应用程序

Basic Analysis I Pub Date : 2020-05-13 DOI:10.1201/9781315166254-31

J. Peterson

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引用次数: 0

摘要

(一)翻译(τx0Th) (f) = Th(τ−x0f), (b)调制(e0Th) (f) =届第九(e·ξ0 f), (c)的重新调节(Th◦)(f) = |侦破(a) | Th (f◦−1),(d)乘法xj: (xjTh) (f) = Th (xjf), (e)的乘法g:(高等)(f) = Th (fg), (f)导数∂xj:(∂xjTh) (f) =−Th(∂xjf),与f (g)卷积:(Th∗f) (x) = Th (f (x−·))(是一个光滑函数),傅里叶变换(h) f: f (Th) (f) = (f (f)),(我)傅里叶反变换f (Th) (f) = Th (f (f)) = (FR (f))。

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Applications

(a) The translation (τx0Th)(f) = Th(τ−x0f), (b) The modulation (e0Th)(f) = Th(e ix·ξ0f), (c) The rescaling (Th ◦A)(f) = |det(A)|Th(f ◦A−1), (d) The multiplication by xj : (xjTh)(f) = Th(xjf), (e) The multiplication by g : (gTh)(f) = Th(fg), (f) The derivative ∂xj : (∂xjTh)(f) = −Th(∂xjf), (g) The convolution with f : (Th ∗ f)(x) = Th(f(x− ·)) (is a smooth function), (h) The Fourier transform F : F(Th)(f) = Th(F(f)), (i) The inverse Fourier transform F(Th)(f) = Th(F(f)) = Th(FR(f)).

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来源期刊

Basic Analysis I

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