## 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 , isp

soo .

The convolution

**integrals**( k * 1 ) ( x ) = Olteni ( 1 ) ( x - y ) / ( 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 , isp

soo .

Page 1046

an

Cauchy principal value as svo er de = lim S + S p de & + 0 J - & + 0Jɛ X poo sin

wy in = lim 2i Exo de x poo sin da = lim 2i | E70 Jeux = 2i sgn ( y ) sin de Do X = ri

...

an

**integral**studied by Hilbert . The**integral**( 2 ) may be interpreted in terms of aCauchy principal value as svo er de = lim S + S p de & + 0 J - & + 0Jɛ X poo sin

wy in = lim 2i Exo de x poo sin da = lim 2i | E70 Jeux = 2i sgn ( y ) sin de Do X = ri

...

Page 1047

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

the multi - dimensional case the convolution

yn ...

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

**integral**f ( x ) dac J - o x - y ! instead of ( 3 ) , all our considerations would fail . Inthe multi - dimensional case the convolution

**integrals**ptoo 2 ( x , y ) ( 4 ) J - \ x -yn ...

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### Contents

IX | 859 |

Eigenvalues and Eigenvectors | 903 |

Spectral Representation | 911 |

Copyright | |

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