Hermitian Analysis by John P. D’Angelo

Author:John P. D’Angelo
Language: eng
Format: epub
Publisher: Springer New York, New York, NY

It is consistent with standard practice not to introduce a complex conjugation when using the inner product notation for the action of a distribution on a function.

Let D denote differentiation. By integration by parts, and because all boundary terms vanish when we are working on the Schwartz space, . We extend differentiation to the dual space by preserving this property. It follows that (D k ) t , the transpose of differentiating k times, is .

By Lemma 3.1, the transpose of the Fourier transform, when acting on functions in , is itself. In Definition 3.7, we will define the Fourier transform of a distribution by setting .

Let us give another example of a distribution and its derivative. Define a function u by u(x) = x for x ≥ 0 and u(x) = 0 for x < 0. This function is sometimes called the ramp function. Then u′, which is not defined at 0 as a function, nonetheless defines a distribution. We have

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