Norm of the Laplace

Norm of the Laplace - The norm of the Laplace transform...

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The norm of the Laplace transform Problem. Let L be the Laplace transform g ( s ) = Lf ( s ) = Z 0 f ( t ) e - st dt Then the operator L : L 2 ( o, ) L 2 ( o, ) is bounded and k L k ≤ π. Proof. | g ( s ) | 2 = ± Z 0 f ( t ) e - st dt ² 2 = ± Z 0 f ( t ) e - st/ 2 t 1 / 4 e - st/ 2 t - 1 / 4 dt ² 2 Z 0 | f ( t ) | 2 e - st t 1 / 2 dt Z 0 e - st t 1 / 2 dt Now Z 0 e - st t 1 / 2 dt = s - 1 / 2 Z 0 e - u u - 1 / 2 du = πs - 1 / 2 . Therefore | g ( s ) | 2 πs - 1 / 2 Z 0 | f ( t ) | 2 e - st t 1 / 2 dt. Integrating this inequality we obtain k g k 2
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