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Ejercicio 8: solve Exercise 7, Page 54: For $f, u \in C_0^\infty(\mathbb{R})$ the space of smooth functions with compact support in $\mathbb{R}$, set $$K_f u(x) = f * u(x) = \int f(y)u(x-y)dy \qquad (1)$$ Show that, for $1 \le p < \infty$, $K_f$ has a unique bounded extension: $$K_f : L^p(\mathbb{R}) \to L^p(\mathbb{R}), \quad ||K_f u||_{L^p(\mathbb{R})} \le ||f||_{L^1} ||u||_{L^p}.$$ The operation in (1) is called convolution.

Name: ejercicio 8 solve exercise 7 page 54 for f u in c0inftymathbbr the space of smooth functions with compact support in mathbbr set kf ux f ux int fyux ydy qquad 1 show that for 1 le p infty 50436
Uploaded: 2024-10-15T08:34:37-08:00
Duration: 6 min
Channel: Shu Naito
Description: ejercicio 8 solve exercise 7 page 54 for f u in c0inftymathbbr the space of smooth functions with compact support in mathbbr set kf ux f ux int fyux ydy qquad 1 show that for 1 le p infty 50436

          Ejercicio 8: solve Exercise 7, Page 54: For $f, u \in C_0^\infty(\mathbb{R})$ the space of smooth functions with compact support in $\mathbb{R}$, set
$$K_f u(x) = f * u(x) = \int f(y)u(x-y)dy \qquad (1)$$
Show that, for $1 \le p < \infty$, $K_f$ has a unique bounded extension:
$$K_f : L^p(\mathbb{R}) \to L^p(\mathbb{R}), \quad ||K_f u||_{L^p(\mathbb{R})} \le ||f||_{L^1} ||u||_{L^p}.$$
The operation in (1) is called convolution.

ejercicio 8 solve exercise 7 page 54 for f u in c0inftymathbbr the space of smooth functions with compact support in mathbbr set kf ux f ux int fyux ydy qquad 1 show that for 1 le p infty 50436

Added by Anthony B.

Calculus: Early Transcendentals

James Stewart 8th Edition

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Solved by Expert Shu Naito

10/15/2024

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Ejercicio 8: solve Exercise 7, Page 54: For $f, u \in C_0^\infty(\mathbb{R})$ the space of smooth functions with compact support in $\mathbb{R}$, set $$K_f u(x) = f * u(x) = \int f(y)u(x-y)dy \qquad (1)$$ Show that, for $1 \le p < \infty$, $K_f$ has a unique bounded extension: $$K_f : L^p(\mathbb{R}) \to L^p(\mathbb{R}), \quad ||K_f u||_{L^p(\mathbb{R})} \le ||f||_{L^1} ||u||_{L^p}.$$ The operation in (1) is called convolution.