I am just learning about inner-products on polynomial space, where the coefficients of the polynomials may be complex: $P_m(\mathbf{F})$

The inner-product given by:

$\langle p,q \rangle = \int_0^1 p(x) \, \overline{q(x)} dx$

I understand that the inner-product can be weighted:

$\langle p,q \rangle = \int_0^1 r(x) \, p(x) \, \overline{q(x)} dx$

What I don't know (haven't been able to find) is whether the function r(x) has restrictions on it such as "it must be a real-valued function", or strictly positive, or strictly non-negative.

Judging by what I know about norm being non-negative, I think that r(x) can't be negative (otherwise the inner product of a function with it self, i.e., the norm, could be less than 0).

Can anyone help?

Regards, Madeleine.

  • $\begingroup$ Can you explain why we need the weight function for this inner product? I am just learning too about polynomial space inner-products. Thanks in advance. $\endgroup$ – user599310 Sep 3 at 17:05

I have learned that r(x) must be real-valued and non-negative.

It must also be integrable on [0, 1].

This guarantees that the norm is >= 0, which is a requirement.


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