Math212a1406 The Fourier Transform The Laplace transform The spectral theorem for bounded PDF
Preview Math212a1406 The Fourier Transform The Laplace transform The spectral theorem for bounded
Outline Conventions, especially about 2π. Basic facts about the Fourier transform acting on S. The Fourier transform on L2. Sampling. The Heisenberg Uncertainty Principle. Tempered distributions. The Laplace transform. The spectral theorem for bounded self-adjoint operators, functional calculus form. The Mellin trransform Math212a1406 The Fourier Transform The Laplace transform The spectral theorem for bounded self-adjoint operators, functional calculus form The Mellin Transform Shlomo Sternberg September 18, 2014 Shlomo Sternberg Math212a1406 The Fourier Transform The Laplace transform The spectral theorem for bounded self-adjoint operators, functional calculus form The Mellin Transform Outline Conventions, especially about 2π. Basic facts about the Fourier transform acting on S. The Fourier transform on L2. Sampling. The Heisenberg Uncertainty Principle. Tempered distributions. The Laplace transform. The spectral theorem for bounded self-adjoint operators, functional calculus form. The Mellin trransform 1 Conventions, especially about 2π. 2 Basic facts about the Fourier transform acting on S. 3 The Fourier transform on L2. 4 Sampling. 5 The Heisenberg Uncertainty Principle. 6 Tempered distributions. ′ Examples of Fourier transforms of elements of S . 7 The Laplace transform. 8 The spectral theorem for bounded self-adjoint operators, functional calculus form. 9 The Mellin trransform Dirichlet series and their special values Shlomo Sternberg Math212a1406 The Fourier Transform The Laplace transform The spectral theorem for bounded self-adjoint operators, functional calculus form The Mellin Transform Outline Conventions, especially about 2π. Basic facts about the Fourier transform acting on S. The Fourier transform on L2. Sampling. The Heisenberg Uncertainty Principle. Tempered distributions. The Laplace transform. The spectral theorem for bounded self-adjoint operators, functional calculus form. The Mellin trransform The space S. The space S consists of all functions on R which are infinitely differentiable and vanish at infinity rapidly with all their derivatives in the sense that m (n) ‖f ‖m,n := sup{|x f (x)|} < ∞. x∈R The ‖ · ‖m,n give a family of semi-norms on S making S into a Frechet space - that is, a vector space space whose topology is determined by a countable family of semi-norms. Shlomo Sternberg Math212a1406 The Fourier Transform The Laplace transform The spectral theorem for bounded self-adjoint operators, functional calculus form The Mellin Transform Outline Conventions, especially about 2π. Basic facts about the Fourier transform acting on S. The Fourier transform on L2. Sampling. The Heisenberg Uncertainty Principle. Tempered distributions. The Laplace transform. The spectral theorem for bounded self-adjoint operators, functional calculus form. The Mellin trransform The measure on R. We use the measure 1 √ dx 2π on R and so define the Fourier transform of an element of S by ∫ 1 fˆ(ξ) := √ f (x)e−ixξdx 2π R and the convolution of two elements of S by ∫ 1 (f ⋆ g)(x) := √ f (x − t)g(t)dt. 2π R Shlomo Sternberg Math212a1406 The Fourier Transform The Laplace transform The spectral theorem for bounded self-adjoint operators, functional calculus form The Mellin Transform d ˆ So the Fourier transform of (−ix)f (x) is f (ξ). dξ Integration by parts (with vanishing values at the end points) gives ∫ ∫ 1 1 ′ −ixξ −ixξ √ f (x)e dx = (iξ)√ f (x)e dx. 2π R 2π R ′ ˆ So the Fourier transform of f is (iξ)f (ξ). Outline Conventions, especially about 2π. Basic facts about the Fourier transform acting on S. The Fourier transform on L2. Sampling. The Heisenberg Uncertainty Principle. Tempered distributions. The Laplace transform. The spectral theorem for bounded self-adjoint operators, functional calculus form. The Mellin trransform ∫ 1 −ixξ We are allowed to differentiate √ f (x)e dx with respect to 2π R ξ under the integral sign since f (x) vanishes so rapidly at ∞. We get ( ∫ ) ∫ d 1 1 −ixξ −ixξ √ f (x)e dx = √ (−ix)f (x)e dx. dξ 2π R 2π R Shlomo Sternberg Math212a1406 The Fourier Transform The Laplace transform The spectral theorem for bounded self-adjoint operators, functional calculus form The Mellin Transform Integration by parts (with vanishing values at the end points) gives ∫ ∫ 1 1 ′ −ixξ −ixξ √ f (x)e dx = (iξ)√ f (x)e dx. 2π R 2π R ′ ˆ So the Fourier transform of f is (iξ)f (ξ). Outline Conventions, especially about 2π. Basic facts about the Fourier transform acting on S. The Fourier transform on L2. Sampling. The Heisenberg Uncertainty Principle. Tempered distributions. The Laplace transform. The spectral theorem for bounded self-adjoint operators, functional calculus form. The Mellin trransform ∫ 1 −ixξ We are allowed to differentiate √ f (x)e dx with respect to 2π R ξ under the integral sign since f (x) vanishes so rapidly at ∞. We get ( ∫ ) ∫ d 1 1 −ixξ −ixξ √ f (x)e dx = √ (−ix)f (x)e dx. dξ 2π R 2π R d ˆ So the Fourier transform of (−ix)f (x) is f (ξ). dξ Shlomo Sternberg Math212a1406 The Fourier Transform The Laplace transform The spectral theorem for bounded self-adjoint operators, functional calculus form The Mellin Transform Outline Conventions, especially about 2π. Basic facts about the Fourier transform acting on S. The Fourier transform on L2. Sampling. The Heisenberg Uncertainty Principle. Tempered distributions. The Laplace transform. The spectral theorem for bounded self-adjoint operators, functional calculus form. The Mellin trransform ∫ 1 −ixξ We are allowed to differentiate √ f (x)e dx with respect to 2π R ξ under the integral sign since f (x) vanishes so rapidly at ∞. We get ( ∫ ) ∫ d 1 1 −ixξ −ixξ √ f (x)e dx = √ (−ix)f (x)e dx. dξ 2π R 2π R d ˆ So the Fourier transform of (−ix)f (x) is f (ξ). dξ Integration by parts (with vanishing values at the end points) gives ∫ ∫ 1 1 ′ −ixξ −ixξ √ f (x)e dx = (iξ)√ f (x)e dx. 2π R 2π R ′ ˆ So the Fourier transform of f is (iξ)f (ξ). Shlomo Sternberg Math212a1406 The Fourier Transform The Laplace transform The spectral theorem for bounded self-adjoint operators, functional calculus form The Mellin Transform Outline Conventions, especially about 2π. Basic facts about the Fourier transform acting on S. The Fourier transform on L2. Sampling. The Heisenberg Uncertainty Principle. Tempered distributions. The Laplace transform. The spectral theorem for bounded self-adjoint operators, functional calculus form. The Mellin trransform The Fourier transform maps S to S. Putting these two facts together gives The Fourier transform is well defined on S and [( )m ] ( )n d d ((−ix)nf ) ˆ= (iξ)m fˆ, dx dξ as follows by differentiation under the integral sign and by integration by parts. This shows that the Fourier transform maps S to S. Shlomo Sternberg Math212a1406 The Fourier Transform The Laplace transform The spectral theorem for bounded self-adjoint operators, functional calculus form The Mellin Transform Outline Conventions, especially about 2π. Basic facts about the Fourier transform acting on S. The Fourier transform on L2. Sampling. The Heisenberg Uncertainty Principle. Tempered distributions. The Laplace transform. The spectral theorem for bounded self-adjoint operators, functional calculus form. The Mellin trransform Convolution goes to multiplication. ∫ ∫ 1 −ixξ (f ⋆ g)ˆ(ξ) = f (x − t)g(t)dte dx 2π ∫ ∫ 1 −i(u+t)ξ = f (u)g(t)e dudt 2π ∫ ∫ 1 1 −iuξ −itξ = √ f (u)e du√ g(t)e dt 2π R 2π R so (f ⋆ g)ˆ= fˆgˆ. Shlomo Sternberg Math212a1406 The Fourier Transform The Laplace transform The spectral theorem for bounded self-adjoint operators, functional calculus form The Mellin Transform Outline Conventions, especially about 2π. Basic facts about the Fourier transform acting on S. The Fourier transform on L2. Sampling. The Heisenberg Uncertainty Principle. Tempered distributions. The Laplace transform. The spectral theorem for bounded self-adjoint operators, functional calculus form. The Mellin trransform Scaling. For any f ∈ S and a > 0 define Saf by (Sa)f (x) := f (ax). Then setting u = ax so dx = (1/a)du we have ∫ 1 −ixξ (Saf )ˆ(ξ) = √ f (ax)e dx 2π R ∫ 1 −iu(ξ/a) = √ (1/a)f (u)e du 2π R so (Saf )ˆ= (1/a)S1/afˆ. Shlomo Sternberg Math212a1406 The Fourier Transform The Laplace transform The spectral theorem for bounded self-adjoint operators, functional calculus form The Mellin Transform
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