Abstract
In this chapter we review basic elements of Fourier analysis on ℝn. Consequently, we introduce spaces of distributions, putting emphasis on the space of tempered distributions S′(ℝn). Finally, we discuss Sobolev spaces and approximation of functions and distributions by smooth functions. Throughout, we fix the measure on ℝn to be Lebesgue measure. For convenience, we may repeat a few definitions in the context of ℝn although they may have already appeared in Chapter C on measure theory. From this point of view, the present chapter can be read essentially independently. The notation used in this chapter and also in Chapter 2 is 〈ξ〉 = (1 + |ξ|2)1/2 where |ξ| = (ξ12 + ξ n 2)1/2, ξ ∈ ℝn.
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© 2010 Birkhäuser Verlag AG
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Ruzhansky, M., Turunen, V. (2010). Fourier Analysis on ℝn. In: Pseudo-Differential Operators and Symmetries. Pseudo-Differential Operators, vol 2. Birkhäuser Basel. https://doi.org/10.1007/978-3-7643-8514-9_6
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DOI: https://doi.org/10.1007/978-3-7643-8514-9_6
Publisher Name: Birkhäuser Basel
Print ISBN: 978-3-7643-8513-2
Online ISBN: 978-3-7643-8514-9
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