Math 139 Fourier Analysis 030325 Introduction to the Fourier Transform ct'd

Recall the basic properties: translation becomes modulation (and vice-versa), scaling become dilation by the reciprocal, differentiation becomes multiplication by a monomial (and vice-versa). Consequence: the Fourier transform maps the Schwartz class to the Schwartz class.

Creating an approximation of the identity. First: introduce the Gaussian. Notice that a scaling of the Gaussian is invariant under the Fourier transform. Important: the family of dilated Gaussians is an approximation of the identity on the real line. Define convolution. Remark: convolution of Schwartz-class functions is still in the Schwartz class; the Fourier Transform of a convolution is the product of the Fourier transforms (as before). Main point: the approximation of the identity behaves as before (we can recover the function uniformly). Proof. Statement of Multiplication theorem. Proof.

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