Ryozi Sakai

Derivatives of Orthonormal Polynomials and Coefficients of Hermite-Fejér Interpolation Polynomials with Exponential-Type Weights

Let , and let be an even function. In this paper, we consider the exponential-type weights , and the orthonormal polynomials of degree with respect to . So, we obtain a certain differential equation of higher order with respect to and we estimate the higher-order derivatives of and the coefficients of the higher-order Hermite-Fejér interpolation polynomial based at the zeros of .

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.

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.

Interpolation Polynomials of Entire Functions for Erdös-Type Weights

Let , and let be an even function. In this paper, we consider some Lagrange interpolation polynomials and the Gauss-Jacobi quadrature formula of entire functions associated with Erdos-type weights , , and we will estimate the error terms.

Higher order Hermite-Fejér interpolation polynomials with Laguerre-type weights

Let ℝ+ = [0, ∞) and R : ℝ+ → ℝ+ be a continuous function which is the Laguerre-type exponent, and pn, ρ(x), ρ>-12 be the orthonormal polynomials with the weight wρ (x) = xρe-R(x). For the zeros {xk,n,ρ}k=1n of pn,ρ(x)=pn(wρ2;x), we consider the higher order Hermite-Fejér interpolation polynomial Ln (l, m, f; x) based at the zeros {xk,n,ρ}k=1n, where 0 ≤ l ≤ m - 1 are positive integers.2010 Mathematics Subject Classification: 41A10.

Some Properties of Orthogonal Polynomials for Laguerre-Type Weights

Let , let be a continuous, nonnegative, and increasing function, and let be the orthonormal polynomials with the weight . For the zeros of we estimate , where is a positive integer. Moreover, we investigate the various weighted -norms () of .