1. Numerical methods for solving differential and integral equations are algorithms for solving algebraic equations (vector solutions) that arise from discretizing infinite-dimensional operator equations (function solutions). When we talk about whether these methods work, we usually do so in terms of their consistency and stability. There is a multistage things that happens here: we start by talking about the well-posedness of the original equation (e.g. the PDE), then the convergence of the mathematical discretization, and then examine what happens when we try to program this thing on a computer. Usually what happens is these algorithms will get implemented "on top" of numerical linear algebra, where algorithms like Gaussian elimination, and different iterative solvers, have been studied very carefully from the perspective of floating point rounding errors etc. This kind of subsumes your concern about "fuzzy" real numbers. Remember that in double precision, if the number "1.0" represents "1 meter", then machin epsilon is atomic scale. So, frequently, you can kind of assume the whole process "just works"...
2/3. I'm not really sure what you mean by these questions... But if you want to do "fourier analysis with a filter preapplied", you'd probably just work with within some space of bandlimited functions. If you only care around N Fourier modes, any time you do an operation which exceeds that number of modes, you need to chop the result back to down to size.
4. In this context, it's really the SVD of an operator you're interested in. In that regard, you can consider trying to extend the various definitions of the SVD to your operator, provided that you carefully think about all spaces involved. I assume at least one "operator SVD" exists and has been studied extensively... For instance, I can imagine trying to extend the variational definition of the SVD... and the algorithms for computing the SVD probably make good sense in a function space, too...
2/3. I'm not really sure what you mean by these questions... But if you want to do "fourier analysis with a filter preapplied", you'd probably just work with within some space of bandlimited functions. If you only care around N Fourier modes, any time you do an operation which exceeds that number of modes, you need to chop the result back to down to size.
4. In this context, it's really the SVD of an operator you're interested in. In that regard, you can consider trying to extend the various definitions of the SVD to your operator, provided that you carefully think about all spaces involved. I assume at least one "operator SVD" exists and has been studied extensively... For instance, I can imagine trying to extend the variational definition of the SVD... and the algorithms for computing the SVD probably make good sense in a function space, too...