For this method you need (n-1)/2 variables so in this case for 5 points we need 2, let's call them x and y....

Consider the following polynomial in 2 variables with 9 a(n) coefficients:

I generate the following table based on the points to be interpolated, x and y can each have values [-1,0,1], and the sum x+y is related to the range of the function. So x=-1 y=-1 x+y = -2, from the list of points the value of the function at -2 is 3 so A is three there.

So I can solve for the 9 unknowns a1..a9 with 9 equations made from the table above:I haven't tried but I imagine wider ranges with more variables would work as well. I also suppose you could narrow the thickness of the band by ranging one variable over a smaller interval like y from -.5...5 and x from -1.5..1.5

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