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Engineer stumped by pure math, need to find some way to get started

  1. Aug 22, 2012 #1
    Hi guys, I am reading some paper on Mellin Transform and its application in signal processing and I am totally stumped by phrases such as lebesgue space, square integrable, isometry, automorphism, measurable and essential supreme and such.

    Of course, when I look them up one by one I sort of get the concept (although still very confused about why people would go through all that mess to define these concepts as such), but I would like to be comprehensive about it.

    Where can I get a systematic understanding of these terms? Do I need a course in real analysis? If possible, can someone elaborate as to what a functional space mean?
     
  2. jcsd
  3. Aug 22, 2012 #2
    Some of those words do imply real analysis.

    A functional space is a set of functions having some particular property, where some "topology" is defined. Which usually means there is a way to measure how "far apart" two functions are. Very frequently functional spaces are linear, which means that for any two numbers a and b, and any two member-functions f and g, af + bg is also a member-function. In those space you typically have a scalar product (f, g), which is used to define the norm ||f|| = |(f, f)|, and the norm is used to define the metric r(f, g) = ||f - g||, which tells you how far apart f and g are.

    Taking the space of square integrable functions on the interval [a, b], the scalar product of f and g is simply the definite integral of the product f(t)g(t) over the interval [a, b]. What is "square integrable"? This a function integral of whose square is finite, which simply means its norm is finite, so all the definitions are self-consistent.
     
  4. Aug 22, 2012 #3

    Bacle2

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    One of the important things about square-integrability is that the space of square-

    integrable functions, laid out as voko did, is a Hilbert space, and Hilbert spaces have

    some very nice properties. To add a bit about norms, there are often many

    different ones used ( and inequivalent topologically), reflecting the different notions

    of what functions being closed to each other may mean.
     
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