Hee Sun Jung

Orthonormal polynomials with exponential-type weights

Let R=(-~,~) and let w@r(x)@?|x|^@rexp(-Q(x)), where @r>-12 and Q(x)@?C^2:R->R^+=[0,~) is an even function. In this paper we consider the properties of the orthonormal polynomials with respect to the weight w@r^2(x), obtaining bounds on the orthonormal polynomials and spacing on their zeros. Moreover, we estimate An(x) and Bn(x) defined in Section 4, which are used in representing the derivative of the orthonormal polynomials with respect to the weight w@r^2(x).

Derivatives of Integrating Functions for Orthonormal Polynomials with Exponential-Type Weights

Let , , where is an even function. In 2008 we have a relation of the orthonormal polynomial with respect to the weight ; , where and are some integrating functions for orthonormal polynomials . In this paper, we get estimates of the higher derivatives of and , which are important for estimates of the higher derivatives of .

The Markov-Bernstein inequality and Hermite-Fejér interpolation for exponential-type weights

We investigate the coefficients of Hermite-Fejer interpolation polynomials based at zeros of orthogonal polynomials with respect to exponential-type weights. First, we obtain the modified Markov-Bernstein inequalities with respect to w@?F(Lip12). Then using the modified Markov-Bernstein inequalities, we estimate the value of |pn^(^r^)(w@r^2,x)/pn^(w@r^2,x)| for r=1,2,... at zeros of pn(w@r^2;x) and we apply this to estimate the coefficients of Hermite-Fejer interpolation polynomials. Here, pn(w@r^2,x) denotes the nth orthogonal polynomial with respect to an exponential-type weight w@r(x)=|x|^@rw(x), x@?R, @r>-1/2.

Mean and uniform convergence of Lagrange interpolation with the Erdős-type weights

Let R=(−∞,∞), and let Q∈C1(R):R→R+:=[0,∞) be an even function. We consider the exponential-type weights w(x)=e−Q(x), x∈R. In this paper, we obtain a mean and uniform convergence theorem for the Lagrange interpolation polynomials Ln(f) in Lp, 1<p⩽∞ with the weight w.MSC:41A05.

Approximation by Lupas-Type Operators and Szász-Mirakyan-Type Operators

Lupas-type operators and Szasz-Mirakyan-type operators are the modifications of Bernstein polynomials to infinite intervals. In this paper, we investigate the convergence of Lupas-type operators and Szasz-Mirakyan-type operators on [0,∞).

Asymptotic Properties of Derivatives of the Stieltjes Polynomials

Let 𝑤𝜆(𝑥)∶=(1−𝑥2)𝜆−1/2 and 𝑃𝜆,𝑛(𝑥) be the ultraspherical polynomials with respect to 𝑤𝜆(𝑥). Then, we denote the Stieltjes polynomials with respect to 𝑤𝜆(𝑥) by 𝐸𝜆,𝑛

Convergence and Divergence of Higher-Order Hermite or Hermite-Fejér Interpolation Polynomials with Exponential-Type Weights

Let R −∞,∞ , and let wρ x |x|ρe−Q x , where ρ > −1/2 and Q ∈ C1 R : R → R 0,∞ is an even function. Then we can construct the orthonormal polynomials pn w2 ρ;x of degree n for w2 ρ x . In this paper for an even integer ν ≥ 2 we investigate the convergence theorems with respect to the higher-order Hermite and Hermite-Fejer interpolation polynomials and related approximation process based at the zeros {xk,n,ρ}k 1 of pn w2 ρ;x . Moreover, for an odd integer ν ≥ 1, we give a certain divergence theorem with respect to the higher-order Hermite-Fejer interpolation polynomials based at the zeros {xk,n,ρ}k 1 of pn w2 ρ;x .

On the Favard-Type Theorem and the Jackson-Type Theorem (II)

Let R=(−∞,∞), and let 𝑄∈ℂ1∶ℝ→[0,∞) be an even function. We consider the exponential weights 𝑤(𝑥)=𝑒−𝑄(𝑥), 𝑥∈ℝ. In this paper we investigate the relations between the Favard-type inequality and the Jackson-type inequality. We also discuss the equivalence of two K-functionals and the modulus of smoothness.

Domain of convergence for a series of orthogonal polynomials

Let { p k } k = 0 ∞ be the orthogonal polynomials with certain exponential weights. In this paper, we prove that under certain mild conditions on exponential weights class, a series of the form Â? b k p k converges uniformly and absolutely on compact subsets of an open strip in the complex plane, and diverges at every point outside the closure of this strip.

Local saturation of a positive linear convolution operator

Let {Hn(t)} be a sequence of non-negative, even, and continuous functions on ℝ. In this paper, we consider a convolution operator Jn(f;x)=∫0∞f(t)Hn(t−x)dt, f∈Lp(R+), and then investigate the local saturation of Jn(f;x).MSC:44A35.