8.4 Fourier transforms of tempered distributions

Fourier transforms of tempered distributions extend classical Fourier analysis to a broader class of objects. This powerful tool allows us to analyze singular functions and generalized functions, opening up new possibilities in signal processing and differential equations.

Tempered distributions, defined on the Schwartz space of rapidly decreasing functions, include both regular functions and more exotic objects like the Dirac delta. Their Fourier transforms preserve important properties, enabling us to solve complex problems in harmonic analysis and beyond.

Tempered Distributions and Schwartz Space

Properties of Tempered Distributions

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• Tempered distributions are continuous linear functionals on the space of rapidly decreasing functions (Schwartz space)
• Act on test functions in the Schwartz space to produce complex numbers
• Generalize the notion of functions and allow for more singular objects like the Dirac delta function
• Include regular distributions and Schwartz functions as special cases

Schwartz Space and Rapidly Decreasing Functions

• Schwartz space consists of smooth functions whose derivatives decay faster than any polynomial at infinity
• Rapidly decreasing functions in Schwartz space have the property that $|x^{\alpha} D^{\beta} f(x)| \leq C_{\alpha,\beta}$ for all multi-indices $\alpha$ and $\beta$
  • $C_{\alpha,\beta}$ is a constant depending on $\alpha$ and $\beta$
  • Ensures functions and their derivatives approach zero faster than any polynomial
• Test functions are elements of the Schwartz space used to define tempered distributions
  • Infinitely differentiable and rapidly decreasing at infinity along with all their derivatives
  • Examples include Gaussian functions like $e^{-x^2}$ and smooth functions with compact support

Fourier Transforms of Distributions

Definition and Properties

• Fourier transform of a tempered distribution $T$ is defined as $\langle \mathcal{F}T, \varphi \rangle = \langle T, \mathcal{F}\varphi \rangle$ for all test functions $\varphi$ in Schwartz space
  • $\mathcal{F}$ denotes the Fourier transform operator
  • Extends the classical Fourier transform to tempered distributions
• Fourier transform is a continuous linear map from the space of tempered distributions to itself
• Inverse Fourier transform of a tempered distribution is well-defined and satisfies $\mathcal{F}^{-1}(\mathcal{F}T) = T$

Convolution and Duality

• Convolution of two tempered distributions $S$ and $T$ is defined as $\langle S * T, \varphi \rangle = \langle S(x), \langle T(y), \varphi(x+y) \rangle \rangle$
  • Generalizes the classical convolution operation to tempered distributions
  • Fourier transform of a convolution is the product of Fourier transforms: $\mathcal{F}(S * T) = \mathcal{F}S \cdot \mathcal{F}T$
• Duality between multiplication and convolution holds for tempered distributions
  • Fourier transform converts convolution to multiplication and vice versa
  • Enables solving differential equations and studying properties of distributions

Support of Distributions

• Support of a tempered distribution $T$ is the smallest closed set outside which $T$ vanishes
  • Analogous to the support of a function
  • Fourier transform of a compactly supported distribution is an entire function (analytic everywhere)
• Distribution with compact support has a Fourier transform that is a smooth function
  • Paley-Wiener theorem characterizes Fourier transforms of compactly supported distributions

Fourier Analysis Theorems

Parseval's and Plancherel's Theorems

• Parseval's theorem for tempered distributions states that $\langle S, T \rangle = \langle \mathcal{F}S, \mathcal{F}T \rangle$
  • Inner product of two tempered distributions equals the inner product of their Fourier transforms
  • Generalizes the classical Parseval's theorem for $L^2$ functions
• Plancherel's theorem is a special case of Parseval's theorem when $S = T$
  • States that the Fourier transform is an isometry on the space of tempered distributions
  • Preserves the $L^2$ norm: $\|\mathcal{F}T\|_{L^2} = \|T\|_{L^2}$ for $T$ in $L^2$

Fourier Inversion and Paley-Wiener Theorem

• Fourier inversion formula for tempered distributions: $T = \mathcal{F}^{-1}(\mathcal{F}T)$
  • Inverse Fourier transform of the Fourier transform recovers the original distribution
  • Holds in the sense of distributions, i.e., when applied to test functions
• Paley-Wiener theorem characterizes Fourier transforms of compactly supported distributions
  • States that a tempered distribution has compact support if and only if its Fourier transform is an entire function satisfying certain growth conditions
  • Provides a necessary and sufficient condition for a function to be the Fourier transform of a compactly supported distribution
  • Example: Fourier transform of the indicator function of an interval is an entire function of exponential type