How to prove Laplace transform is bounded on L_2(RR_+)?

Pavukol

Pavukol

Answered question

2022-09-07

The Laplace transform is defined by
( L f ) ( s ) 0 e s x f ( x ) d x , s > 0 ,
then how can we check that the Laplace transform L is bounded as an operator from L 2 ( R + ) to L 2 ( R + ) with norm π ?

Answer & Explanation

cerfweddrq

cerfweddrq

Beginner2022-09-08Added 15 answers

Observe
L ( f ) L 2 ( 0 , ) 2 =   0 0 0 d s d x d y   e s ( x + y ) f ( x ) f ( y ) =   0 0 d x d y   f ( x ) f ( y ) x + y = 0 d x   f ( x ) H ( x )
where
H ( f χ [ 0 , ) ) ( x ) = d y   f ( y ) χ [ 0 , ) ( y ) x + y = d y   g ( y ) x + y
for all x > 0. Then it follows
L ( f ) L 2 ( 0 , ) 2 f L 2 ( 0 , ) H ( g ) L 2 ( 0 , ) .
Using the L 2 L 2 boundedness of the Hilbert transform, we have the estimate
L ( f ) L 2 ( 0 , ) 2 π f L 2 ( 0 , ) 2 .

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