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user139844
user139844

How can I build (i.e. find an explicit formula) a smooth function $f_\epsilon: \mathbb R \to \mathbb R$ depending on a parameter $\epsilon >0$ which is "almost the identity" but constant around a (fixed) value $\bar x \in \mathbb R$ in the following sense:

  • $f_\epsilon \in C^2(\mathbb R)$;
  • $L_\epsilon > f'_\epsilon \ge 0$;
  • $|f''_\epsilon| \le C_\epsilon $;
  • $|xf''_\epsilon(x) |\le f'_\epsilon(x)$;
  • $f_\epsilon(\cdot) = \bar x$ in $(\bar x - \epsilon/2, \bar x + \epsilon/2)$;
  • $\|f_\epsilon(\cdot) - \mathrm{Id}\|_{L^p([a,b])} \lesssim \epsilon $ for any $p \in [1,\infty]$ (here $Id(x) = x$ is the identity and $[a,b]$ is any compact interval of $\mathbb R$)

How can I build (i.e. find an explicit formula) a smooth function $f_\epsilon: \mathbb R \to \mathbb R$ depending on a parameter $\epsilon >0$ which is "almost the identity" but constant around a (fixed) value $\bar x \in \mathbb R$ in the following sense:

  • $f_\epsilon \in C^2(\mathbb R)$;
  • $L_\epsilon > f'_\epsilon \ge 0$;
  • $|f''_\epsilon| \le C_\epsilon $;
  • $|xf''_\epsilon(x) |\le f'_\epsilon(x)$;
  • $f_\epsilon(\cdot) = \bar x$ in $(\bar x - \epsilon/2, \bar x + \epsilon/2)$;
  • $\|f_\epsilon(\cdot) - \mathrm{Id}\|_{L^p([a,b])} \lesssim \epsilon $ for any $p \in [1,\infty]$ (here $Id(x) = x$ is the identity and $[a,b]$ is any compact interval of $\mathbb R$)

How can I build (i.e. find an explicit formula) a smooth function $f_\epsilon: \mathbb R \to \mathbb R$ depending on a parameter $\epsilon >0$ which is "almost the identity" but constant around a (fixed) value $\bar x \in \mathbb R$ in the following sense:

  • $f_\epsilon \in C^2(\mathbb R)$;
  • $L_\epsilon > f'_\epsilon \ge 0$;
  • $|f''_\epsilon| \le C_\epsilon $;
  • $f_\epsilon(\cdot) = \bar x$ in $(\bar x - \epsilon/2, \bar x + \epsilon/2)$;
  • $\|f_\epsilon(\cdot) - \mathrm{Id}\|_{L^p([a,b])} \lesssim \epsilon $ for any $p \in [1,\infty]$ (here $Id(x) = x$ is the identity and $[a,b]$ is any compact interval of $\mathbb R$)
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user139844
user139844

How can I build (i.e. find an explicit formula) a smooth function $f_\epsilon: \mathbb R \to \mathbb R$ depending on a parameter $\epsilon >0$ which is "almost the identity" but constant around a (fixed) value $\bar x \in \mathbb R$ in the following sense:

  • $f_\epsilon \in C^2(\mathbb R)$;
  • $L_\epsilon > f'_\epsilon \ge 0$;
  • $|f''_\epsilon| \le C_\epsilon $;
  • $|xf''_\epsilon(x) |\le f'_\epsilon(x)$;
  • $f_\epsilon(\cdot) = \bar x$ in $(\bar x - \epsilon/2, \bar x + \epsilon/2)$.;
  • $\|f_\epsilon(\cdot) - \mathrm{Id}\|_{L^p([a,b])} \lesssim \epsilon $ for any $p \in [1,\infty]$ (here $Id(x) = x$ is the identity and $[a,b]$ is any compact interval of $\mathbb R$)

How can I build (i.e. find an explicit formula) a smooth function $f_\epsilon: \mathbb R \to \mathbb R$ depending on a parameter $\epsilon >0$ which is "almost the identity" but constant around a (fixed) value $\bar x \in \mathbb R$ in the following sense:

  • $f_\epsilon \in C^2(\mathbb R)$;
  • $L_\epsilon > f'_\epsilon \ge 0$;
  • $|f''_\epsilon| \le C_\epsilon $;
  • $|xf''_\epsilon(x) |\le f'_\epsilon(x)$;
  • $f_\epsilon(\cdot) = \bar x$ in $(\bar x - \epsilon/2, \bar x + \epsilon/2)$.
  • $\|f_\epsilon(\cdot) - \mathrm{Id}\|_{L^p([a,b])} \lesssim \epsilon $ for any $p \in [1,\infty]$ (here $Id(x) = x$ is the identity and $[a,b]$ is any compact interval of $\mathbb R$)

How can I build (i.e. find an explicit formula) a smooth function $f_\epsilon: \mathbb R \to \mathbb R$ depending on a parameter $\epsilon >0$ which is "almost the identity" but constant around a (fixed) value $\bar x \in \mathbb R$ in the following sense:

  • $f_\epsilon \in C^2(\mathbb R)$;
  • $L_\epsilon > f'_\epsilon \ge 0$;
  • $|f''_\epsilon| \le C_\epsilon $;
  • $|xf''_\epsilon(x) |\le f'_\epsilon(x)$;
  • $f_\epsilon(\cdot) = \bar x$ in $(\bar x - \epsilon/2, \bar x + \epsilon/2)$;
  • $\|f_\epsilon(\cdot) - \mathrm{Id}\|_{L^p([a,b])} \lesssim \epsilon $ for any $p \in [1,\infty]$ (here $Id(x) = x$ is the identity and $[a,b]$ is any compact interval of $\mathbb R$)
Source Link
user139844
user139844

Build an explicit "small perturbation" of the identity satisfying some properties

How can I build (i.e. find an explicit formula) a smooth function $f_\epsilon: \mathbb R \to \mathbb R$ depending on a parameter $\epsilon >0$ which is "almost the identity" but constant around a (fixed) value $\bar x \in \mathbb R$ in the following sense:

  • $f_\epsilon \in C^2(\mathbb R)$;
  • $L_\epsilon > f'_\epsilon \ge 0$;
  • $|f''_\epsilon| \le C_\epsilon $;
  • $|xf''_\epsilon(x) |\le f'_\epsilon(x)$;
  • $f_\epsilon(\cdot) = \bar x$ in $(\bar x - \epsilon/2, \bar x + \epsilon/2)$.
  • $\|f_\epsilon(\cdot) - \mathrm{Id}\|_{L^p([a,b])} \lesssim \epsilon $ for any $p \in [1,\infty]$ (here $Id(x) = x$ is the identity and $[a,b]$ is any compact interval of $\mathbb R$)