weight functions for (numerical) integration

Bart · science forum beginner Joined: 07 Jul 2005 Posts: 39

Could anybody comment on the following:

Suppose you want to calculate the integral

I = \int_{0}^{1} f(x)dx

over the unit interval or unit hypercube. Then using Monte Carlo
methods, this can be done by simply taking the average of the
function values at certain randomly chosen x_k:

I_N = 1/N * sum(f(x_k), k=1..N)

Now suppose you introduce a weightfunction w(x), and as an
alternative, you approximate the integral by

I_N = 1/N * sum(w(k/N)*f(x_k), k=1..N)

Then my question is: in how far is it necessary for the weights
to be normalized such that the integral still converges to I?
That is, in how far must we have that

1/N * sum(w(k/N), k=1..N) = 1

or

lim_{N->infty} 1/N * sum(w(k/N), k=1..N) = 1

and what happens if e.g.

1/N * sum(w(k/N), k=1..N) = C

or

lim_{N->infty} 1/N * sum(w(k/N), k=1..N) = C

with C a certain constant.

Can anybody comment on this or give me some usefull references
that are related to this question?

Thanks,
Bart

--

Peter Spellucci · science forum Guru Joined: 29 Apr 2005 Posts: 702

In article <1151065959.64787@seven.kulnet.kuleuven.ac.be>,
Bart <Reply.In@This.Group> writes:

Bart · science forum beginner Joined: 07 Jul 2005 Posts: 39

On 2006-06-23, Peter Spellucci <spellucci@fb04373.mathematik.tu-darmstadt.de> wrote:

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