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Does any one know any bounded integral operator?

To put it in another way, I am given the following equation

$$ \int_{a}^{b} f(x,y) X(y) dy = b(x) $$

where $f(x,y)$ and $b(x)$ are known.

  • What conditions do I need in order to have a unique solution for $X(y)$?
  • How can I find $X(x)$?

Thank you

  • 0
    I suppose complete uniqueness is out of the question: Let $X$ be a solution, $X'$ a function that differs from $X$ only in a zero set of points. Then $X'$ is also a solution. For every reasonable integral I can think of, finite sets will have measure zero. Hence, the result will not be unique, since you can change every solution in a finite numbers of points and get another solution.2012-03-10
  • 0
    Yeah, you are right. Let's put it in this way: Give one solution which holds for any arbitrary f(.,.) and b(.)? What condition(s) are needed to guarantee the uniqueness?2012-10-31
  • 0
    Do you want absolute uniqueness or uniqueness up to some equivalence relation (e.g., equivalent almost everywhere)?2012-10-31
  • 0
    Any kind of information could be useful. Do you have any ideas about such equivalence relation?2012-10-31
  • 0
    Try "equivalent almost everywhere": Two functions $f, g$ are equivalent almost everywhere if the set $\{x | f(x) \neq g(x) \}$ has measure zero. This is still a very weak condition here, for example: Choose $f(x,y) = 1$ and $b(x) = 0$. Then you want to solve the equation $\int_a^b X(y) \operatorname{d}y = 0$, which does have infinitely many solutions, e.g., all functions of the form $X(y) = c \cdot \left(y - \frac{a+b}{2}\right)$. I am no expert in higher analysis, but I doubt there is any integral operator that will give you unique solutions except when adding some other stronger constraints.2012-10-31
  • 0
    Any other idea? How can I create stronger condition to get a unique solution? (Compare the question at hand with solution and conditions of continuous Fourier transform and wavelet transform.)2012-11-04
  • 0
    Here is a [reference](http://www.tdx.cat/bitstream/handle/10803/6189/10Jvl10de11.pdf;jsessionid=66561E61887ED93E887D00F4575A1A8C.tdx2?sequence=10).2013-02-19

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