How Can I Fit Functions to Both Time and Pixel Number in a Matrix Animation?

[elegysix]

If I have a set of n numbers, (n=0,1,2, ... n-1), what is the maximum number coefficients in a series I would need in order to find y=f(n) true on the integers up to n-1? Or is that something I just have to check after fitting it? ( does it need an nth degree polynomial?)


Another question:

If I have a finite set of functions, F_t(x), and these functions may or may not be similar (i.e. F_t(x) = x, x², x³... x^t ), how can I fit a plane to a set of functions such that F(1,x)=F₁(x), F(2,x)=F₂(x) ... in order to find an equation F(t,x)?

can/how is this done if each function is a summation? (i.e. F₁=\suma₁(n)*xⁿ, F₂=\suma₂(n)*xⁿ)

can I find an equation like F(t,x) = \sumG(t)*a_t(n)*xⁿ that will satisfy this?

thanks

 

[theorem4.5.9]

For the first question, n points uniquely identifies a degree n-1 polynomial, which will have n coefficients.

As for your second question I'm not sure but it really depends on the inputs $F_t$ if it's possible to write $F$ in a closed form. If I'm reading you correctly, you want to construct a function based on level set information. Interesting question and I'd like to see what others have to say on this.

 

[elegysix]

If anyone is interested, the problem I'm trying to solve is this:

say we have an NxN (3x3 for this example) matrix that changes over time t. say we have 3 different frames, 1 NxN matrix representing the value of each pixel in a screen for each frame. so 3 matrices total

If I count from left to right and down across the matrix, for a 3x3 matrix, the last (bottom right pixel) will be number 9. in mod 3 (or mod N) , the value is 33 or (NN), which is also the coordinates of the pixel (omitting the single values like 01, 02, 03 by adding 3 (N) ).

matrix numbered as:
|1 2 3 | --> | 11 12 13 |
|4 5 6 | --> | 21 22 23 |
|7 8 9 | --> | 31 32 33 |

So I want to take the values in the matrix and make a set {a1,a2,a3...a9}
and fit a function to them, F₁(x), which is no problem so far.
And do the same for each frame, so that I have a function for each frame, F₁(x), F₂(x), F₃(x). no problem this far.

Now comes the problem: I want to combine these functions together so that I can find any pixel's value by entering the time, t, and the pixel number, x. So that F_t(x) can be given by F(t,x).

that's where if F₁ = \sum a(n)*xⁿ and F₂ = \sum b(n)*xⁿ and similarly for F₃
can they be combined somehow into a function F(t,x) such as \sum G(t)*Z(n)*xⁿ
so that G(1)*Z(n) gives a(n), and G(2)*Z(n) = b(n) and so forth ?
How can I solve for what G(t) and Z(n) are? or is it possible?


The point is to be able to have an animation or video be in a formula, and not need so much data to transfer. the quality I believe would depend on the expansion of n, taking n out to infinity would produce perfect quality.

thanks