## 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

Eps 00 .

The convolution

**integrals**( 1 ) ( k * f ) ( x ) = Senk ( 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

Eps 00 .

Page 1046

an

Cauchy principal value as poo eixu - e - ixy = lim - dx EJE X poo sin xy , = lim E70

2i Ję X poo sin a do = lim E — 0 " 2i | Ley X 8 = 2i sgn ( ) sina da = ni sgn ( y ) .

an

**integral**studied by Hilbert . The**integral**( 2 ) may be interpreted in terms of aCauchy principal value as poo eixu - e - ixy = lim - dx EJE X poo sin xy , = lim E70

2i Ję X poo sin a do = lim E — 0 " 2i | Ley X 8 = 2i sgn ( ) sina da = 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

4 ) ...

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

**integral**pto f ( x ) , - dx J - 20 lx - y instead of ( 3 ) , all our considerations wouldfail . In the multi - dimensional case the convolution

**integrals**p + oo 2 ( x − y ) , (4 ) ...

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### Contents

BAlgebras | 859 |

Commutative BAlgebras | 869 |

Commutative BAlgebras | 877 |

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