How can I Prove the following Integral Inequality?

In summary, the conversation discusses an inequality involving integrals and norms of functions. It is stated that this inequality is different from the Cauchy-Schwarz inequality and an example is provided to show its failure. The original inequality is also discussed and it is concluded that it is almost trivially true.
  • #1
zarei175
3
0
I want to prove the following inequality:

$$\sum\limits_{k\in\mathbb{N}}\Big(\int \big|f(x)\big|\big|g(x-k)\big|dx\Big)^2 \leq \big\|f\big\|^2 \sum\limits_{k\in\mathbb{N}}\Big (\int\big|g(x-k)\big|dx\Big)^2$$
where

$$\|f\|^2=\int |f(x)|^2dx.$$

My attempt: Just prove the following inequality

$$\Big(\int \big|f(x)\big|\big|g(x-k)\big|dx\Big)^2 \leq \|f\|^2\Big(\int \big|g(x-k)\big|dx\Big)^2$$
I think that this inequality is different from the Cauchy–Schwarz inequality.
Cauchy–Schwarz inequality is
$$\Big|\int f(x)\overline{g(x)}dx\Big|^2 \leq \int |f(x)|^2dx ~\cdot~\int |g(x)|^2dx $$
 
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  • #2
[tex]|\int f(x)\bar{g(x)}dx|^2 \le (\int|f(x)||g(x)|dx)^2[/tex], so Cauchy-Schwarz applies.
 
  • #3
No, Cauchy-Schwarz Inequality does not apply in the OP.

If it had been

$$\Big(\int \big|f(x)\big|\big|g(x-k)\big|dx\Big)^2 \leq \|f\|^2\Big(\int \big|g(x-k)\big|^2dx\Big)$$

then this is an application of Cauchy-Schwarz, but in the OP we have:

$$\Big(\int \big|f(x)\big|\big|g(x-k)\big|dx\Big)^2 \leq \|f\|^2\Big(\int \big|g(x-k)\big|dx\Big)^2$$

and here, Cauchy-Schwarz does not apply.

In fact, the latter inequality fails if e.g. ##k=0## and ##f(x)=g(x)=1## for ##0\le x\le 1/2## and ##0## otherwise: in this case, the left side of the inequality is ##1/4## and the right side is ##1/8## (assuming that the interval of integration is all of ##\mathbb R##).

Here, ##f=g## is not continuous, but it can easily be appoximated by a ##C^\infty##-function for which the inequality still fails.

On the other hand, the original inequality

$$\sum\limits_{k\in\mathbb{N}}\Big(\int \big|f(x)\big|\big|g(x-k)\big|dx\Big)^2 \leq \big\|f\big\|^2 \sum\limits_{k\in\mathbb{N}}\Big (\int\big|g(x-k)\big|dx\Big)^2$$

is almost trivially true: If ##f(x)=0## a.e or ##g(x)=0## a.e. then both sides are ##0##, otherwise, the right side is ##\infty## (all the terms in the sum are equal and positive).

So, it doesn't seem to me that this inequality could be very useful...
 

Related to How can I Prove the following Integral Inequality?

What is an integral inequality?

An integral inequality is a mathematical statement that compares the values of two integrals. It states that the integral of one function is greater than or less than the integral of another function, over a given interval.

Why do we need to prove an integral inequality?

Proving an integral inequality is important because it provides a rigorous mathematical justification for the inequality. It also helps to establish the conditions under which the inequality holds, which can be useful in solving other related problems.

What is the process for proving an integral inequality?

The process for proving an integral inequality involves breaking down the problem into smaller, more manageable steps. This usually involves using known mathematical properties and theorems to manipulate the given integrals and show that they are equivalent or that one is greater than the other. It is also important to clearly state any assumptions and conditions that are necessary for the inequality to hold.

What are some common techniques used to prove integral inequalities?

Some common techniques used to prove integral inequalities include the comparison test, the Cauchy-Schwarz inequality, and the mean value theorem for integrals. Other techniques may involve using substitutions, integration by parts, or applying known inequalities such as the AM-GM inequality.

What are some tips for successfully proving an integral inequality?

Some tips for proving an integral inequality include carefully analyzing the given integrals and identifying any patterns or relationships between them. It is also helpful to use known mathematical theorems and techniques, and to clearly explain each step in the proof. Additionally, it is important to check for any assumptions or restrictions on the variables or functions involved in the inequality.

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