Skip to main content

Halil Oruc

Followers

0

Following

1

Public Views

Interests

Uploads

Papers by Halil Oruc

Generating functions for B-Splines with knots in geometric or affine progression

Calcolo, Dec 5, 2013

We derive explicit formulas for the generating functions of B-splines with knots in either geomet... more We derive explicit formulas for the generating functions of B-splines with knots in either geometric or affine progression. To find generating functions for B-splines whose knots have geometric or affine ratio q, we construct a PDE for these generating functions in which classical derivatives are replaced by q-derivatives. We then solve this PDE for the generating functions using q-exponential functions. We apply our generating functions to derive some known and some novel identities for B-splines with knots in geometric or affine progression, including a generalization of the Schoenberg identity, formulas for sums and alternating sums, and an explicit expression for the moments of these B-splines. Special cases include both the uniform B-splines with knots at the integers and the nonuniform B-splines with knots at the q-integers.

Tensor product q-Bernstein polynomials

Bit Numerical Mathematics, Oct 8, 2008

ABSTRACT An affine de Casteljau type algorithm to compute q-Bernstein Bézier curves is introduced... more ABSTRACT An affine de Casteljau type algorithm to compute q-Bernstein Bézier curves is introduced and its intermediate points are obtained explicitly in two ways. Furthermore we define a tensor product patch, based on this algorithm, depending on two parameters. Degree elevation procedure is studied. The matrix representation of tensor product patch is given and we find the transformation matrix between a classical tensor product Bézier patch and a tensor product q-Bernstein Bézier patch. Finally, q-Bernstein polynomials B n,m (f;x,y) for a function f(x,y), (x,y)∈[0,1]×[0,1] are defined and fundamental properties are discussed.

q-Bernstein polynomials and Bézier curves

Journal of Computational and Applied Mathematics, Feb 1, 2003

We define q-Bernstein polynomials, which generalize the classical Bernstein polynomials, and show... more We define q-Bernstein polynomials, which generalize the classical Bernstein polynomials, and show that the difference of two consecutive q-Bernstein polynomials of a function f can be expressed in terms of second order divided differences of f. It is also shown that the approximation to a convex function by its q-Bernstein polynomials is one sided. A parametric curve is represented using a generalized Bernstein basis and the concept of total positivity is applied to investigate the shape properties of the curve. We study the nature of degree elevation and degree reduction for this basis and show that degree elevation is variation diminishing, as for the classical Bernstein basis.

Further Properties of Quantum Spline Spaces

Mathematics, Oct 14, 2020

We construct q-B-splines using a new form of truncated power functions. We give basic properties ... more We construct q-B-splines using a new form of truncated power functions. We give basic properties to show that q-B-splines form a basis for quantum spline spaces. On the other hand, we derive algorithmic formulas for 1/q-integration and 1/q-differentiation for q-spline functions. Moreover, we show a way to find the polynomial pieces on each interval of a q-spline function.

Symmetric functions and the Vandermonde matrix

Journal of Computational and Applied Mathematics, Nov 1, 2004

This work deduces the lower and the upper triangular factors of the inverse of the Vandermonde ma... more This work deduces the lower and the upper triangular factors of the inverse of the Vandermonde matrix using symmetric functions and combinatorial identities. The L and U matrices are in turn factored as bidiagonal matrices. The elements of the upper triangular matrices in both the Vandermonde matrix and its inverse are obtained recursively. The particular value x i = 1 + q + • • • + q i−1 in the indeterminates of the Vandermonde matrix is investigated and it leads to q-binomial and q-Stirling matrices. It is also shown that q-Stirling matrices may be obtained from the Pascal matrix.

q-Peano Kernel and its Applications

arXiv (Cornell University), Aug 11, 2015

We introduce a q-analogue of the Peano kernel theorem by replacing ordinary derivatives and integ... more We introduce a q-analogue of the Peano kernel theorem by replacing ordinary derivatives and integrals by quantum derivatives and quantum integrals. In the limit q → 1, the q-Peano kernel reduces to the classical Peano kernel. We also give applications to polynomial interpolation and construct examples in which classical remainder theory fails whereas q-Peano kernel works. Furthermore we derive a relation between q-B-splines and divided differences via the q-Peano kernel.

Probabilistic Approach to the Schoenberg Spline Operator and Unimodal Density Estimator

Using Chebyshev's inequality, we provide a probabilistic proof of the uniform convergence for... more Using Chebyshev's inequality, we provide a probabilistic proof of the uniform convergence for continuous functions on a closed interval by Schoenberg's variation diminishing spline operator. Furthermore, we introduce a unimodal density estimator based on this spline operator and thus generalize that of Bernstein polynomials and beta density. The advantage of this method is the local property. That is, re ning the knots while keeping the degree xed of B-splines yields better estimates. We also give a numerical example to verify our results.

A generalization of the peano kernel and its applications

Communications Faculty of Sciences University of Ankara. Series A1: mathematics and statistics, 2018

Based on the q-Taylor Theorem, we introduce a more general form of the Peano kernel (q-Peano) whi... more Based on the q-Taylor Theorem, we introduce a more general form of the Peano kernel (q-Peano) which is also applicable to non-di¤erentiable functions. Then we show that quantum B-splines are the q-Peano kernels of divided di¤erences. We also give applications to polynomial interpolation and construct examples in which classical remainder theory fails whereas q-Peano kernel works.

Tensor Product q −Bernstein Bézier Patches

Springer eBooks, 2009

An affine de Casteljau type algorithm to compute q−Bernstein Bézier curves is introduced and its ... more An affine de Casteljau type algorithm to compute q−Bernstein Bézier curves is introduced and its intermediate points are obtained explicitly in two ways. Furthermore we define a tensor product patch, based on this algorithm, depending on two parameters. Degree elevation procedure is studied. The matrix representation of tensor product patch is given and we find the transformation matrix between a classical tensor product Bézier patch and a tensor product q−Bernstein Bézier patch. Finally, q−Bernstein polynomials B n,m (f ; x, y) for a function f (x, y), (x, y) ∈ [0, 1] × [0, 1] are defined and fundamental properties are discussed.

Generalized Bernstein polynomials and total positivity

This thesis submitted for Ph.D. degree deals mainly with geometric properties of generalized Bern... more This thesis submitted for Ph.D. degree deals mainly with geometric properties of generalized Bernstein polynomials which replace the single Bernstein polynomial Bn! by a one-parameter family of polynomials B~!. It also provides a triangular decomposition and l-banded factorization of the Vandermonde matrix. We first establish the generalized Bernstein polynomials for monomials, which leads to a definition of Stirling polynomials of the second kind. These are q-analogues of Stirling numbers of the second kind. Some of the properties of the Stirling numbers are generalized to their q-analogues. We show that the generalized Bernstein polynomials are monotonic in degree n, that is B~! ~ B~+d, when the function! is convex. There is also a representation of B~! B~+1!involving second order divided differences of the function j'. Shape preserving properties of the generalized Bernstein polynomials are studied by making use of the concept of total positivity. It is proved that monotonic ...

q-Peano Kernel and its Applications

arXiv: Classical Analysis and ODEs, 2015

We introduce a q-analogue of the Peano kernel theorem by replacing ordinary derivatives and integ... more We introduce a q-analogue of the Peano kernel theorem by replacing ordinary derivatives and integrals by quantum derivatives and quantum integrals. In the limit q \to 1, the q-Peano kernel reduces to the classical Peano kernel. We also give applications to polynomial interpolation and construct examples in which classical remainder theory fails whereas q-Peano kernel works. Furthermore we derive a relation between q-B-splines and divided differences via the q-Peano kernel.

Further Properties of Quantum Spline Spaces

Mathematics, 2020

We construct q-B-splines using a new form of truncated power functions. We give basic properties ... more We construct q-B-splines using a new form of truncated power functions. We give basic properties to show that q-B-splines form a basis for quantum spline spaces. On the other hand, we derive algorithmic formulas for 1/q-integration and 1/q-differentiation for q-spline functions. Moreover, we show a way to find the polynomial pieces on each interval of a q-spline function.

A generalization of rational Bernstein�B�zier curves

Bit, 2007

Tensor Product q-Bernstein B�zier Patches

Numerical Analysis and Its Applications, 2008

On the q-Bernstein Polynomials

We discuss here recent developments on the convergence of the q-Bernstein polynomials B(sub n)f w... more We discuss here recent developments on the convergence of the q-Bernstein polynomials B(sub n)f which replaces the classical Bernstein polynomial with a one parameter family of polynomials. In addition, the convergence of iterates and iterated Boolean sum of q-Bernstein polynomial will be considered. Moreover a q-difference operator D(sub q)f defined by D(sub q)f = fX,QX is applied to q-Bernstein polynomials. This gives us some results which complement those concerning derivatives of Bernstein polynomials. It is shown that, with the parameter 0 is less than q is less than or equal to 1, if the change in k, f(sub r) is greater than or equal to 0 then D(k)(sub q)B(sub n)f is greater than or equal to 0. If f is monotonic so is D(sub q)B(sub n)f. If f is convex then D(2)(sub q)B(sub n)f is greater than or equal to 0.

Tensor Product q −Bernstein Bézier Patches

Lecture Notes in Computer Science, 2009

An affine de Casteljau type algorithm to compute q−Bernstein Bézier curves is introduced and its ... more An affine de Casteljau type algorithm to compute q−Bernstein Bézier curves is introduced and its intermediate points are obtained explicitly in two ways. Furthermore we define a tensor product patch, based on this algorithm, depending on two parameters. Degree elevation procedure is studied. The matrix representation of tensor product patch is given and we find the transformation matrix between a classical tensor product Bézier patch and a tensor product q−Bernstein Bézier patch. Finally, q−Bernstein polynomials B n,m (f ; x, y) for a function f (x, y), (x, y) ∈ [0, 1] × [0, 1] are defined and fundamental properties are discussed.

Bernstein–Schoenberg operator with knots at the q-integers

Mathematical and Computer Modelling, 2012

ABSTRACT We consider a special knot sequence ui+1=qui+1ui+1=qui+1 and define a one parameter fami... more ABSTRACT We consider a special knot sequence ui+1=qui+1ui+1=qui+1 and define a one parameter family of Bernstein–Schoenberg operators. We prove that this operator converges to ff uniformly for all ff in C[0,1]C[0,1]. This operator also inherits the geometric properties of the classical Bernstein–Schoenberg operator. Moreover we show that the error function Em,nEm,n has a particular symmetry property, that is Em,n(f;x;q)=Em,n(f;1−x,1/q)Em,n(f;x;q)=Em,n(f;1−x,1/q) provided that ff is symmetric on [0,1][0,1].

Convexity and generalized Bernstein polynomials

Proceedings of the Edinburgh Mathematical Society, 1999

In a recent generalization of the Bernstein polynomials, the approximated function f is evaluated... more In a recent generalization of the Bernstein polynomials, the approximated function f is evaluated at points spaced at intervals which are in geometric progression on [0, 1], instead of at equally spaced points. For each positive integer n, this replaces the single polynomial Bnf by a one-parameter family of polynomials , where 0 < q ≤ 1. This paper summarizes briefly the previously known results concerning these generalized Bernstein polynomials and gives new results concerning when f is a monomial. The main results of the paper are obtained by using the concept of total positivity. It is shown that if f is increasing then is increasing, and if f is convex then is convex, generalizing well known results when q = 1. It is also shown that if f is convex then, for any positive integer n This supplements the well known classical result that when f is convex.

Symmetric functions and the Vandermonde matrix

Journal of Computational and Applied Mathematics, 2004

This work deduces the lower and the upper triangular factors of the inverse of the Vandermonde ma... more This work deduces the lower and the upper triangular factors of the inverse of the Vandermonde matrix using symmetric functions and combinatorial identities. The L and U matrices are in turn factored as bidiagonal matrices. The elements of the upper triangular matrices in both the Vandermonde matrix and its inverse are obtained recursively. The particular value x i = 1 + q + • • • + q i−1 in the indeterminates of the Vandermonde matrix is investigated and it leads to q-binomial and q-Stirling matrices. It is also shown that q-Stirling matrices may be obtained from the Pascal matrix.

q-Bernstein polynomials and Bézier curves

Journal of Computational and Applied Mathematics, 2003

We define q-Bernstein polynomials, which generalize the classical Bernstein polynomials, and show... more We define q-Bernstein polynomials, which generalize the classical Bernstein polynomials, and show that the difference of two consecutive q-Bernstein polynomials of a function f can be expressed in terms of second order divided differences of f. It is also shown that the approximation to a convex function by its q-Bernstein polynomials is one sided. A parametric curve is represented using a generalized Bernstein basis and the concept of total positivity is applied to investigate the shape properties of the curve. We study the nature of degree elevation and degree reduction for this basis and show that degree elevation is variation diminishing, as for the classical Bernstein basis.