Suppose while conducting experiments, I measure a finite number of variables with some constants like temperature, etc. We get a table of finite number measurements (numerical values to some decimal digits of accuracy). Like the table shown here:
where constant1, constant2, etc. are boundary conditions.
My question is whether we can always find infinitely many $C^{\infty}$ functions that fit data of this type in an equation that can give measured values from the boundary conditions of the experiment?
If yes, how can we prove it?
Note:
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To prove there are an infinite number of polynomials that fit, you can just find the interpolating polynomial. Then you can add $a\prod(x-x_i)$ where the $x_i$ are your measurement points for any value of $a$ to it.