ISSN:
1432-0940
Keywords:
Primary 41A55
;
65D30
;
65D32
;
Secondary 42C05
;
Integration rules
;
Interpolatory integration rules
;
Convergence
;
Distribution of points
;
Weak convergence
;
Potential theory
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
Notes:
Abstract Suppose that, forn≥1, $$I_n [f]: = \sum\limits_{j = 1}^n {w_{jn} f(x_{jn} )} $$ is aninterpolatory integration rule of numerical integration, that is, $$I_n [f]: = \int\limits_{ - 1}^1 {P(x)dx,} degree(P)〈 n.$$ Suppose, furthermore, that, for each continuousf:[−1, 1]→R, $$\mathop {\lim }\limits_{n \to \infty } I_n [f] = \int\limits_{ - 1}^1 {f(x)dx.} $$ What can then be said about thedistribution of the points $$\{ x_{jn} \} _{1 \leqslant j \leqslant n} $$ n→∞? In all the classical examples they havearcsin distribution. More precisely, if $$\mu _n : = \frac{1}{n}\sum\limits_{j = 1}^n {\delta _{x_{jn} } } $$ is the unit measure assigning mass 1/n to each pointx jn, then, asn→∞ $$d\mu _n (x)\mathop \to \limits^* \upsilon (x)dx: = \frac{1}{\pi }(\arcsin x)'dx = \frac{{dx}}{{\pi (1 - x^2 )^{1/2} }}.$$ Surprisingly enough, this isnot the general case. We show that the set of all possible limit distributions has the form 1/2(v(x) dx+dv(x)), wherev is an arbitrary probability measure on [−1, 1]. Moreover, given any suchv, we may find rulesI n,n≥1, with positive weights, yielding the limit distribution 1/2v(x) dx+dv(x)). We also consider generalizations when the quadratures have precision other thann−1, and when we place a weight σ in our integral.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/BF01229335
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