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Is there a field of mathematics that considers multiplying functions in a manner analogous to matrix multiplication? For instance,

  1. Let $\mathbf{x}$ is an $n$-dimensional vector such that $x_i=\sin(2\pi \frac {i} {n}))$, for $i=1,\ldots, n$.
  2. Let $\mathbf{A}$ be an $n\times n$ matrix where $A_{i,j}=\cos(4\pi \frac {i} {n} - 2\pi \frac {j} {n})$ for each $i,j$.
  3. Let $\mathbf{y}=\mathbf{A}\mathbf{x}$ so that $y_i=\frac 1 2 \sin(4\pi \frac {i} {n})$.

If we imagine $n$ going to infinity, then we can define an operator, $\text{prod_fcn}$, such that $\text{prod_fcn}(x(t), A(t, s)) = y(t)$. This example is illustrated graphically below: enter image description here

Obviously this is just one simple example of a curve being "warped" by a plane. I'm curious to know if this sort of functional operation is a commonly studied thing. If so, any information about this would be appreciated.

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    $\begingroup$ How familiar are you with functional analysis? Much of it deals with linear operators on infinite-dimensional spaces, which is a direct generalization of multiplication of vectors by matrices (not by "planes", of course). Very often, these infinite dimensional linear operators can be approximated by matrix multiplication. $\endgroup$ Dec 3, 2017 at 21:46
  • $\begingroup$ Not very...my background is in statistics. Good lead though. And of course, the "planes" part was misleading. I just needed the title to sound catchier. $\endgroup$
    – jjet
    Dec 4, 2017 at 2:06

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Looks as if you are missing a factor $1/n$?. But otherwise, your matrix sum corresponds to a discrete approximation of the following integral operator: $$ L \phi(y) = \int_0^1 k(y,x) \phi(x) \; dx $$ where $\phi(x)=\sin(2\pi x)$ and $k(y,x) = \cos(2\pi(2y-x))$. An operator of the above form and with continuous (even better $C^1$) kernel $k(y,x)$ has very nice properties, in particular may be well approximated by the discrete matrix version as in your example.

More general versions are in $L^2$ where you are dealing with Hilbert-Schmidt integral operators. But you have to be more careful when approximating by a matrix.

One may construct Fredholm determinant for such operators (which in the continuous case, you may easily approximate by your matrix version). This enables to calculate e.g. eigenvalues for the operator.

You may also have a look at Integral transforms.

These are just some examples of the use of the limiting case of your matrix multiplication.

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  • $\begingroup$ Thanks, this is helpful. The integral transform is the most obvious match for my problem. And clearly it has tons of applications. $\endgroup$
    – jjet
    Dec 9, 2017 at 19:56

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