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I want to construct a polynomial basis for the space of $d-$variate polynomials of degree $Q$ for numerical purposes.

Let's pick for example $Q = 2$ and $d = 3$.

Now, if $d = 1$ it would be easy: I would just pick: ${1,x,x^2}$ they are linearly independent and span the space of polynomials of max degree 2. However if d = 3 then I am looking for polynomials of the form: $p_1(x,y,z),p_2(x,y,z),p_3(x,y,z),...,$

I found that I would need $Q+d$ polynomials to have a basis, is that true? Moreover, my application is a numerical one, so I would need a basis that works well numerically, like for example Chebyshev polynomials: they work well in 1D.

Can I get some pointers as where to look?

EDIT: Just to be more clear, in 1D on an interval [a,b] I have the following "formula" for Chebyshev polynomials: $$Cheb(x,k) = \cos{(k*\arccos{[-1 + \frac{2}{(b-a)}*(x-a)]})}$$

Which gives me the value at the point x of the k-th Chebyshev polynomial. I was wondering if there existed something similar for degree other than one.

EDIT 2: I also heard about centralized and scaled monomials of the form $$p_\alpha(x)=(\frac{x-c}{d})^\alpha$$ where d is half diameter of the domain specified and c is the "center" of the domain. In dimension 1 it is pretty clear.

My question then is: in dimension $d = 3$ can this idea be generalized as: $$p_\alpha(x,y,z) = (\frac{xyz- \Vert c \Vert}{d})^\alpha$$

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  • $\begingroup$ There are $\binom{Q+d}{d}$ monomials of degree $Q$ or smaller in $d$ variables. Think of a peg-hole situation, $Q+d$ holes and $d$ pegs, the number of holes in segment $i$ before peg $i$ gives the degree of the $i$-th variable in the monomial. There can be holes left after the last peg for monomials of non-maximal degree. $\endgroup$ – LutzL May 31 at 11:52

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