Accelerated polynomial approximation of finite order entire functions by growth reduction

@article{Mller1997AcceleratedPA,
  title={Accelerated polynomial approximation of finite order entire functions by growth reduction},
  author={J{\"u}rgen M{\"u}ller},
  journal={Math. Comput.},
  year={1997},
  volume={66},
  pages={743-761}
}
  • J. Müller
  • Published 1 April 1997
  • Mathematics
  • Math. Comput.
Let f be an entire function of positive order and finite type. The subject of this note is the convergence acceleration of polynomial approximants of f by incorporating information about the growth of f(z) for z → ∞. We consider near polynomial approximation on a compact plane set K, which should be thought of as a circle or a real interval. Our aim is to find sequences (f n ) n of functions which are the product of a polynomial of degree < n and an easy computable second factor and such that… 

Figures and Tables from this paper

Generalized growth and best approximation of entire functions in Lp-norm in several complex variables

AbstractThe paper deals with the characterization of generalized order and generalized type of entire functions in several complex variables in terms of the coefficients of the development with

Devendra Kumar ON THE FAST GROWTH OF ANALYTIC FUNCTIONS BY MEANS OF LAGRANGE POLYNOMIAL

The present paper is concerned with the fast growth of analytic functions in the sets of the form {z C : φK(z) < R} (where φK(z) is the Siciak extremal function of a compact set K) by means of the

A Jentzsch-Theorem for Kapteyn, Neumann and General Dirichlet Series

  • F. Bornemann
  • Mathematics
    Computational Methods and Function Theory
  • 2022
Comparing phase plots of truncated series solutions of Kepler’s equation by Lagrange’s power series with those by Bessel’s Kapteyn series strongly suggests that a Jentzsch-type theorem holds true not

Linear semi-infinite programming theory: An updated survey

Reduced Cancellation in the Evaluation of Entire Functions and Applications to the Error Function

A general concept for the reduction of cancellation problems in the evaluation of Taylor sections of certain entire functions is proposed. The resulting method is applied to and tested in the case of

Accuracy and Stability of Computing High-order Derivatives of Analytic Functions by Cauchy Integrals

TLDR
There is a unique radius that minimizes the loss of accuracy caused by round-off errors, and for large classes of functions, this radius actually gives about full accuracy; a remarkable fact that is explained by the theory of Hardy spaces, by the Wiman–Valiron and Levin–Pfluger theory of entire functions, and by the saddle-point method of asymptotic analysis.

References

SHOWING 1-10 OF 51 REFERENCES

Accelerating the convergence of power series of certain entire functions

SummaryDue to cancellation, the numerical evaluation of an entire function by its Taylor series expansion may become a difficult task whenever terms of large modulus are required to evaluate a small

Truncation Errors in Pade Approximations to Certain Functions: An Alternative Approach

TLDR
It is shown that a similar approach can be used for certain rational approximations, to give excellent a priori estimates of the degree of the approximation to be used, on the basis of the truncation error of this function.

The Degree of Convergence for Entire Functions

Consider a Doint set C in the complex plane whose complement K is connected and regular (i .e. K possesses a Green ' s function with pole at infini ty). Let denote the transfini te diameter of C .

An acceleration method for the power series of entire functions of order 1

When f(z) is given by a known power series expansion, it is possible to construct the power series expansion for f(z; p) = e/sup -p/z f(z). We define p/sub opt/ to be the value of p for which the

The Faber polynomials for annular sectors

A conformai mapping of the exterior of the unit circle to the exterior of a region of the complex plane determines the Faber polynomials for that region. These polynomials are of interest in

Computation of Faber series with application to numerical polynomial approximation in the complex plane

TLDR
This paper discusses efficient Fast Fourier Transform and recursive methods for the computation of Faber polynomials, and points out that the FFT method described by Geddes, for computing Chebyshev coefficients can be generalized to compute Faber coefficients.

Lectures on complex approximation

I: Approximation by Series Expansions and by Interpolation.- I. Representation of complex functions by orthogonal series and Faber series.- 1. The Hilbert space L2(G).- A. Definition of L2(G).- B.

On two methods for accelerating convergence of series

SummaryThe Euler-Knopp transformation and a recently considered transformation, effective for entire function of order 1, are applied to series involving completely monotonic coefficients. Some

Explicit Faber polynomials on circular sectors

We present explicit and precise expressions for the Faber polynomials on circular sectors up to degree 20. The precision is obtained by modifying (and simultaneously speeding up) an algorithm of
...