Order of uniform approximation by polynomial interpolation in the complex plane and beyond (original) (raw)

On the Lagrange Complex Interpolation

Upb Scientific Bulletin Series a Applied Mathematics and Physics, 2010

In lucrare prezint unele rezultate legate de interpolarea Lagrange în domeniul complex (cor. prop. 1şi prop. 2). Formula (6) este o extindere a formulei lui Shannon de eşantionare (7) , pentru cazul momentelor echidistante de eşantionare. In §2 am adăugat un rezultat privind eşantionarea in domeniul frecvenţelor. In §3 am dat o extindere multidimensională a formulei de eşantionare , care nu foloseşte interpolarea Lagrange, ci o abordare distribuţională. In this work, I present some results regarding the Lagrange interpolation in the complex domain (cor. prop. 1 and prop. 2). Formula (6) is an extension of the well-known Shannon's sampling formula (7) , for sampling equidistant moments. In §2 I present a simple result regarding the sampling in frequency. In §3 I give a multidimensional extension , which does not use the Lagrange interpolation but a distributional approach.

On the Divergence of Polynomial Interpolation in the Complex Plane

Constructive Approximation, 2001

We extend the results in [1] and [2] from the divergence of Hermite-Fejér interpolation in the complex plane to the divergence of arbitrary polynomial interpolation in the complex plane. In particular, we prove the following theorem: Let n = −1 ≤ t (n)

Interpolatory properties of best L2-approximants

Indagationes Mathematicae, 1990

Let f be a continuous function and s, be the polynomial of degree at most n of best Lz(p)approximation to f on [ -1, I]. Let Z,(f):= {XE [ -1, 1] :f(x) -s,,(x) =O}. Under mild conditions on the measure p, we prove that U Z,(f) is dense in [ -1, 11. This answers a question posed independently by A. Kroo and V. Tikhomiroff. It also provides an analogue of the results of Kadec and Tashev (for 15,) and Kroo and Peherstorfer (for L,) for least squares approximation.

Interpolation and the Laguerre-Pólya class

A long standing open problem, known as the Karlin-Laguerre problem, in the study of the distribution of real zeros of a polynomial is to characterize all real sequences T={γ_k}_{k=0}^∞ such that they satisfy the property Z_c (T[p(x)])≤Z_c (p(x)), where Z_c(p(x)) denotes the number of non-real zeros of the real polynomial p(x)=∑_{k=0}^{n} a_k x_k and T(p(x))=∑_{k=0}^{n}γ_k a_k x_k. The main result of this paper shows that under a mild growth restriction, an entire function of exponential type f(z) for which the sequence T={f(k)}_{k=0}^{∞} satisfies the above condition must have only real zeros. The paper concludes with some applications to the Riemann hypothesis.

Interpolatory Pointwise Estimates for Polynomial Approximation X1. Introduction and Main Results

2007

We discuss whether or not it is possible to have interpolatory pointwise estimates in the approximation of a function f 2 C 0; 1], by polynomials. For the sake of completeness as well as in order to strengthen some existing results, we discuss brieey the situation in unconstrained approximation. Then we deal with positive and monotone constraints where we show exactly when such interpolatory estimates are achievable by proving aarmative results and by providing the necessary counterexamples in all other cases. The eeect of the endpoints of the nite interval on the quality of approximation of continuous functions by algebraic polynomials, was rst observed by Nikolski Nik46]. Later pointwise estimates of this phenomenon were given by Timan Tim51] (k = 1), Dzjadyk Dzj58, Dzj77] (k = 2), Freud Fre59] (k = 2), and Brudny Bru63] (k 2), who proved that if f 2 C r 0; 1], then for each n N = r + k ? 1, a polynomial p n 2 n exists, such that (1.1) jf(x) ? p n (x)j c(r; k) r n (x)! k (f (r) ; ...

Polynomial interpolation, an L-function, and pointwise approximation of continuous functions

Journal of Approximation Theory, 2008

We show that if {s k } ∞ k=1 is the sequence of all zeros of the L-function L(s,) := ∞ k=0 (−1) k (2k + 1) −s satisfying Re s k ∈ (0, 1), k = 1, 2,. .. , then any function from span {|x| s k } ∞ k=1 satisfies the pointwise rapid convergence property, i.e. there exists a sequence of polynomials Q n (f, x) of degree at most n such that f − Q n C[−1,1] C(f)E n (f), n=1, 2,. .. , and for every x ∈ [−1, 1], lim n→∞ (|f (x)−Q n (f, x)|)/E n (f)= 0, where E n (f) is the error of best polynomial approximation of f in C[−1, 1]. The proof is based on Lagrange polynomial interpolation to |x| s , Re s > 0, at the Chebyshev nodes. We also establish a new representation for |L(x,

Shape Preserving Approximation by Complex Polynomials in the Unit Disk

2013

The purpose of this paper is to obtain new results concerning the preservation of some properties in Geometric Function Theory, in approximation of analytic functions by polynomials, with best approximation types of rates. In addition, the approximating polynomials satisfy some interpolation conditions too.

Generalized polynomial approximation

Israel Journal of Mathematics, 1973

We estimate the rate of covergence to functions in the spaces L p 10,1] and C[0,1] by polynomial of the form Eta~x t, where the ;t's are positive real numbers and 0.