## Linear Operators: Spectral theory |

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Page 1074

8 Show, with the hypotheses and notation of Exercise 6, that if b is in L„(— oo, +

oo), then 9 Let A be a real function of a real variable such that A(-)F(-) is the

transform of ...

8 Show, with the hypotheses and notation of Exercise 6, that if b is in L„(— oo, +

oo), then 9 Let A be a real function of a real variable such that A(-)F(-) is the

**Fourier**transform of a function in L^ — oo, + oo) whenever F is the**Fourier**transform of ...

Page 1075

15 Show that there exists a function / in Lx(— oo, +°o) for which the family of

functions F denoting the

(x)\dx < oo. ^>o J 16 Show that not every continuous function, defined for — oo < t

...

15 Show that there exists a function / in Lx(— oo, +°o) for which the family of

functions F denoting the

**Fourier**transform of /, fails to satisfy the inequality sup \fA(x)\dx < oo. ^>o J 16 Show that not every continuous function, defined for — oo < t

...

Page 1160

95], Loomis [1; p. 151], Pontrjagin [1; Sec. 31—35], Raikov [1], and Weil [1; Sec.

28]. The results of Section 3 are, as we have seen, generalizations both of the

theory of

these ...

95], Loomis [1; p. 151], Pontrjagin [1; Sec. 31—35], Raikov [1], and Weil [1; Sec.

28]. The results of Section 3 are, as we have seen, generalizations both of the

theory of

**Fourier**series and of the**Fourier**integral. For the classical results inthese ...

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### Contents

BAlgebras | 859 |

Commutative BAlgebras | 860 |

Commutative BAlgebras | 874 |

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Acad adjoint extension adjoint operator algebra Amer analytic B-algebra B*-algebra Banach spaces Borel set boundary conditions boundary values bounded operator closed closure coefficients complex numbers constant 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 g Haar measure Hence Hilbert space Hilbert-Schmidt operator hypothesis identity inequality integral interval kernel Lemma linear operator linearly independent mapping Math matrix measure Nauk SSSR N. S. neighborhood norm open set operators in Hilbert orthogonal orthonormal partial differential operator Plancherel's theorem positive Proc prove real axis real numbers representation satisfies second order Section sequence singular solution spectral set spectral theory square-integrable subspace Suppose symmetric operator topology transform unique unitary vanishes vector zero