Convolution of a polynomial with itself

In summary, the convolution of a polynomial with itself is just a summation of the binomial coefficients.
  • #1
Char. Limit
Gold Member
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Again, in my quest to learn things I won't use in a class for at least a year, I've been looking at convolutions. Specifically, after finishing the multiple choice section of an AP Chemistry test 50 minutes early, I looked at the convolution of a polynomial with itself. I'm confused about one thing, though.

Here's what I have:

[tex]t^m \star t^m = t^{2m+1} \sum_{n=1}^{m+1} \frac{\binom{m}{n}}{n+m}[/tex]

However, I don't like that sum in the solution, I don't think I computed it right, and I want it gone. Is there any way to do that?
 
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  • #2
That doesn't look like the convolution I learned quite a few years ago, which was this:
[tex](f \star g)(t) = \int_{-\infty}^{\infty} f(\tau)g(t - \tau)d\tau[/tex]
 
  • #3
Hmm... I used the same formula, but the bounds I learned were from 0 to t. Using that, I came up with the OP formula for the convolution of a polynomial (really a monomial, now that I think about it, but I can't change the title) with itself.
 
  • #4
I got the formula I posted from Wikipedia, but it's the same formula that I remember from when I studied convolution way back when. If you can show us how you got your summation formula from the definition it would be helpful.
 
  • #5
Mark44 said:
I got the formula I posted from Wikipedia, but it's the same formula that I remember from when I studied convolution way back when. If you can show us how you got your summation formula from the definition it would be helpful.

It was just the pattern I noticed. For example...

For t:

[tex]t^3(\frac{1}{2}-\frac{1}{3})[/tex]

For t^2:

[tex]t^5(\frac{1}{3}-\frac{2}{4}+\frac{1}{5})[/tex]

For t^3:

[tex]t^7(\frac{1}{4}-\frac{3}{5}+\frac{3}{6}-\frac{1}{7})[/tex]

For t^4:

[tex]t^9(\frac{1}{5}-\frac{4}{6}+\frac{6}{7}-\frac{4}{8}+\frac{1}{9})[/tex]

The summation was my attempt at phrasing this pattern, which of course I forgot to put in a (-1)^(n+1). I just noticed that the denominators for t^m's convolution with itself were in consecutive integers from m+1 to 2m+1, and the numberators were each a row of Pascal's triangle, which if I remember right is also the binomial coefficients.
 

What is the definition of convolution of a polynomial with itself?

The convolution of a polynomial with itself is a mathematical operation that combines two functions to produce a third function. In this case, the two functions are the same polynomial, and the resulting function is the convolution of the polynomial with itself.

Why is the convolution of a polynomial with itself important in science?

The convolution of a polynomial with itself has many applications in science, particularly in signal processing and image processing. It allows us to analyze and manipulate signals and images in a more efficient and accurate way.

How is the convolution of a polynomial with itself calculated?

The convolution of a polynomial with itself is calculated by multiplying the coefficients of the two polynomials and then adding them together. This process is repeated for each term in the polynomials, and the resulting terms are combined to form the convolution polynomial.

What are some properties of the convolution of a polynomial with itself?

Some important properties of the convolution of a polynomial with itself include commutativity, associativity, and distributivity. These properties allow us to manipulate the convolution polynomial in various ways to solve problems and analyze data.

How is the convolution of a polynomial with itself used in real-world applications?

The convolution of a polynomial with itself is used in a variety of real-world applications, such as image and signal processing, data analysis, and machine learning. It allows us to extract important features from data and make predictions based on past patterns.

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