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I Trigonometric series with normalised coefficients

  1. Mar 19, 2017 #1
    Hi all,
    I have a trigonometric function series
    $$f(x)={1 \over 2}{\Lambda _0} + \sum\limits_{l = 1}^\infty {{\Lambda _l}\cos \left( {lx} \right)} $$
    with the normalization condition
    $$\Lambda_0 + 2\sum\limits_{l = 1}^\infty {{\Lambda _l} = 1} $$
    and ##\Lambda_l## being monotonic decrescent weights, i.e. ##\Lambda_0>\Lambda_1>\Lambda_2...##
    Clearly from these two latter conditions one can prove ##f(x)## exists, but my problem is to characterise this function. Which theorems do you suggest to be helpful to this aim?
     
  2. jcsd
  3. Mar 19, 2017 #2

    Charles Link

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    Homework Helper

    This is a Fourier series without the sine terms. Can you say anything about f(x) being even or odd, i.e. symmetric or anti-symmetric about x=0? Also, because it is a Fourier series, doesn't that say something about f(x) being periodic? How long is the interval over which the function repeats? ## \\ ## Also, generally on the Physics Forums, questions like this should be put in the homework section. Welcome to Physics Forums. :) :)
     
  4. Mar 19, 2017 #3
    Thank you for you reply. I'm not sure this is homework, because it's actually a very tiny part of a research I'm doing in quantum optics.
    I'm aware this is a special case of Fourier series, that's why it is called trigonometric series.
    f(x) is a function defined like above, it has no other specifications.
    Moreover f(x) being even is clear from its definition: x -> -x, f(x)=f(-x).
    Periodicity is very clear as well: being a sum of cosines, the lowest harmonic l=1 dictates the period 2π.
    The question is about formalisation of other properties (like its sign or zeros, or amplitude, or derivative) of this function through theorems analysing the possible trend for weights Λ (for instance power distribution, Gaussian distribution, ...).
    Hope to hear from you soon, thanks again
     
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