How does distributional derivatives work in the context of linear mappings?

[Kalidor]

When do derivatives in the sense of distributions and classical derivative coincide?
Of course f needs to be differentiable. What else? Any reference?

 

[strangerep]

When do derivatives in the sense of distributions and classical derivative coincide?
Of course f needs to be differentiable. What else? Any reference?

If by "derivatives in the sense of distributions" you simply mean "derivative of
a distribution", then... well... the space of test functions is a subspace of the
space of distributions so the question becomes "is this distribution also a test
function, or not?"

Sorry if this answer sounds obscure, but I'm not really sure what you're asking.
You could maybe try the Wiki page on distributions:

http://en.wikipedia.org/wiki/Distributional_derivative

to get some more background which might help you rephrase your question
more clearly...

 

[Stevan White]

The distributional derivative can be defined in terms of a Fréchet derivative on general normed linear spaces.

For a mapping f from one normed linear space to another, the derivative T at an element x in the domain is the linear operator with the same domain and range that satisfies

lim_δ→0 sup_0<||h||<δ || Th - ( f(x+h) - f(x) ) || / ||h|| = 0

if the limit exists.

If the domain and range are real finite-dimensional vector spaces, this is exactly the Jacobian.

If the domain and range are both the real numbers, this is the familiar derivative of elementary calculus.

This is not the complex derivative though!

Hope that helps!

 

[Stevan White]

A different interpretation of your question:

For what distribution is the distributional derivative the same as the elementary derivative of a given function?

Observe that in the above formula for the distributional derivative, if the mapping f is linear, then T is the same as f. (The elementary calculus analog is: if the function is linear, f(t) = at for some number a, then the derivative is a.)

In particular, the derivative operator d/dt is a linear mapping defined on spaces of differentiable functions. So the distributional derivative of the derivative operator, acting on a function g, is the derivative of g. In this sense, the derivative operator is the only mapping with this property.

This is hard to grasp, I know. It took me a long time, and lots of work.

The main issues are:
1) the distributional derivative acts on operators on the function space.
2) there are an awful lot of operators on function spaces in common use, and the standard notations for them is very scattered. I mentioned differential operators, but there are also pointwise multiplication, convolution, inner product, integral operators (these are all linear ones!) and then you get into the nonlinear ones... such as multiplying a function by itself pointwise etc etc. It's hard ot devise a consistent notation.