## Linear Operators: Spectral Theory : Self Adjoint Operators in Hilbert Space, Volume 2 |

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Page 1983

... the Hilbert space of

N dimensions . We shall show that this algebra of all convolutions defined on H

regardless of whether they be defined in terms of bounded additive set functions

...

... the Hilbert space of

**square integrable**functions on real Euclidean space RN ofN dimensions . We shall show that this algebra of all convolutions defined on H

regardless of whether they be defined in terms of bounded additive set functions

...

Page 2037

... For ye D ( B ) , we have , by definition , lim 0 ) _ " Î y = Bys . n + 0 + h On the

other hand , it follows from ( 55 ) that for each s in RN if ( h ) - Î lim = Â ( s ) * ( s ) ,

h O + which shows that Â ( s ) * ( 8 ) is

.

... For ye D ( B ) , we have , by definition , lim 0 ) _ " Î y = Bys . n + 0 + h On the

other hand , it follows from ( 55 ) that for each s in RN if ( h ) - Î lim = Â ( s ) * ( s ) ,

h O + which shows that Â ( s ) * ( 8 ) is

**square integrable**on RN and that By = Âys.

Page 2332

If « = iß is purely imaginary , then we decompose f into a finite sum of

and immediately apply the ordinary theory of orthogonal expansions in Hilbert

space ...

If « = iß is purely imaginary , then we decompose f into a finite sum of

**square****integrable**functions , each vanishing outside an interval of length at most 1 / 8 ,and immediately apply the ordinary theory of orthogonal expansions in Hilbert

space ...

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