# Integral question on a polynomial

#### joshmccraney

Problem Statement
Does there exist a function $f \in L_2[0,1]$ such that for all natural $n$, we have $$\int_0^1 x^n f(x) \, dx = 1$$
Relevant Equations
Nothing comes to mind.
At first I was thinking about using the dirac delta function $\delta(x-1)$, but then I recalled $\delta \notin L_2[0,1]$. Any ideas? I'm thinking no such function exists.

Related Calculus and Beyond Homework News on Phys.org

#### Mark44

Mentor
Problem Statement: Does there exist a function $f \in L_2[0,1]$ such that for all natural $n$, we have $$\int_0^1 x^n f(x) \, dx = 1$$
Relevant Equations: Nothing comes to mind.

At first I was thinking about using the dirac delta function $\delta(x-1)$, but then I recalled $\delta \notin L_2[0,1]$. Any ideas? I'm thinking no such function exists.
Please remind me (it's been a really long time since I've dealt with this stuff) what $f \in L_2[0,1]$ means. One of my topology books used $\mathcal l^2$, but this had to do with Hilbert spaces and sequences that were square-integrable.

Regarding your problem, I'm not sure that such a function exists. If we ignore the f(x) part, we get $\int_0^1 x dx = \frac 1 2, \int_0^1 x^2 dx = \frac 1 3, \dots, \int_0^1 x^n dx = \frac 1 {n + 1}$, and so on.

#### LCKurtz

Science Advisor
Homework Helper
Gold Member
Have you thought about using the Cauchy-Schwartz inequality to disprove it?

#### joshmccraney

Please remind me (it's been a really long time since I've dealt with this stuff) what $f \in L_2[0,1]$ means.
A function $f$ is in $L_2[0,1]$ if $\int_0^1 |f|^2\, dx$ exists.

Have you thought about using the Cauchy-Schwartz inequality to disprove it?
The Cauchy-Schwartz inequality for this case would be $$\int_0^1| f x^n| \, dx \leq \int_0^1|f|^2\, dx\int_0^1|x^n|^2\, dx = \frac{1}{2n+1}\int_0^1|f|^2\, dx$$ and if we assume such a function $f$ exists, then we have $$1 \leq \frac{1}{2n+1}\int_0^1|f|^2\, dx.$$
But I'm unsure how this is not true.

#### LCKurtz

Science Advisor
Homework Helper
Gold Member
What happens if n is large?

• joshmccraney

#### joshmccraney

What happens if n is large?
But isn't it possible $n\to\infty \implies f \to \infty$, and so balances? Like if $f = n$, or would this imply $f \notin L_2$?

#### LCKurtz

Science Advisor
Homework Helper
Gold Member
But isn't it possible $n\to\infty \implies f \to \infty$, and so balances? Like if $f = n$, or would this imply $f \notin L_2$?
No. You are using an indirect argument, supposing there such an $f$. It doesn't get to change and it has a finite $\|f\|$.

#### joshmccraney

Gotcha! Thanks. I liked your above comment so if anyone searches their attention is drawn to that one!

### Want to reply to this thread?

"Integral question on a polynomial"

### Physics Forums Values

We Value Quality
• Topics based on mainstream science
• Proper English grammar and spelling
We Value Civility
• Positive and compassionate attitudes
• Patience while debating
We Value Productivity
• Disciplined to remain on-topic
• Recognition of own weaknesses
• Solo and co-op problem solving