy of Sciences