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Page 1045
The convolution integrals ( 1 ) ( k * f ) ( x ) = Senk ( x − y ) f ( 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 , 1
Eps 00 .
The convolution integrals ( 1 ) ( k * f ) ( x ) = Senk ( x − y ) f ( 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 , 1
Eps 00 .
Page 1046
an integral studied by Hilbert . The integral ( 2 ) may be interpreted in terms of a
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 a
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 ) .
Page 1047
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 would
fail . In the multi - dimensional case the convolution integrals p + oo 2 ( x − y ) , (
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 would
fail . 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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additive adjoint adjoint operator algebra analytic assume B-algebra basis belongs Borel set boundary conditions boundary values bounded called clear closed closure coefficients commutative compact complex Consequently consider constant contains converges Corollary corresponding defined Definition denote dense determined domain eigenvalues element equal equation essential spectrum evident Exercise exists extension fact finite follows formal differential operator formula function function f give given Hence Hilbert space ideal identity independent indices inequality integral interval isometric isomorphism Lemma linear mapping matrix measure multiplicity neighborhood norm normal operator obtained positive preceding present projection proof properties prove range regular remark representation respectively restriction result satisfies seen sequence shown singular solution spectral square-integrable statement subset subspace sufficiently Suppose symmetric Theorem theory topology transform unique unit vanishes vector zero