## Linear Operators: Spectral theory |

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Page 1045

The convolution

considered as operators in L ( En ) , 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 ) t ( y ) dy will beconsidered as operators in L ( En ) , 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 +00 eiry eixy dx lim ( + ) dr r r E - 0 . city -e dx lim E -

OJE sin ry dx lim 2i E - 0 ir E sin x dx Jim 2i E - 0 x EY . sin x 2i sgn ( y ) S. dx r 0 ni

...

an

**integral**studied by Hilbert . The**integral**( 2 ) may be interpreted in terms of aCauchy principal value as +00 eiry eixy dx lim ( + ) dr r r E - 0 . city -e dx lim E -

OJE sin ry dx lim 2i E - 0 ir E sin x dx Jim 2i E - 0 x EY . sin x 2i sgn ( y ) S. dx r 0 ni

...

Page 1047

If we tried to take 101-1 as the convolution kernel , i.e. , if we considered the

multi - dimensional case the convolution

If we tried to take 101-1 as the convolution kernel , i.e. , if we considered the

**integral**+00 f ( x ) \ x- y dx instead of ( 3 ) , all our considerations would fail . In themulti - dimensional case the convolution

**integrals**++ 2 ( x - y ) ( 4 ) f ( y ) dy \ x - y !### What people are saying - Write a review

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