## Linear Operators: Spectral theory |

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Page 1074

is the

transform of a function in L.,(– co, + oc), the

in Exercise 6. 10 Let à be a function defined on (– 00, + 00) which is of finite total

...

is the

**Fourier**transform of a function in L.,(–oo, -i- oc) whenever F is the**Fourier**transform of a function in L.,(– co, + oc), the

**Fourier**transforms being defined asin Exercise 6. 10 Let à be a function defined on (– 00, + 00) which is of finite total

...

Page 1075

F(t)e-* dt, - 27 J_A F denoting the

inequality Sup ... A > 0 16 Show that not every continuous function, defined for –

o – t < 00 and approaching zero as t approaches + o or — o, is the

transform of a ...

F(t)e-* dt, - 27 J_A F denoting the

**Fourier**transform of f, fails to satisfy theinequality Sup ... A > 0 16 Show that not every continuous function, defined for –

o – t < 00 and approaching zero as t approaches + o or — o, is the

**Fourier**transform of a ...

Page 1176

Let p, q, k, be as in the preceding lemma, and, for each N, let X^N be the

transformation in L,(l,) which maps the vector whose nth component has the

transform k,($)f ...

Let p, q, k, be as in the preceding lemma, and, for each N, let X^N be the

transformation in L,(l,) which maps the vector whose nth component has the

**Fourier**transform f,($) into the vector whose nth component has the**Fourier**transform k,($)f ...

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### Contents

BAlgebras | 859 |

Bounded Normal Operators in Hilbert Space | 887 |

Spectral Representation | 909 |

Copyright | |

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adjoint extension adjoint operator algebra Amer analytic B-algebra Banach Borel set boundary conditions boundary values bounded operator closed closure Cº(I coefficients complete complex numbers continuous function converges Corollary deficiency indices Definition denote dense differential equations Doklady Akad domain eigenfunctions eigenvalues element essential spectrum exists finite dimensional follows from Lemma follows from Theorem follows immediately formal differential operator formally self adjoint formula Fourier function f Haar measure Hence Hilbert space Hilbert-Schmidt operator identity inequality integral interval kernel Lemma Let f linear operator linearly independent mapping Math matrix measure Nauk SSSR N.S. neighborhood norm open set operators in Hilbert orthogonal orthonormal Plancherel's theorem positive Proc PRoof prove real numbers satisfies sequence singular ſº solution spectral spectral set spectral theory square-integrable subspace Suppose theory To(r topology transform unique unitary vanishes vector zero