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I was looking for approaches on how to adequately interpolate the values for a continuous 3D function for which I have the exact values in a 3D grid of equidistant points. I found that linear interpolation in 3D works well but I lose some precision. So I found Gauss interpolation, mentioned here

http://mathworld.wolfram.com/GausssInterpolationFormula.html

my first question; for the coefficient expression, are the two terms multiplied or summed?

I was looking for an online reference with all the information but found anything. Lastly I wonder how could I use this idea for the 3D mesh of points mentioned before. I guess I would get first the individual values in one dimension for the function, and then get the 3D value directly from them.

Last, I wonder which similar and accurate methods could you recommend me. I heard about Chebyshev method and some other, but I am not sure abut their accuracy.

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    To your first question: The two terms are multiplied. That ensures that $\zeta_k(x_l)=\delta_{kl}$. To your second question: Since you have a set of functions with $\zeta_k(x_l)=\delta_{kl}$, you can form all possible products of the form $\zeta_{k_x}(x)\zeta_{k_y}(y)\zeta_{k_z}(z)$ to get a set of functions, each of which is $1$ at one of your mesh points and $0$ at all others. Then you can write the interpolation as a linear combination of these.2011-03-30
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    Did you try tricubic interpolation as I suggested in your previous question?2011-03-30
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    @lhf, yes I tried it! but I would like to try with some other additional methods that can increase precision in my calculations. thanks a lot!2011-03-31
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    Page 895 of Abramowitz and Stegun http://people.math.sfu.ca/~cbm/aands/page_895.htm has a couple formulas using very few points.2011-04-03

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