3 ms·
Sampling has nothing to do with this. The inner product of f (x) and g (x) simply is the integral with respect to x of their pointwise scalar products (I suppos
by bandrami 11y ago
Sampling has nothing to do with this. The inner product of f (x) and g (x) simply is the integral with respect to x of their pointwise scalar products (I suppose in a very broad sense "pointwise" could be seen as analogous to sampling, as could "with respect to x", but the Calculus Gods will smite any who think of dx as a sample of x). The "ah ha" moment was in seeing inner product as the fundamental operation and integration as derived from it.
- JadeNB 11y ago> I suppose in a very broad sense "pointwise" could be seen as analogous to sampling Indeed, I don't understand how it could be otherwise. To multiply functions pointwise, you need to know their values at points. It seems to me that 'sampling' is a very good word to describe the process of evaluating a function at a lot of points. > as could "with respect to x", but the Calculus Gods will smite any who think of dx as a sample of x Indeed not! It is the spacing between sample points. That is, the `dx` in an integral literally stands for the "ghost of [the] departed quantity" `x_{i + 1} - x_i` (and, in an infinitesimal approach to calculus, it doesn't just stand for but literally is such a difference).