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Page 1045
The convolution integrals ( k * 1 ) ( x ) = Olteni ( 1 ) ( x - y ) / ( y ) dy will be
considered 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 .
The convolution integrals ( k * 1 ) ( x ) = Olteni ( 1 ) ( x - y ) / ( y ) dy will be
considered 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 integral studied by Hilbert . The integral ( 2 ) may be interpreted in terms of a
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 a
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
...
Page 1047
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 . In
the multi - dimensional case the convolution integrals ptoo 2 ( x , y ) ( 4 ) J - \ x -
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 . In
the 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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