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I need to write MATLAB program (or use an existing one) to obtain Fourier series coefficients. Let's say the series is going to approximate a 1D curve. The boundary conditions are:

  • value of the curve's function in a few places
  • curve's first derivative known in a few places
  • maximum allowed value of curve's second derivative
  • number of harmonics
  • the value of curve's integral over a specified interval should be as big as possible <- optimization criterion.

All the functions and tools I've seen so far can obtain series coefficients in much more simple cases. Any hints will be helpful, especially:

  • literature, articles with descriptions of useful methods
  • hints on objective functions useful for Fourier Series
  • info on useful MATLAB Tools
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  • $\begingroup$ It seems the same question asked here. $\endgroup$ – Stefano M Aug 9 '12 at 13:09
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    $\begingroup$ @Jann - You might have better luck on here explaining how you are formulating your optimization problem and asking for specific advice on each bit. I don't know of a general-purpose MATLAB tool that will solve this problem for you. $\endgroup$ – Aron Ahmadia Aug 10 '12 at 13:35
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Write your function as a linear combination of harmonics and treat the coefficients as variables. This gives a semi-infinite linear program. The Matlab optimization toolbox has a routine linprog to solve linear programs (LPs) only; so you need to discretize the curvature constraint.

Replace the curvature constraint by $N$ constraints evaluated at $N$ equidistant points, solve the LP, add more points close to the points where near-worst case curvature occurred, and repeat until the desired accuracy is reached.

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