Suppose f(x) in L^2(Omega), then we know that simply ||f(x)||^{2} = int_{Omega} f^2(x)dx
Now consider the situation that the analytic form of the f(x) is unknow, but its values at a set of sampling position x_{1}, x_{2}, cdots, x_{n}, i.e., what we have are f(x_{1}), f(x_{2}), cdots, f(x_{n}).
The question is, does there exist a better way to approximate the ||f(x)||^{2} than the below form?
Suppose f(x) in L^2(Omega), then we know that simply
||f(x)||^{2} = int_{Omega} f^2(x)dx
Now consider the situation that the analytic form of the f(x) is unknow,
but its values at a set of sampling position x_{1}, x_{2}, cdots, x_{n},
i.e., what we have are f(x_{1}), f(x_{2}), cdots, f(x_{n}).
The question is, does there exist a better way to approximate the ||f(x)||^{2} than the below form?
||f(x)||^{2} = sum_{i}^{n} f^2(x_{i})