## Linear Operators: Spectral theory |

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Page 1045

The convolution

considered as operators in L , ( E " ) , and conditions ... If Senk ( y ) | dy < oo , then

it follows from Lemma 3.1 that the convolution

and ...

The convolution

**integrals**( 1 ) ( k * f ) ( x ) = Senk ( x − y ) f ( y ) dy will beconsidered as operators in L , ( E " ) , and conditions ... If Senk ( y ) | dy < oo , then

it follows from Lemma 3.1 that the convolution

**integral**( 1 ) exists for almost all x ,and ...

Page 1046

an

Cauchy principal value as otoo eity eixy dx lim r + So dx ir E - 0 X oo poo pixy - e -

ixu lim dx E - OJE 100 sin xy lim 2i dx X E sin x dix lim 2i E - 0 Ey sin x 2i sgn ( y )

...

an

**integral**studied by Hilbert . The**integral**( 2 ) may be interpreted in terms of aCauchy principal value as otoo eity eixy dx lim r + So dx ir E - 0 X oo poo pixy - e -

ixu lim dx E - OJE 100 sin xy lim 2i dx X E sin x dix lim 2i E - 0 Ey sin x 2i sgn ( y )

...

Page 1047

If we tried to take | x | -1 as the convolution kernel , i.e. , if we considered the

multi - dimensional case the convolution

dy ...

If we tried to take | x | -1 as the convolution kernel , i.e. , if we considered the

**integral**too f ( x ) \ — y dx instead of ( 3 ) , all our considerations would fail . In themulti - dimensional case the convolution

**integrals**too ( 4 ) 2 ( x - y ) 1cc - y " [ ( y )dy ...

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### Contents

SPECTRAL THEORY Self Adjoint Operators in Hilbert Space | 858 |

BAlgebras | 859 |

Commutative BAlgebras | 868 |

Copyright | |

57 other sections not shown

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