Problem 11. Prove that the space C^∞ (smooth functions of compact support) is dense in L^1 (with

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Problem 11. Prove that the space $C^\infty$ (smooth functions of compact support) is dense in $L^1$ (with respect to the $L^1$ topology) by following the steps below. 1. Given $f \in L^1$ consider (convolution with a rescaled bump function) $$f_\epsilon(x) := \int f(x - y) \frac{1}{\epsilon} \varphi(y/\epsilon) dy$$ MAT 357 - FINAL PRACTICE PROBLEMS 5 where $\varphi$ is a non-negative $C^\infty$-smooth bump function supported in $[-1, 1]$, equal to 1 on $[-1/4, 1/4]$ and with $\int \varphi = 1$. Prove that $f_\epsilon$ is smooth for all $\epsilon > 0$, and check that if $f$ has compact support, then $f_\epsilon$ has compact support too. 2. Approximate any $f \in L^1$ by an $L^1$ function of compact support $g$ (use Dominated Convergence) and then show that $g_\epsilon \to g$ in $L^1$ as $\epsilon \to 0$. Hint: Review the proof about pointwise approximation using good kernels, and use continuity of translations... What would you need for the same approximation result to hold for $L^p$ functions, $p \in (1, \infty)$?

          Problem 11. Prove that the space $C^\infty$ (smooth functions of compact support) is dense in $L^1$ (with respect to the $L^1$ topology) by following the steps below.
1. Given $f \in L^1$ consider (convolution with a rescaled bump function)
$$f_\epsilon(x) := \int f(x - y) \frac{1}{\epsilon} \varphi(y/\epsilon) dy$$
MAT 357 - FINAL PRACTICE PROBLEMS
5
where $\varphi$ is a non-negative $C^\infty$-smooth bump function supported in $[-1, 1]$, equal to 1 on $[-1/4, 1/4]$ and with $\int \varphi = 1$.
Prove that $f_\epsilon$ is smooth for all $\epsilon > 0$, and check that if $f$ has compact support, then $f_\epsilon$ has compact support too.
2. Approximate any $f \in L^1$ by an $L^1$ function of compact support $g$ (use Dominated Convergence) and then show that $g_\epsilon \to g$ in $L^1$ as $\epsilon \to 0$.
Hint: Review the proof about pointwise approximation using good kernels, and use continuity of translations...
What would you need for the same approximation result to hold for $L^p$ functions, $p \in (1, \infty)$?

Problem 11. Prove that the space C^∞ (smooth functions of compact support) is dense in L^1 (with respect to the L^1 topology) by following the steps below.
1. Given f ∈ L^1 consider (convolution with a rescaled bump function)

fϵ(x) := ∫ f(x - y) (1)/(ϵ)φ(y/ϵ) dy

MAT 357 - FINAL PRACTICE PROBLEMS
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where φ is a non-negative C^∞-smooth bump function supported in [-1, 1], equal to 1 on [-1/4, 1/4] and with ∫φ = 1.
Prove that fϵ is smooth for all ϵ > 0, and check that if f has compact support, then fϵ has compact support too.
2. Approximate any f ∈ L^1 by an L^1 function of compact support g (use Dominated Convergence) and then show that gϵ→ g in L^1 as ϵ→ 0.
Hint: Review the proof about pointwise approximation using good kernels, and use continuity of translations...
What would you need for the same approximation result to hold for L^p functions, p ∈ (1, ∞)?

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