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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:

enter image description 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?


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
up vote 3 down vote accepted

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

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