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Is this question acceptable?: I want to ask about linear operatiors

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    • CommentAuthorAnixx
    • CommentTimeJan 12th 2011 edited
     
    I want to ask a question about linear operators over domain of numerical functions, but in light of a similar question of mine http://mathoverflow.net/questions/51829/are-there-linear-operators-which-do-not-belong-to-the-following-classes has been closed I am afraid it will be rejected. The same answer asked on Math SE did not receive answers so far.

    That is I want to ask the following.

    What categories linear operators on the domain of numerical functions, of one variable are usually classified into?

    I encountered with the following categories: differintegral, discrete differintegral, right composition, multiplication by coefficient, combination and sum of linear operators.

    What are other well-established categories?

    I wonder, if it is possible to construct other linear operator U, which is not finite combination or sums of the above mentioned and exists such f that |(Uf)(z)-f(z)|>0 for more than countable set of z?

    If such linear operators exist, are they as well studied as those mentioned above?

    Can anybody give an example?

    Is there a common name for operators that belong to the mentioned group by analogy with elementary functions?

    It is possible to determine if a given linear operator belongs to the above mentioned group if it is defined by

    1) expression involving non-linear operators with no obvious way to reduce the expression to linear operators only

    2) infinite transformation matrix
    • CommentAuthorHarry Gindi
    • CommentTimeJan 12th 2011 edited
     

    When there's a typo in the meta thread title, you know that the stars must be conspiring against the creation of this question on MO (have not read the body text).

    • CommentAuthorAnixx
    • CommentTimeJan 13th 2011
     
    I think the typo means nothing, do not believe in stars.
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    Sorry to say, but your question is way too unclear to be answered (you don't say whether your complex functions need to be differentiable or holomorphic, you don't explain how a k-th order derivative for complex k is defined, you use incorrect notation) and has, by its look, way too little chance to be true (quick estimate: the number of functions 2^continuum, so the number of operators is 2^2^continuum, but your classes are more like 2^continuum; this is not a proof but a good argument not to waste one's time with questions like that).
  5.  

    I think the typo means nothing, do not believe in stars.

    I'm... speechless.

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    It seems clear that Math Overflow is not the right forum for these sorts of questions, but to try to address Anixx's rough question more constructively: have you (Anixx) considered integral operators? I didn't see those in your list. Examples include Fourier transform, Mellin transform, etc.

    The rough impression I get (moving beyond the debate about well-formedness of $f(z) \circ g(z)$ etc.) is that Anixx is chiefly interested in definable linear operators on various function spaces. But for that, one needs to specify the language in which definitions are made, and here we get into issues of mathematical logic. At the very least, we could try to open Anixx's mind to the reality that classes of definable operators typically go far beyond the ambit of what he has so far considered. For example, he has so far excluded any sort of multi-line definition like f |--> [x |--> f(g(x)) if x > 0, else f(x)h(x)], where g, h are given functions of some sort.

    Overall, the impression I get is that Anixx hasn't acquired a great deal of training in modern-day mathematics (as taught in graduate schools) and the relentless precision that is required to pin down what one really wants (for example, maybe he wants to exclude the multi-line definitions as a kind of Lakatosian monster-barring, but it's not clear). But I don't see that a bunch of professionals piling on and his reacting defensively and combatively is leading to any useful result.

  7.  
    Todd, you are absolutely correct. I realized last night that I was wasting my time. Anixx is more concerned with arguing than with actually understanding anyone's comments. So I won't let him draw me in to these useless debates any more.
    • CommentAuthorHarry Gindi
    • CommentTimeJan 13th 2011 edited
     

    @Spiro: You only realized this last night? I thought it was pretty obvious after the first 50+ comment thread on meta..

    • CommentAuthorAnixx
    • CommentTimeJan 13th 2011 edited
     
    > he has so far excluded any sort of multi-line definition like f |--> [x |--> f(g(x)) if x > 0, else f(x)h(x)], where g, h are given functions of some sort.

    This can be constructed with use of a piece-wise function. That is U[f(x)]=f(g(x))r1(x)+f(x)h(x)r2(x) where r1(x) is 1 for x>0 otherwise 0 and r2(x) is 1 for x<=0 otherwise 0. I already considered this. This is not mentioned explicitely because I thought it is evident how to constriuct it with the mentioned operators.
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    Fair enough, Anixx. (By the way, "you're welcome" with regard to my mentioning integral operators!) My problem I guess is the same problem others are having: your question as posted is really too vague to deal with, since you don't specify what functions you have in mind. Surely not completely arbitrary functions?

    I don't think meta MO is the right place to iron out what you mean, however, and therefore I shan't discuss it further.

    • CommentAuthorAnixx
    • CommentTimeJan 13th 2011 edited
     
    > By the way, "you're welcome" with regard to my mentioning integral operators

    Fourier transform can be reduced to convolution. Well indeed it seems that convolution operation is another basic one which is missing from the list. I would like to see more such examples.

    Here are my derivations:

    • CommentAuthorMariano
    • CommentTimeJan 13th 2011
     

    Pick a basis of the space of all functions C --> C. Now permute it.

    • CommentAuthorAnixx
    • CommentTimeJan 13th 2011 edited
     
    And there is even a formula for convolution which expresses it as product of integrals



    I only fail to see how to apply it to diverging integrals. As far as only one of the integrals is operator of f function, we can consider the other a coefficient, so that the whole expression belongs to the case 3 mentioned in the first post.

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