Linear Operators, Part 2 |
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Page 1074
... Fourier transform of a function in ∞ ) , the Fourier transforms being defined as in Exercise 6 . 10 Let 2 be a function defined on ( − ∞ , ∞ ) which is of finite total variation . Show that if 1 < p ≤ 2 , and if F is the Fourier ...
... Fourier transform of a function in ∞ ) , the Fourier transforms being defined as in Exercise 6 . 10 Let 2 be a function defined on ( − ∞ , ∞ ) which is of finite total variation . Show that if 1 < p ≤ 2 , and if F is the Fourier ...
Page 1075
... Fourier Show that if f is the integral of a function in L2 ( − ∞ , ∞ ) , this is impossible . 18 Let f be in L1 ( − ∞ , ∞ ) and let F be its Fourier transform . Then 1 • + ∞0 f ( x ) = lim A → ∞ 2π √ F ( t ) e - itz a ( ) t dt ...
... Fourier Show that if f is the integral of a function in L2 ( − ∞ , ∞ ) , this is impossible . 18 Let f be in L1 ( − ∞ , ∞ ) and let F be its Fourier transform . Then 1 • + ∞0 f ( x ) = lim A → ∞ 2π √ F ( t ) e - itz a ( ) t dt ...
Page 1160
... Fourier series and of the Fourier integral . For the classical results in these closely related subjects the reader should consult Zygmund [ 1 ] for Fourier series and Bochner [ 6 ] , Titchmarsh [ 3 ] and Wiener [ 4 ] for the Fourier ...
... Fourier series and of the Fourier integral . For the classical results in these closely related subjects the reader should consult Zygmund [ 1 ] for Fourier series and Bochner [ 6 ] , Titchmarsh [ 3 ] and Wiener [ 4 ] for the Fourier ...
Contents
BAlgebras | 859 |
Commutative BAlgebras | 868 |
Commutative BAlgebras | 874 |
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adjoint extension adjoint operator algebra analytic B-algebra Borel set boundary conditions boundary values bounded operator C₁ closed closure coefficients compact operator complex numbers continuous function converges Corollary deficiency indices Definition denote dense eigenvalues element equation essential spectrum Exercise exists finite dimensional follows from Lemma follows from Theorem follows immediately formal differential operator formally self adjoint formula Fourier function defined function f Haar measure Hence Hilbert space Hilbert-Schmidt operator identity inequality integral interval isometric isomorphism kernel L₁ L₁(R L₂(I L₂(R Lemma Let f linearly independent mapping matrix measure neighborhood non-zero norm operators in Hilbert orthogonal orthonormal basis Plancherel's theorem positive preceding lemma PROOF prove real axis real numbers satisfies sequence solution spectral spectral theorem square-integrable subspace Suppose T₁ T₂ theory To(t topology tr(T transform unique unitary vanishes vector zero