## Linear Operators: Spectral theory |

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Page 1045

The convolution

considered as operators in L ( E " ) , and ... 1 that the convolution

exists for almost all x , and defines a bounded mapping of L , ( En ) into itself , Isp

soo .

The convolution

**integrals**( 1 ) ( k * } ) ( x ) = Senk ( x − y ) f ( y ) dy will beconsidered as operators in L ( E " ) , and ... 1 that the convolution

**integral**( 1 )exists for almost all x , and defines a bounded mapping of L , ( En ) into itself , Isp

soo .

Page 1046

an

Cauchy principal value as J - 20 X X 1 - 0 JE o eixy - e - ixy E - 0 Jɛ X poo sin xy ,

= lim 2i EVO JE X = lim 2i & Jay & = 2i sgn ( y ) sin x - dx JO X = ni sgn ( y ) .

an

**integral**studied by Hilbert . The**integral**( 2 ) may be interpreted in terms of aCauchy principal value as J - 20 X X 1 - 0 JE o eixy - e - ixy E - 0 Jɛ X poo sin xy ,

= lim 2i EVO JE X = lim 2i & Jay & = 2i sgn ( y ) sin x - dx JO X = ni sgn ( y ) .

Page 1047

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

fail . In the multi - dimensional case the convolution

...

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

**integral**ptoo f ( x ) - dx J - o | « — y ! instead of ( 3 ) , all our considerations wouldfail . In the multi - dimensional case the convolution

**integrals**pto 2 ( x − y ) - [ ( y )...

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### Contents

IX | 859 |

Bounded Normal Operators in Hilbert Space | 887 |

Compact Groups | 945 |

Copyright | |

46 other sections not shown

### Other editions - View all

Linear Operators, Part 1 Nelson Dunford,Jacob T. Schwartz,William G. Bade,Robert G. Bartle Snippet view - 1958 |

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