T. M. Mills

On Lagrange interpolation with equidistant nodes

A quantitative version of a classical result of S.N. Bernstein concerning the divergence of Lagrange interpolation polynomials based on equidistant nodes is presented. The proof is motivated by the results of numerical computations.

A Variational Approach to Splines

This is an expository paper in which we present an introduction to a variational approach to spline interpolation. We present a sequence of theorems which starts with Holladays classical result concerning natural cubic splines and culminates in some general abstract results.

The Lebesgue function for generalized Hermite-Fejér interpolation on the Chebyshev nodes

This paper presents a short survey of convergence results and properties of the Lebesgue function λ m,n (x) for(0, 1, …, m )Hermite-Fejer interpolation based on the zeros of the n th Chebyshev polynomial of the first kind. The limiting behaviour as n → ∞ of the Lebesgue constant Λ m,n = max{λ m,n (x) : −1 ≤ x ≤ 1} for even m is then studied, and new results are obtained for the asymptotic expansion of Λ m,n . Finally, graphical evidence is provided of an interesting and unexpected pattern in the distribution of the local maximum values of λ m,n (x) if m ≥ 2 is even.

MARKOFF TYPE INEQUALITIES FOR CURVED MAJORANTS

(x) = fo(x).We make the following remarks concerning Theorems 1 and 2.REMARK 1. For the parabolic majorant, the corresponding problems in the uniformnorm have been solved by Pierre and Rahman [11] and Rahman and Schmeisser [13].REMARK 2. Problems of this type also occur in approximation theory, most notablyin the work of Dzyadyk [3].For the second aim of this paper, we recall a well known inequality of S. Bern-stein [1]. According to this result, in(x)f p is a real algebraic polynomial of degree nor less that satisfies(1.12) \p

On Hermite-Fejér type interpolation on the Chebyshev nodes

Given f ∈ C [−1, 1], let H n , 3 ( f , x ) denote the (0,1,2) Hermite-Fejer interpolation polynomial of f based on the Chebyshev nodes. In this paper we develop a precise estimate for the magnitude of the approximation error | H n , 3 ( f , x ) − f ( x )|. Further, we demonstrate a method of combining the divergent Lagrange and (0,1,2) interpolation methods on the Chebyshev nodes to obtain a convergent rational interpolatory process.

The risk of being diagnosed with cancer.

There is widespread interest in cancer statistics. The following statement appeared recently: “In Australia in 2007, the risk of being diagnosed with cancer before the age of 85 was 1 in 2 for males and 1 in 3 for females.” Similar reports appear in publications from some cancer organisations and in the media. How do policy makers use these reports, and how does the general public interpret these statistics? The method for calculating these risks can be summarised as follows. The incidence of cancer in a particular region and year is the number of new cases of cancer diagnosed in that region and year. The incidence rate is the incidence per 100,000 head of population. The concept of cumulative risk was developed to be a simple measure that can be used to compare incidence rates in different populations, or in the same population at different times. It is an alternative to the more common age-standardised incidence rate. The cumulative rate (R) up to age 85 is defined to be five times the sum of the age-specific incidence rates divided by 100,000. The factor – five – stems from the fact that, usually, data are collated in five-year age groups 0–4, 5–9, etc. This cumulative rate is normally expressed as a percentage. If all causes of death other than cancer are ignored, then the estimated probability that a person in this population will be diagnosed with cancer by age 85 (or cumulative risk) is given by 1-exp(-R). A mathematical justification of this result can be found elsewhere. The point of this letter is to draw attention to the fact that the calculation is based on the assumption that “no other cause of death [was] in operation”. In other words, it is assumed that everyone will be diagnosed with cancer at some stage during their lifetime. When estimates of cumulative risks are presented, the general reader may be better informed if this assumption were stated explicitly. Cumulative risk is not only an epidemiological measure; it can be a persuasive tool. It is a measure that may be used by the public (in interpreting advertising about the risk of cancer) and by oncology decision makers (in making policies or allocating resources). For example, at the policy level, different regions will have different cumulative risks. Assessing regional variation requires a sound understanding before cumulative risk is used to inform decisions. This is due to implicit value judgements concerning people’s welfare in measures of disparities that are age-related. Hence, it is important that all assumptions that underpin the calculations are clear. Williams and Doessel offer further discussion of measurement issues in health care.

A note on edge-connectivity of the Cartesian product of graphs

The main aim of this paper is to establish conditions that are necessary and sufficient for the edge-connectivity of the Cartesian product of two graphs to equal the sum of the edge-connectivities of the factors. The paper also clarifies an issue that has arisen in the literature on Cartesian products of graphs.

Demonstrating the Durbin–Watson Statistic

The paper describes one way to demonstrate the Durbin-Watson statistic to a large class of students.

ON THE DIVERGENCE OF HERMITE-FEJER TYPE INTERPOLATION WITH EQUIDISTANT NODES

If f ( x ) is defined on [−1, 1], let H¯ 1 n ( f, x ) denote the Lagrange interpolation polynomial of degree n (or less) for f which agrees with f at the n +1 equally spaced points x k, n = −1 + (2 k )/ n (0 ≤ k ≤ n ). A famous example due to S . Bernstein shows that even for the simple function h ( x ) = │ x │, the sequence H¯ 1 n ( h, x ) diverges as n → ∞ for each x in 0 H¯ mn ( f, x ), which is the unique polynomial of degree no greater than m ( n + 1) – 1 which satisfies ( f, X k , n ) = δ o, p f ( x k , n ) (0 ≤ p ≤ m − 1, 0 ≤ k ≤ n ). In general terms, if m is an even number, the polynomials H¯ mn ( f, x ) seem to possess better convergence properties than the H¯ 1 n ( f, x ). Nevertheless, D.L. Berman was able to show that for g ( x ) ≡ x , the sequence H¯ 2n ( g, x ) diverges as n → ∞ for each x in 0 x │. In this paper we extend Bermans result by showing that for any even m, H¯ mn ( g, x ) diverges as n → ∞ for each x in 0 x │ H¯ mn ( g, x ) – g ( x )│.

Applications of the cumulative rate to kidney cancer statistics in Australia

Cancer incidence and mortality statistics in two populations are usually compared by using either the age-standardised rate or the cumulative risk by a certain age. We argue that the cumulative rate is a superior measure because it obviates the need for a standard population, and is not open to misinterpretation as is the case for cumulative risk. Then we illustrate the application of the cumulative rate by analysing incidence and mortality data for kidney cancer in Australia using the cumulative rate. Kidney cancer, which is also known as malignant neoplasm of kidney, is one of the less common cancers in Australia. In 2012, approximately 2.5% of all new cases of cancer were kidney cancer, and approximately 2.1% of all cancer related deaths in Australia were due to kidney cancer. There is variation in incidence and mortality by sex, age, and geographical location in Australia. We examine how the cumulative rate performs in measuring the variation of this disease across such sub-populations. This is part of our e ort to promote the use of the cumulative rate as an alternative to the age-standardised rates or cumulative risk. In addition we hope that this statistical investigation will contribute to the aetiology of the disease from an Australian perspective.