Revisions to Inequality with convolution

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edited Jan 15 at 16:04

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I have some troubles with the following problem:

A definition

Let $\sigma_1$ and $\sigma_2$ two positive numbers. We denote for all $x\in\mathbb{R}$, >$G_\sigma\left[ \phi \right](x)$ the gaussian convolution (with std $\sigma>0$) of a function $\phi$ at point $x$, in other words:

$$\forall x \in \mathbb{R},~ G_\sigma\left[ \phi \right](x) :=\frac{1}{\sqrt{2\pi}\sigma} \int_\mathbb{R}\phi(x-s)e^{-\frac{s^2}{2\sigma^2}}ds.$$

My question

Let $f$ and $g$ be two continuous functions on $\mathbb{R}$. Is there a way to bound the functionreal number

$$\Delta_{\sigma_1,\sigma_2}[f;g]:= |G_{\sigma_1}[f]-G_{\sigma_2}[g]|$$$$\Delta_{\sigma_1,\sigma_2}[f;g]:= \|G_{\sigma_1}[f]-G_{\sigma_2}[g]\|_\infty$$

with a functionsomething depending on $|f-g|$$\|f-g\|_\infty$.

Illustration of my post

What I would like is something like:

$$\exists (a,b,k)\in\mathbb{R}^3,~\forall x \in \mathbb{R}, \Delta_{\sigma_1,\sigma_2}\left[f;g\right](x) \leq a+b|f(x)-g(x)|^k$$$$\exists (a,b,k)\in\mathbb{R}^3,~\forall x \in \mathbb{R}, \Delta_{\sigma_1,\sigma_2}\left[f;g\right](x) \leq a+b\|f-g\|_\infty^k$$

with constants that can depend on $\sigma_1$ and $\sigma_2$.

Any hint would be highly appreciated, thank you!

I have some troubles with the following problem:

A definition

Let $\sigma_1$ and $\sigma_2$ two positive numbers. We denote for all $x\in\mathbb{R}$, >$G_\sigma\left[ \phi \right](x)$ the gaussian convolution (with std $\sigma>0$) of a function $\phi$ at point $x$, in other words:

$$\forall x \in \mathbb{R},~ G_\sigma\left[ \phi \right](x) :=\frac{1}{\sqrt{2\pi}\sigma} \int_\mathbb{R}\phi(x-s)e^{-\frac{s^2}{2\sigma^2}}ds.$$

My question

Let $f$ and $g$ be two continuous functions on $\mathbb{R}$. Is there a way to bound the function

$$\Delta_{\sigma_1,\sigma_2}[f;g]:= |G_{\sigma_1}[f]-G_{\sigma_2}[g]|$$

with a function depending on $|f-g|$.

Illustration of my post

What I would like is something like:

$$\exists (a,b,k)\in\mathbb{R}^3,~\forall x \in \mathbb{R}, \Delta_{\sigma_1,\sigma_2}\left[f;g\right](x) \leq a+b|f(x)-g(x)|^k$$

with constants that can depend on $\sigma_1$ and $\sigma_2$.

Any hint would be highly appreciated, thank you!

I have some troubles with the following problem:

A definition

Let $\sigma_1$ and $\sigma_2$ two positive numbers. We denote for all $x\in\mathbb{R}$, >$G_\sigma\left[ \phi \right](x)$ the gaussian convolution (with std $\sigma>0$) of a function $\phi$ at point $x$, in other words:

$$\forall x \in \mathbb{R},~ G_\sigma\left[ \phi \right](x) :=\frac{1}{\sqrt{2\pi}\sigma} \int_\mathbb{R}\phi(x-s)e^{-\frac{s^2}{2\sigma^2}}ds.$$

My question

Let $f$ and $g$ be two continuous functions on $\mathbb{R}$. Is there a way to bound the real number

$$\Delta_{\sigma_1,\sigma_2}[f;g]:= \|G_{\sigma_1}[f]-G_{\sigma_2}[g]\|_\infty$$

with a something depending on $\|f-g\|_\infty$.

Illustration of my post

What I would like is something like:

$$\exists (a,b,k)\in\mathbb{R}^3,~\forall x \in \mathbb{R}, \Delta_{\sigma_1,\sigma_2}\left[f;g\right](x) \leq a+b\|f-g\|_\infty^k$$

with constants that can depend on $\sigma_1$ and $\sigma_2$.

Any hint would be highly appreciated, thank you!

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Source Link

edited Jan 15 at 15:48

NancyBoy

393
1
10

I have some troubles with the following problem:

A definition

Let $\sigma_1$ and $\sigma_2$ two positive numbers. We denote for all $x\in\mathbb{R}$, >$G_\sigma\left[ \phi \right](x)$ the gaussian convolution (with std $\sigma>0$) of a function $\phi$ >atat point $x$, in other words:

$$\forall x \in \mathbb{R},~ G_\sigma\left[ \phi \right](x) :=\frac{1}{\sqrt{2\pi}\sigma} \int_\mathbb{R}\phi(x-s)e^{-\frac{s^2}{2\sigma^2}}ds.$$

My question

Let $f$ and $g$ be two continuous functions on $\mathbb{R}$. Is there a way to bound the function

$$\Delta_{\sigma_1,\sigma_2}[f;g]:= |G_{\sigma_1}[f]-G_{\sigma_2}[g]|$$

with a function depending on $|f-g|$.

Illustration of my post

What I would like is something like:

$$\exists (a,b,k)\in\mathbb{R}^3,~\forall x \in \mathbb{R}, \Delta_{\sigma_1,\sigma_2}\left[f;g\right](x) \leq a+b|f(x)-g(x)|^k$$

with explicit constants that can depend on $\sigma_1$ and $\sigma_2$.

Any hint would be highly appreciated, thank you!

I have some troubles with the following problem:

A definition

Let $\sigma_1$ and $\sigma_2$ two positive numbers. We denote for all $x\in\mathbb{R}$, >$G_\sigma\left[ \phi \right](x)$ the gaussian convolution (with std $\sigma>0$) of a function $\phi$ >at point $x$, in other words:

$$\forall x \in \mathbb{R},~ G_\sigma\left[ \phi \right](x) :=\frac{1}{\sqrt{2\pi}\sigma} \int_\mathbb{R}\phi(x-s)e^{-\frac{s^2}{2\sigma^2}}ds.$$

My question

Let $f$ and $g$ be two continuous functions on $\mathbb{R}$. Is there a way to bound the function

$$\Delta_{\sigma_1,\sigma_2}[f;g]:= |G_{\sigma_1}[f]-G_{\sigma_2}[g]|$$

with a function depending on $|f-g|$.

Illustration of my post

What I would like is something like:

$$\exists (a,b,k)\in\mathbb{R}^3,~\forall x \in \mathbb{R}, \Delta_{\sigma_1,\sigma_2}\left[f;g\right](x) \leq a+b|f(x)-g(x)|^k$$

with explicit constants.

Any hint would be highly appreciated, thank you!

I have some troubles with the following problem:

A definition

Let $\sigma_1$ and $\sigma_2$ two positive numbers. We denote for all $x\in\mathbb{R}$, >$G_\sigma\left[ \phi \right](x)$ the gaussian convolution (with std $\sigma>0$) of a function $\phi$ at point $x$, in other words:

$$\forall x \in \mathbb{R},~ G_\sigma\left[ \phi \right](x) :=\frac{1}{\sqrt{2\pi}\sigma} \int_\mathbb{R}\phi(x-s)e^{-\frac{s^2}{2\sigma^2}}ds.$$

My question

Let $f$ and $g$ be two continuous functions on $\mathbb{R}$. Is there a way to bound the function

$$\Delta_{\sigma_1,\sigma_2}[f;g]:= |G_{\sigma_1}[f]-G_{\sigma_2}[g]|$$

with a function depending on $|f-g|$.

Illustration of my post

What I would like is something like:

$$\exists (a,b,k)\in\mathbb{R}^3,~\forall x \in \mathbb{R}, \Delta_{\sigma_1,\sigma_2}\left[f;g\right](x) \leq a+b|f(x)-g(x)|^k$$

with constants that can depend on $\sigma_1$ and $\sigma_2$.

Any hint would be highly appreciated, thank you!

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edited Jan 15 at 15:40

NancyBoy

393
1
10

I have some troubles with the following problem:

A definition

Let $\sigma_1$ and $\sigma_2$ two positive numbers. We denote for all $x\in\mathbb{R}$, >$G_\sigma\left[ \phi \right](x)$ the gaussian convolution (with std $\sigma>0$) of a function $\phi$ >at point $x$, in other words:

$$\forall x \in \mathbb{R},~ G_\sigma\left[ \phi \right](x) :=\frac{1}{\sqrt{2\pi}\sigma} \int_\mathbb{R}\phi(x-s)e^{\frac{s^2}{2\sigma^2}}ds.$$$$\forall x \in \mathbb{R},~ G_\sigma\left[ \phi \right](x) :=\frac{1}{\sqrt{2\pi}\sigma} \int_\mathbb{R}\phi(x-s)e^{-\frac{s^2}{2\sigma^2}}ds.$$

My question

Let $f$ and $g$ be two continuous functions on $\mathbb{R}$. Is there a way to bound the function

$$\Delta_{\sigma_1,\sigma_2}[f;g]:= |G_{\sigma_1}[f]-G_{\sigma_2}[g]|$$

with a function depending on $|f-g|$.

Illustration of my post

What I would like is something like:

$$\exists (a,b,k)\in\mathbb{R}^3,~\forall x \in \mathbb{R}, \Delta_{\sigma_1,\sigma_2}\left[f;g\right](x) \leq a+b|f(x)-g(x)|^k$$

with explicit constants.

Any hint would be highly appreciated, thank you!

I have some troubles with the following problem:

A definition

Let $\sigma_1$ and $\sigma_2$ two positive numbers. We denote for all $x\in\mathbb{R}$, >$G_\sigma\left[ \phi \right](x)$ the gaussian convolution (with std $\sigma>0$) of a function $\phi$ >at point $x$, in other words:

$$\forall x \in \mathbb{R},~ G_\sigma\left[ \phi \right](x) :=\frac{1}{\sqrt{2\pi}\sigma} \int_\mathbb{R}\phi(x-s)e^{\frac{s^2}{2\sigma^2}}ds.$$

My question

Let $f$ and $g$ be two continuous functions on $\mathbb{R}$. Is there a way to bound the function

$$\Delta_{\sigma_1,\sigma_2}[f;g]:= |G_{\sigma_1}[f]-G_{\sigma_2}[g]|$$

with a function depending on $|f-g|$.

Illustration of my post

What I would like is something like:

$$\exists (a,b,k)\in\mathbb{R}^3,~\forall x \in \mathbb{R}, \Delta_{\sigma_1,\sigma_2}\left[f;g\right](x) \leq a+b|f(x)-g(x)|^k$$

with explicit constants.

Any hint would be highly appreciated, thank you!

I have some troubles with the following problem:

A definition

Let $\sigma_1$ and $\sigma_2$ two positive numbers. We denote for all $x\in\mathbb{R}$, >$G_\sigma\left[ \phi \right](x)$ the gaussian convolution (with std $\sigma>0$) of a function $\phi$ >at point $x$, in other words:

$$\forall x \in \mathbb{R},~ G_\sigma\left[ \phi \right](x) :=\frac{1}{\sqrt{2\pi}\sigma} \int_\mathbb{R}\phi(x-s)e^{-\frac{s^2}{2\sigma^2}}ds.$$

My question

Let $f$ and $g$ be two continuous functions on $\mathbb{R}$. Is there a way to bound the function

$$\Delta_{\sigma_1,\sigma_2}[f;g]:= |G_{\sigma_1}[f]-G_{\sigma_2}[g]|$$

with a function depending on $|f-g|$.

Illustration of my post

What I would like is something like:

$$\exists (a,b,k)\in\mathbb{R}^3,~\forall x \in \mathbb{R}, \Delta_{\sigma_1,\sigma_2}\left[f;g\right](x) \leq a+b|f(x)-g(x)|^k$$

with explicit constants.

Any hint would be highly appreciated, thank you!

Source Link

asked Jan 15 at 15:30

NancyBoy

393
1
10

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