Linear Operators, Part 2Interscience Publishers, 1963 - Algebra, Universal |
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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
IX | 859 |
Bounded Normal Operators in Hilbert Space | 887 |
Miscellaneous Applications | 937 |
Copyright | |
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adjoint extension adjoint operator algebra analytic B-algebra B*-algebra Borel set boundary conditions boundary values bounded operator C₁ closed closure Co(I coefficients compact subset complex numbers continuous function converges Corollary deficiency indices Definition denote dense domain eigenvalues element essential spectrum 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(R L₂(I L₂(R Lemma Let f linear linearly independent mapping matrix measure neighborhood norm open set open subset orthonormal partial differential operator Plancherel's theorem positive PROOF prove real axis real numbers satisfies Section sequence solution spectral spectral theory square-integrable subspace Suppose T₁ T₁(t theory To(t topology unique unitary vanishes vector zero