In a recent paper we showed that error curves in polynomial Chebyshev approximation of analytic functions on the unit disk tend to approximate perfect circles about the origin. Making use of a theorem of Caratheodory and Fejer, we derived in the process a method for calculating near-best approximations rapidly by finding the principal singular value and corresponding singular vector of a complex Hankel matrix. This paper extends these developments to the problem of Chebyshev approximation by rational functions, where non-principal singular values and vectors of the same matrix turn out to be required. The theory is based on certain extensions of the Caratheodory-Fejer result which are also currently finding application in the fields of digital signal processing and linear systems theory. It is shown among other things that if f($\epsilon z$) is approximated by a rational function of type (m,n) for $\epsilon$ < 0, then under weak assumptions the corresponding error curves deviate from perfect circles of winding number M + N + 1 by a relative magnitude O(${\epsilon}^{m+n+2}$) as $\epsilon\ \rightarrow\ 0$. The "CF approximation" that our method computes approximates the true best approximation to the same high relative order. A numerical procedure for computing such approximations is described and shown to give results that confirm the asymptotic theory. Approximation of $e^z$ on the unit disk is taken as a central computational example.

/pdf/rational-chebyshev-approximation-on-the-unit-disk-59y9bklg1u.pdf

Rational Chebyshev approximation on the unit disk

https://people.maths.ox.ac.uk/trefethen/publication/PDF/1981_3.pdf

Near-circularity of the error curve in complex Chebyshev approximation

Rational minimax approximation of real functions on real intervals is an established topic, but when it comes to complex functions or domains, there appear to be no algorithms currently in use. Suc...

/pdf/an-algorithm-for-real-and-complex-rational-minimax-44fwbf9l79.pdf

An Algorithm for Real and Complex Rational Minimax Approximation

Non-strong uniqueness in real and complex Chebyshev approximation

Intestinal absorption of strontium (Sr) in thirteen healthy adult German volunteers has been investigated by simultaneous oral and intravenous administration of two stable tracer isotopes, i.e. 84Sr and 86Sr. The measured Sr tracer concentration in plasma was analyzed using the convolution integral technique to obtain the intestinal absorption rate. The results showed that the Sr labeled in different foodstuffs was absorbed into the body fluids in a large range of difference. The maximum Sr absorption rates were observed within 60–120 min after administration. The rate of absorption is used to evaluate the intestinal absorption fraction, i.e. the f1 value for various foodstuffs. The equivalent and effective dose coefficients for ingestion of 90Sr were calculated using these f1 values, and they were compared with those recommended by the International Commission on Radiological Protection (ICRP). The geometric and arithmetic means of the f1 values are 0.38 and 0.45 associated with a geometric standard deviation and a standard deviation of 1.88 and 0.22, respectively. The 90% confidence interval of the f1 values obtained in the present study ranges from 0.13 to 0.98. Expressed as the ratio of the 95 and 50% percentiles of the estimated probability, the uncertainty for the f1 value corresponds to a factor of 2.58. The effective dose coefficients of 90Sr after ingestion are 6.1×10−9 Sv Bq−1 for an f1 value of 0.05, 1.0×10−8 Sv Bq−1 for 0.1, 1.9×10−8 Sv Bq−1 for 0.2, 2.8×10−8 Sv Bq−1 for 0.3, 3.6×10−8 Sv Bq−1 for 0.4, 5.3×10−8 Sv Bq−1 for 0.6, 7.1×10−8 Sv Bq−1 for 0.8, and 7.9×10−8 Sv Bq−1 for 0.9, respectively. Taking the effective dose coefficient of 2.8×10−8 Sv Bq−1 for an f1 value of 0.3, which is recommended by the ICRP, as a reference, the effective dose coefficient of 90Sr after ingestion varies by a factor of 2.8 when the f1 value changes by a factor of 3, i.e. it decreases from 0.3 to 0.1 or increases from 0.3 to 0.9, respectively.

Human biokinetics of strontium. Part I: intestinal absorption rate and its impact on the dose coefficient of 90Sr after ingestion.

Gewisse rationale tschebyscheff-approximationen in der komplexen ebene

Ein approximationsproblem aus der pharmakokinetik

For the case of best Tschebyscheff approximation to a real continuous function on an interval [a,b] by polynomials of degree ≤ n it is well known that the error function has at least n + 1 zeros on (a,b). In general no analogous result holds for the case of Tschebyscheff approximation to an analytic function on the disk S ≔ {z∣ ∣z∣ ≤ 1} in the complex plane. However Saff [4] exhibits a class of entire function with the property that for each n sufficiently large the polynomial of degree ≤ n of best Tschebyscheff approximation to f on S interpolates to f in precisely n + 1 points in S. The purpose of this paper is to extend this interpolation property to a class of analytic, not necessarily entire functions, which includes (z−a)−m, ln(z−a) (∣a∣ > 1, m∈N).

Volker Klotz

Papers

Gewisse rationale tschebyscheff-approximationen in der komplexen ebene

Ein approximationsproblem aus der pharmakokinetik

Zur Anzahl der Interpolationspunkte polynomialer Tschebyscheff-Approximationen im Einheitskreis