Skip to main content
MathOverflow

Return to Question

Minor Math Jaxing, formatting and grammar.
Source Link
LSpice
LSpice
  • 12.9k
  • 4
  • 45
  • 69

Let $f$ be an even continuous function with compact support such that $$ \int f(t)\,\mathrm{d}t=1, $$ and let $g$ be a bounded continuous function such that the convolution $f\star g$ satisfies the following equality $$ (f\star g)(x)=g(x). $$ How to prove that if $g$ has global minimum then $g$ is constant?
I thought about using the Convolution Theorem, but it seems it doesn't work in this case. May it be thatMaybe, without the requirement that $g$ has global minimum, there is a way to prove that $g$ is linear?

Let $f$ be an even continuous function with compact support such that $$ \int f(t)\,\mathrm{d}t=1, $$ and let $g$ be a bounded continuous function such that the convolution $f\star g$ satisfies the following equality $$ (f\star g)(x)=g(x). $$ How to prove that if $g$ has global minimum then $g$ is constant?
I thought about using the Convolution Theorem, but it seems it doesn't work in this case. May it be that, without the requirement that $g$ has global minimum, there is a way to prove that $g$ is linear?

Let $f$ be an even continuous function with compact support such that $$ \int f(t)\,\mathrm{d}t=1, $$ and let $g$ be a bounded continuous function such that the convolution $f\star g$ satisfies the following equality $$ (f\star g)(x)=g(x). $$ How to prove that if $g$ has global minimum then $g$ is constant?
I thought about using the Convolution Theorem, but it seems it doesn't work in this case. Maybe, without the requirement that $g$ has global minimum, there is a way to prove that $g$ is linear?

Minor Math Jaxing, formatting and grammar.
Source Link
edit approved Apr 24, 2020 at 19:30
Daniele Tampieri
edit approved Apr 24, 2020 at 19:30
Daniele Tampieri
  • 6.4k
  • 7
  • 30
  • 45

If the convolution of two functions f*g$f\star g$ is equal gto $g$, f$f$ is even with compact support and g$g$ is bounded, then gimplies that $g$ is constant?

Let $f$ isbe an even continuous function with compact support such that $\int f(t)dt=1 $, $$ \int f(t)\,\mathrm{d}t=1, $$ and let $g$ isbe a bounded continuous function andsuch that the convolution (f*g)(x)=g(x). How$f\star g$ satisfies the following equality $$ (f\star g)(x)=g(x). $$ How to prove that if $g$ has global minimum then $g$ is constant.?
I thought about using the Convolution Theorem, but it seems it doesn't work in this case. May it be that, without restrictionthe requirement that $g$ has global minimum, there is a way to prove that $g$ is linear.?

If the convolution of two functions f*g is equal g, f is even with compact support and g is bounded, then g is constant

Let $f$ is even continuous function with compact support such that $\int f(t)dt=1 $, $g$ is bounded continuous function and convolution (f*g)(x)=g(x). How to prove that if $g$ has global minimum then $g$ is constant. I thought about Convolution Theorem but it seems it doesn't work in this case. May be without restriction that $g$ has global minimum there is a way to prove that $g$ is linear.

If the convolution of two functions $f\star g$ is equal to $g$, $f$ is even with compact support and $g$ is bounded, implies that $g$ is constant?

Let $f$ be an even continuous function with compact support such that $$ \int f(t)\,\mathrm{d}t=1, $$ and let $g$ be a bounded continuous function such that the convolution $f\star g$ satisfies the following equality $$ (f\star g)(x)=g(x). $$ How to prove that if $g$ has global minimum then $g$ is constant?
I thought about using the Convolution Theorem, but it seems it doesn't work in this case. May it be that, without the requirement that $g$ has global minimum, there is a way to prove that $g$ is linear?

Source Link
Anton Sorokovskiy
Anton Sorokovskiy
  • 137
  • 5

If the convolution of two functions f*g is equal g, f is even with compact support and g is bounded, then g is constant

Let $f$ is even continuous function with compact support such that $\int f(t)dt=1 $, $g$ is bounded continuous function and convolution (f*g)(x)=g(x). How to prove that if $g$ has global minimum then $g$ is constant. I thought about Convolution Theorem but it seems it doesn't work in this case. May be without restriction that $g$ has global minimum there is a way to prove that $g$ is linear.