The Riemann-Lebesgue lemma states that the Fourier transform of any absolutely integrable (L¹) function must be a bounded function that tends to zero as the frequency variable approaches infinity. This means any valid Fourier transform must be a bounded function — an unbounded function (like x, which grows without limit) cannot be the Fourier transform of any well-behaved function.
By the Riemann-Lebesgue lemma and general properties of Fourier transforms of integrable functions, the Fourier transform F(ω) of any function f(x) must be bounded (|F(ω)|...
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