The arithmetic mean of $y_i \ldots y_n$ is: $$\frac{1}{n}\sum_{i=1}^n~y_i $$
For a smooth function $f(x)$, we can find the arithmetic mean of $f(x)$ from $x_0$ to $x_1$ by taking $n$ samples and using the above formula. As $n$ tends to infinity, it becomes an integration: $$\int_{x_0}^{x_1} f(x)~dx \over x_1 – x_0$$
On the other hand, the geometric mean of $y_i \ldots y_n$ is: $$\left( \prod_{i=1}^n~{y_i}\right)^{1/n}$$
Similarly, we can find the geometric mean of $f(x)$ by taking $n$ samples.
Here is my question: As $n$ tends to infinity, what do we call the resultant mathematical object? The geometric integration?
The geometric mean and the arithmetic mean, along with the quadratic mean (root mean square), the harmonic mean, etc, are special case of the generalized mean (with $p=0,1,2,-1$, respectively).
$$\left( \frac{1}{n}\sum_{i=1}^n~x_i^p\right)^{1/p}$$
Do we have a generalized integration for different values of $p$?
Best Answer
You don't need to introduce a new concept for this. The geometric mean of $y_i$ is nothing but $\exp$ of the arithmetic mean of $\log y_i$, and this generalizes in the straightforward way to integration: $$\exp\left(\frac{\int_{x_0}^{x_1} \log f(x)dx}{\int_{x_0}^{x_1} dx}\right).$$ You can do the same thing with the generalized mean, replacing $\log$ and $\exp$ with raising to the power of $p$ and $1/p$ respectively. I'm not aware of whether this has a specific name, but it is quite similar to the concept of the $L^p$ norm of a function.