5 ms·

>replicate you mean construct the periodic extension? >You can force a description of it using pi don't know what about the fourier series is forced...? >bu

>replicate you mean construct the periodic extension? >You can force a description of it using pi don't know what about the fourier series is forced...? >but the most natural descriptions won't have any reference to it at all can you give an example? when comparing piece-wise definitions and fourier series definitions i think the fourier series is the more natural.

klyrs 7y ago
> can you give an example? How about a sawtooth; no pi needed, and the period is 1. f(x) = x-floor(x) The Gibbs phenomenon is my proof that Fourier series is not more natural.
- mhh__ 7y ago
  Why? The Gibbs phenomenon is just an exhibit of a finite Fourier series, no?
  - kragen 7y ago
    It depends on which kind of convergence you like to think about.
  - throwlaplace 7y ago
    fourier series converges pointwise but not uniformly. for many use cases you would like uniform convergence (e.g. approximation in an entire neighborhood of a point) and a fourier series won't give you that no matter how many terms you add. but like i said above - gibbs only occurs at non-removable discontinuities and so you shouldn't expect it to be well behaved at those points regardless of the representation.
- deleted 7y ago
  [deleted]
- throwlaplace 7y ago
  gibbs phenomenon only occurs for discontinuous functions. so it's a reflection of a pathology, not a pathology per se. what i mean is the problem is with the function itself (it's got non-removable discontinuities!) not the fourier series representation.
hansvm 7y ago
>you mean construct the periodic extension? Yes. A more common language felt appropriate given the broad background of the audience, and I also didn't handle any boundaries or edge cases in the functions whose existence I invoked -- if you'd like to poke holes. >don't know what about the fourier series is forced...? Does it not feel at least a little unnatural to require infinitely many coefficients to describe a process that can otherwise be understood with a small, finite number of elementary operations? Simplicity is in the eye of the beholder, and I can't fault you if that's your viewpoint, but I do not agree. >can you give an example? when comparing piece-wise definitions and fourier series definitions i think the fourier series is the more natural. Another comment mentioned the sawtooth function. It seems your viewpoint is that the fourier series is the more natural description of periodic phenomena, perhaps because of its ability to represent many (but definitely not all) periodic functions. One can, however, capture _all_ periodic functions with a piecewise description, and depending on the function (e.g. the sawtooth) this alternative approach might be considerably simpler to write out and explain. That doesn't necessarily make it more elegant in your eyes (and IMO theorems which are too general purpose can lose sight of what made them worthwhile in the first place), but if not then I'm not sure that either of us have enough material to convince the other of their chosen view.
- throwlaplace 7y ago
  >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.

klyrs 7y ago

> can you give an example? How about a sawtooth; no pi needed, and the period is 1. f(x) = x-floor(x) The Gibbs phenomenon is my proof that Fourier series is not more natural.

mhh__ 7y ago
Why? The Gibbs phenomenon is just an exhibit of a finite Fourier series, no?
- kragen 7y ago
  It depends on which kind of convergence you like to think about.
- throwlaplace 7y ago
  fourier series converges pointwise but not uniformly. for many use cases you would like uniform convergence (e.g. approximation in an entire neighborhood of a point) and a fourier series won't give you that no matter how many terms you add. but like i said above - gibbs only occurs at non-removable discontinuities and so you shouldn't expect it to be well behaved at those points regardless of the representation.
deleted 7y ago
[deleted]
throwlaplace 7y ago
gibbs phenomenon only occurs for discontinuous functions. so it's a reflection of a pathology, not a pathology per se. what i mean is the problem is with the function itself (it's got non-removable discontinuities!) not the fourier series representation.

hansvm 7y ago

>you mean construct the periodic extension? Yes. A more common language felt appropriate given the broad background of the audience, and I also didn't handle any boundaries or edge cases in the functions whose existence I invoked -- if you'd like to poke holes. >don't know what about the fourier series is forced...? Does it not feel at least a little unnatural to require infinitely many coefficients to describe a process that can otherwise be understood with a small, finite number of elementary operations? Simplicity is in the eye of the beholder, and I can't fault you if that's your viewpoint, but I do not agree. >can you give an example? when comparing piece-wise definitions and fourier series definitions i think the fourier series is the more natural. Another comment mentioned the sawtooth function. It seems your viewpoint is that the fourier series is the more natural description of periodic phenomena, perhaps because of its ability to represent many (but definitely not all) periodic functions. One can, however, capture _all_ periodic functions with a piecewise description, and depending on the function (e.g. the sawtooth) this alternative approach might be considerably simpler to write out and explain. That doesn't necessarily make it more elegant in your eyes (and IMO theorems which are too general purpose can lose sight of what made them worthwhile in the first place), but if not then I'm not sure that either of us have enough material to convince the other of their chosen view.

throwlaplace 7y ago
>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.