## Linear Operators: Spectral theory |

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Page 1045

The convolution

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

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

sps oo .

The convolution

**integrals**( 1 ) ( k * f ) ( x ) = Sg - k ( x , y ) f ( y ) dy will beconsidered as operators in L , ( En ) , and ... 1 that the convolution

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

sps oo .

Page 1046

an

Cauchy principal value as ptoo piry so die = lim { + 13 de poo eixy - e - ixy - dx E -

01 € X E= 2i sgn ( x ) * sin e dit JOX = ri sgn ( y ) . This is a bounded function .

an

**integral**studied by Hilbert . The**integral**( 2 ) may be interpreted in terms of aCauchy principal value as ptoo piry so die = lim { + 13 de poo eixy - e - ixy - dx E -

01 € X E= 2i sgn ( x ) * sin e dit JOX = ri sgn ( y ) . This is a bounded function .

Page 1047

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

the multi - dimensional case the convolution

dy ...

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

**integral**ptoo f ( x ) Te x - y da instead of ( 3 ) , all our considerations would fail . Inthe multi - dimensional case the convolution

**integrals**( 4 ) ptoo 2 ( x , y ) = f ( y )dy ...

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### Contents

BAlgebras | 861 |

Commutative BAlgebras | 868 |

Commutative BAlgebras | 874 |

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