## Linear Operators: Self Adjoint Operators in Hilbert Space. Spectral theory. Part II |

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Page 951

When integration is with respect to Haar measure , as is generally the case , we

write dx instead of a ( dr ) . ... ( b ) For f , geLy ( R ) the

integrable in y for almost all x and the convolution f * g

defined by ...

When integration is with respect to Haar measure , as is generally the case , we

write dx instead of a ( dr ) . ... ( b ) For f , geLy ( R ) the

**function f**( x , y ) g ( y ) isintegrable in y for almost all x and the convolution f * g

**of f**and g , which isdefined by ...

Page 986

that f is invariant under translations . This shows that I 21 and completes the proof

of the lemma . Q.E.D. 7 THEOREM . ( Wiener L , -closure theorem ) . Linear

combinations of the translates of a

only ...

that f is invariant under translations . This shows that I 21 and completes the proof

of the lemma . Q.E.D. 7 THEOREM . ( Wiener L , -closure theorem ) . Linear

combinations of the translates of a

**function f**in Li ( R ) are dense in L ( R ) if andonly ...

Page 1646

It is clear that if F corresponds to the

g in the sense of the above definition , then aF + BG corresponds to of + Bg . Thus

the linear space of functions integrable on each compact subset of I may be ...

It is clear that if F corresponds to the

**function f**and G corresponds to the functiong in the sense of the above definition , then aF + BG corresponds to of + Bg . Thus

the linear space of functions integrable on each compact subset of I may be ...

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### Contents

SPECTRAL THEORY Self Adjoint Operators in Hilbert Space | 858 |

BAlgebras | 859 |

Preliminary Notions | 865 |

Copyright | |

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additive Akad algebra Amer analytic assume Banach spaces basis belongs Borel boundary conditions boundary values bounded called clear closed closure coefficients compact complex Consequently constant contains continuous converges Corollary corresponding defined Definition denote dense determined domain eigenvalues element equal equation essential spectrum evident Exercise exists extension finite follows formal differential operator formula function function f given Hence Hilbert space identity independent indices inequality integral interval Lemma limit linear mapping Math matrix measure multiplicity Nauk neighborhood norm obtained partial positive preceding present problem projection proof properties prove range regular remark representation respectively restriction result satisfies seen sequence singular solution spectral square-integrable statement subset subspace sufficiently Suppose symmetric Theorem theory topology transform unique vanishes vector zero