Fourier Transform On Other Function Spaces
The definition of the Fourier transform by the integral formula
is valid for Lebesgue integrable functions ƒ; that is, ƒ ∈ L1(R). The image of L1 a subset of the space C0(R) of continuous functions that tend to zero at infinity (the Riemann–Lebesgue lemma), although it is not the entire space. Indeed, there is no simple characterization of the image.
It is possible to extend the definition of the Fourier transform to other spaces of functions. Since compactly supported smooth functions are integrable and dense in L2(R), the Plancherel theorem allows us to extend the definition of the Fourier transform to general functions in L2(R) by continuity arguments. Further : L2(R) → L2(R) is a unitary operator (Stein & Weiss 1971, Thm. 2.3). In particular, the image of L2(R) is itself under the Fourier transform. The Fourier transform in L2(R) is no longer given by an ordinary Lebesgue integral, although it can be computed by an improper integral, here meaning that for an L2 function ƒ,
where the limit is taken in the L2 sense. Many of the properties of the Fourier transform in L1 carry over to L2, by a suitable limiting argument.
The definition of the Fourier transform can be extended to functions in Lp(R) for 1 ≤ p ≤ 2 by decomposing such functions into a fat tail part in L2 plus a fat body part in L1. In each of these spaces, the Fourier transform of a function in Lp(R) is in Lq(R), where is the Hölder conjugate of p. by the Hausdorff–Young inequality. However, except for p = 2, the image is not easily characterized. Further extensions become more technical. The Fourier transform of functions in Lp for the range 2 < p < ∞ requires the study of distributions (Katznelson 1976). In fact, it can be shown that there are functions in Lp with p>2 so that the Fourier transform is not defined as a function (Stein & Weiss 1971).
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