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I encountered one problem which is to find $F(\lambda)$ which satisfies $1 = \int_{-\pi}^\pi e^{i k \lambda} dF(\lambda), \forall k \in \mathbb{Z}$ and the answer is $F(\lambda) = 1$ if $\lambda \ge 0$ and 0 otherwise.

I would like to solve it using mathematica, but a naive approach such as RSolve[1 == Integrate[Exp[I k x], {F[x], -Pi, Pi}], F, x, Assumptions -> Integers[k]] doesn't seem to work. The error message says

Supplied equations are not difference equations of the given functions.

Any help will be highly appreciated!

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If F[x_]:=HeavisideTheta[x] then $dF=\delta(x)dx$, hence $\int_{-\pi}^{\pi}e^{ikx}dF=1$. The code is

F = HeavisideTheta[x]; Integrate[Exp[I k x] D[F, x], {x, -Pi, Pi}, 
 Assumptions -> k > 0] 
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  • $\begingroup$ @yarchik Any way we can't solve it with Mathematica as it is. May be you mean some numerical solution then it could be interesting to see how solution turns to HeavisideTheta[x]. $\endgroup$ Apr 28 at 10:21
  • $\begingroup$ To have a constructive solution one may Fourier transform with respect to $k$ the left and right hand sides. $\endgroup$
    – yarchik
    Apr 28 at 10:49

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