# Proof that infinite functions can fit a table of numerical values

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:

If information provided is ambiguous or insufficient, please comment below so that I can provide. Thank you!

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If your question is: can we find an arbitrary amount of functions that fit values in a certain table, the answer would be yes. No need to frame it as a pseudo-question about science. That's certainly not what science is about. – Raskolnikov May 15 '12 at 18:14
Yes, I changed it now – user5198 May 15 '12 at 18:21
@user5198: Are there any conditions on what functions we can use? A finite number of points can already be interpolated by infinitely many polynomials, or by infinitely many piecewise constant functions, or ... – Johannes Kloos May 15 '12 at 18:38
I added the constraint on functions. No piecewise functions are allowed. – user5198 May 15 '12 at 19:36

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.

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yes! because you are adding zero to the interpolating polynomial for all values of x_i. Thanks – user5198 May 16 '12 at 1:30