Integrate Polynomial f(x^2): Reduce Sampling Points to N/2?

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kuan
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Let f(x^2) be a polynomial. I would like to carry out the integration

\int_a^b f(x^2) dx

using quadrature rule. Suppose a and b > 0 and are arbitrary and the degree of f(x^2) is 2N.

I would like to know if there is a possibility to reduce the sampling points down to N/2?
 
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Yes. This is an open question.

The integration of the polynomial f(x^2) can be computed accurately up to machine precision using the quadrature rule. For example, using the Gauss-Legendre rule, we need approximately N sampling points.

My question is that if we can reduce the number of the sampling point further by a factor of 2 or not, since f(y) is the polynomial too, where y=x^2.

In another word, I wonder if there is a way to change variable, x, S.T., the integration is reduced to degree N with different weight function and can be carried out accurately using the Gauss integration.

Also notice that there are several special cases, one can do it. If a=0, we can map this integration into Gauss-Jacobi system, where alpha = -1/2 and beta = 0.