ISSN:
0945-3245
Keywords:
AMS(MOS). 65D30
;
CR: 5.16
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
Notes:
Summary This paper is concerned with the theoretical properties of a productintegration method for the integral $$\int\limits_{ - 1}^1 {k(x)f(x)dx}$$ , wherek is absolutely integrable andf is continuous. The integral is approximated by $$\sum\limits_{i = 0}^n {w_{ni} f(x_{ni} )}$$ , where the points are given byx ni =cos(iπ/n, 0≦i≦n, and where the weightsw ni are chosen to make the rule exact iff is any polynomial of degree ≦n. The principal result is that ifk∈L p [−1, 1] for somep〉1, then the rule converges to the exact result asn→∞ for all continuous (or indeed R-integrable) functionsf, and moreover that the sum of the absolute values of the weights converges to the least possible value, namely $$\int\limits_{ - 1}^1 {|k(x)|dx}$$ . A limiting expression for the individual weights is also obtained, under certain assumptions. The results are exteded to other point sets of a similar kind, including the classical Chebyshev points.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/BF01398509
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