Seon-Hong Kim

An approximation of circular arcs by quartic Bézier curves

In this paper we propose an approximation method for circular arcs by quartic Bezier curves. Using an alternative error function, we give the closed form of the Hausdorff distance between the circular arc and the quartic Bezier curve. We also show that the approximation order is eight. By subdivision of circular arcs with equi-length, our method yields the curvature continuous spline approximation of the circular arc. We confirm that the approximation proposed in this paper has a smaller error than previous quartic Bezier approximations.

Some properties of Chebyshev polynomials

In this paper we obtain some new bounds for Chebyshev polynomials and their analogues. They lead to the results about zero distributions of certain sums of Chebyshev polynomials and their analogues. Also we get an interesting property about the integrals of certain sums of Chebyshev polynomials.

SELF-RECIPROCAL POLYNOMIALS WITH RELATED MAXIMAL ZEROS

For each real number > 6, we prove that there is a sequence of fourth degree self-reciprocal polynomials such that the zeros of are all simple and real, and every has the largest (in modulus) zero where and are the first and the second largest (in modulus) zeros of , respectively. One such sequence is given by so that , where and other are polynomials in n defined by the severely nonlinear recurrence for , with the usual empty product conventions, i.e., .

ON SOME COMBINATIONS OF SELF-RECIPROCAL POLYNOMIALS

Let Pn be the set of all monic integral self-reciprocal poly- nomials of degree n whose all zeros lie on the unit circle. In this paper we study the following question: For P(z), Q(z) 2 Pn, does there exist a continuous mapping r ! Gr(z) 2 Pn on (0;1) such that G0(z) = P(z) and G1(z) = Q(z)?

CERTAIN TRINOMIAL EQUATIONS AND LACUNARY POLYNOMIALS

Abstract. We estimate the positive real zeros of certain trinomial equa-tions and then deduce zeros bounds of some lacunary polynomials. 1. Introduction and statement of resultsMany of classical inequalities of analysis have been obtained from trinomialequations, and there have been a number of literatures about zero distributionsof trinomial equations and lacunary polynomials. See, for example, [1], [2], [3]and [4]. In this paper, we investigate positive real zeros distributions of certaintrinomial equations and, using this, we estimate zeros bounds for some lacunarypolynomials. While studying these, we will need a new generalized upper boundof the exponential function: for 0 ≤ x < 1 and 1 ≤ n ≤ 2 we have(1) e x ≤ U ( n,x ) = 1 − 1 n +1 n ˆ1+i1 − 1 n ¢ x 1 − xn ! n ≤ 11 −x, where U (1 ,x ) = 11 −x . For the details about this, see [5]. The first resultabout trinomial equations follows from the lemma below that will be proved inSection 2.Lemma 1. Let n be an integer ≥ 4 , and (2)12 n

CERTAIN REPRESENTATION OF A REAL POLYNOMIAL

We show that every real polynomial of degree n can be represented by the sum of two real polynomials of degree n, each having only real zeros. As an example of this, we consider a real polynomial with even degree whose all zeros lie on the imaginary axes except the origin.