4 ms·

>but definitely not all integrable is quite a general class of functions (especially if you define the coefficients in terms of the lebesgue integral instead o

>but definitely not all integrable is quite a general class of functions (especially if you define the coefficients in terms of the lebesgue integral instead of the riemann integral). >a small, finite number of elementary operations piece-wise definition is not an elementary operation. you can't easily add/multiply/compose piece-wise defined functions. you can't easily differentiate/integrate them either. fourier space representations don't suffer from any of these issues.

hansvm 7y ago
>integrable is quite a general class of functions (especially if you define the coefficients in terms of the lebesgue integral instead of the riemann integral). Quite true, but it's still just a subset of all functions, and you're still left with some sticky points with respect to pointwise convergence vs convergence almost everywhere. >piece-wise definition is not an elementary operation. you can't easily add/multiply/compose piece-wise defined functions. you can't easily differentiate/integrate them either. fourier space representations don't suffer from any of these issues. I'll give you composition as a tricky operation, but fourier series aren't particularly amenable to composition either. Addition, multiplication, differentiation, integration, and whatnot are very straightforward though. You use a refinement of the two respective piecewise partitions, perform the desire operations, and potentially have to reconcile misbehaviors along the boundaries. Fourier series have a different set of formulaic approaches to the same operations, but they have roughly the same pitfalls (e.g. regions of non-differentiability) and additionally have problems like the aforementioned Gibbs phenomenon if finite approximations are desired or else some difficulty in deciphering which elementary functions might describe the series.

hansvm 7y ago

>integrable is quite a general class of functions (especially if you define the coefficients in terms of the lebesgue integral instead of the riemann integral). Quite true, but it's still just a subset of all functions, and you're still left with some sticky points with respect to pointwise convergence vs convergence almost everywhere. >piece-wise definition is not an elementary operation. you can't easily add/multiply/compose piece-wise defined functions. you can't easily differentiate/integrate them either. fourier space representations don't suffer from any of these issues. I'll give you composition as a tricky operation, but fourier series aren't particularly amenable to composition either. Addition, multiplication, differentiation, integration, and whatnot are very straightforward though. You use a refinement of the two respective piecewise partitions, perform the desire operations, and potentially have to reconcile misbehaviors along the boundaries. Fourier series have a different set of formulaic approaches to the same operations, but they have roughly the same pitfalls (e.g. regions of non-differentiability) and additionally have problems like the aforementioned Gibbs phenomenon if finite approximations are desired or else some difficulty in deciphering which elementary functions might describe the series.