Difference between transforms and integrals/derivatives?

[onethatyawns]

I've looked around and am having a hard time finding common terms in definitions to know how they relate. Are all transforms based upon integrals/derivatives? Are integrals/derivatives a type of transform?

Are integrals/derivatives measurements of transforms? I believe that is correct, but I just want to make sure.

It also appears that there are multiple types of transforms, and some of them are called integral transforms. These are the Fourier, Laplace, Hilbert, etc transforms.

 

[Svein]

Are all transforms based upon integrals/derivatives?

No.

Are integrals/derivatives measurements of transforms?

I am not sure what you are asking about here. But - certain types of integrals and certain types of derivatives can be regarded as transforms.

It also appears that there are multiple types of transforms, and some of them are called integral transforms. These are the Fourier, Laplace, Hilbert, etc transforms.

Yes, but observe that these transforms are very special types of integrals. These integrals are of the type G(s)=\int K(x, s)f(x)dx, where K(x, s) is called a kernel and must observe certain conditions.

 

[onethatyawns]

I am not sure what you are asking about here. But - certain types of integrals and certain types of derivatives can be regarded as transforms.

Are there integrals or derivatives that would not be transforms? Or are all integrals and derivatives a type of transforms?

Because, on a basic level, transforms are things like shrinking/expanding an area, displacing a set of points, and rotating a set of points. It would then make sense to call a derivative a form of transform or measurement of a transform because it measures how fast these transforms are taking place between two variables. It would make sense to call an integral a transform because, as defined by a Riemann sum with infinitesimal intervals, we're talking about shrinking sections of areas and adding them up. Or, maybe shrinking is not the correct term. Maybe slicing is better.

Does that make any sense?

Yes, but observe that these transforms are very special types of integrals. These integrals are of the type G(s)=\int K(x, s)f(x)dx, where K(x, s) is called a kernel and must observe certain conditions.

Yes, I found that very interesting.

 

[Svein]

Are there integrals or derivatives that would not be transforms?

For example \int_{-1}^{1}xdx and \frac{d}{dx}2x. Both of these end up in a number, not another function.

 

[Stephen Tashi]

Both of these end up in a number, not another function.

But the result of \frac{d}{dx} (2x) can be viewed as the constant function c(x) = 2.