• Bulletin of the Korean Mathematical Society
• /
• v.36 no.3
• /
• pp.543-564
• /
• 1999

Consider a symmetric bilinear form defined on $\prod\times\prod$ by $_{\lambda\mu}$ = $<\sigma,fg>\;+\;\lambdaL[f](a)L[g](a)\;+\;\muM[f](b)m[g](b)$ ,where $\sigma$ is a quasi-definite moment functional, L and M are linear operators on $\prod$, the space of all real polynomials and a,b,$\lambda$ , and $\mu$ are real constants. We find a necessary and sufficient condition for the above bilinear form to be quasi-definite and study various properties of corresponding orthogonal polynomials. This unifies many previous works which treated cases when both L and M are differential or difference operators. finally, infinite order operator equations having such orthogonal polynomials as eigenfunctions are given when $\mu$=0.

• Communications of the Korean Mathematical Society
• /
• v.21 no.4
• /
• pp.675-678
• /
• 2006

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.

• Communications of the Korean Mathematical Society
• /
• v.12 no.3
• /
• pp.603-617
• /
• 1997

Consider a Sobolev inner product on the space of polynomials such as $$ \phi(p,q) = \lambda p(c)q(c) + <\tau,p'(x)q'(x)> $$ where $\tau$ is a moment functional and c and $\lambda$ are real constants. We investigate properties of orthogonal polynomials relative to $\phi(\cdot,\cdot)$ and give necessary and sufficient conditions under which such Sobolev orthogonal polynomials satisfy a spectral type differential equation with polynomial coefficients.

• Communications of the Korean Mathematical Society
• /
• v.14 no.1
• /
• pp.121-134
• /
• 1999

Generalized nonnegative polynomials are defined as the products of nonnegative polynomials raised to positive real powers. The generalized degree can be defined in a natural way. We extend some results on Infinite-Finite range inequalities, Christoffel functions, and Nikolski type inequalities corresponding to weights W\ulcorner(x)=exp(-｜x｜\ulcorner), $\alpha$>0, to those for generalized nonnegative polynomials.

• Journal of the Korean Mathematical Society
• /
• v.59 no.4
• /
• pp.775-787
• /
• 2022

Sufficient conditions for the Jensen polynomials of the derivatives of a real entire function to be hyperbolic are obtained. The conditions are given in terms of the growth rate and zero distribution of the function. As a consequence some recent results on Jensen polynomials, relevant to the Riemann hypothesis, are extended and improved.

• 제어로봇시스템학회:학술대회논문집
• /
• 2004.08a
• /
• pp.421-424
• /
• 2004

This paper presents a sufficient condition for the real weighted diamond polynomials to be Schur stable using bilinear transformation and Kharitonov's theorem.

• Bulletin of the Korean Mathematical Society
• /
• v.52 no.2
• /
• pp.467-481
• /
• 2015

Choi and Park introduced an invariant of a finite simple graph, called signed a-number, arising from computing certain topological invariants of some specific kinds of real toric manifolds. They also found the signed a-numbers of path graphs, cycle graphs, complete graphs, and star graphs. We introduce a signed a-polynomial which is a generalization of the signed a-number and gives a-, b-, and c-numbers. The signed a-polynomial of a graph G is related to the $Poincar\acute{e}$ polynomial $P_{M(G)}(z)$, which is the generating function for the Betti numbers of the real toric manifold M(G). We give the generating functions for the signed a-polynomials of not only path graphs, cycle graphs, complete graphs, and star graphs, but also complete bipartite graphs and complete multipartite graphs. As a consequence, we find the Euler characteristic number and the Betti numbers of the real toric manifold M(G) for complete multipartite graphs G.

• Bulletin of the Korean Mathematical Society
• /
• v.47 no.6
• /
• pp.1189-1194
• /
• 2010

Kim and Park investigated the distribution of zeros around the unit circle of real self-reciprocal polynomials of even degrees with five terms, where the absolute value of middle coefficient equals the sum of all other coefficients. In this paper, we extend some of their results to the same kinds of polynomials with arbitrary many nonzero terms.

• Bulletin of the Korean Mathematical Society
• /
• v.33 no.1
• /
• pp.135-170
• /
• 1996

Recently many people have studied the Sobolev orthogonal polynomials, that is, polynomials which are orthogonal relative to a symmetric bilinear form $\phi(\cdot,\cdot)$ defined by $$ (1.1) $\phi(p,q) := (p,q)_N = \sum_{k=0}^{N} \int_{R}p^(k) (x)q^(k) (x) d\mu_k, $$ where each $d\mu_k$ is a signed Borel measure on the real line $R$ with finite moments of all orders. For the brief history on this subject, we refer to the survey article Ronveaux [13] and Marcellan and et al [10].

• Communications of the Korean Mathematical Society
• /
• v.12 no.4
• /
• pp.935-950
• /
• 1997

When $\tau$ is a quasi-definite moment functional on P, the vector space of all real polynomials, we consider a symmetric bilinear form $\phi(\cdot,\cdot)$ on $P \times P$ defined by $$ \phi(p,q) = \lambad p(a)q(a) + \mu p(b)q(b) + <\tau,p'q'>, $$ where $\lambda,\mu,a$, and b are real numbers. We first find a necessary and sufficient condition for $\phi(\cdot,\cdot)$ and show that such orthogonal polynomials satisfy a fifth order differential equation with polynomial coefficients.