Function transforms (providing a broader picture of Laplace Transforms)
I’ll begin this article by brushing up a few definitions:
A function is a mapping between two sets: the domain D and the codomain C.
It’s very important to note here that the function is the mapping itself, and
not an element in the codomain or the domain. The function operates on an element
in the domain to give an element in the codomain. Formally, we would write this
function as
I’ll also brush up on what a Vector Space is: a vector space
- this is called Addition - this is called Scalar Multiplication.
There are some other requisites of a vector space, such as the existence of an additive inverse and identity elements, but I won’t get into the nitty-gritties of those. Just keep in mind that addition and scalar multiplication over a vector space are closed.
Function Spaces
Suppose for a function
Consider another function
Let’s try proving the same for scalar multiplication. For any
From the above two statements, we can see that the maps themselves are linear:
that is, they can be added and scalarly multiplied, with the resulting map
preserving the domain and codomain. We can thus say that the functions thus
defined form a vector space themselves, which we call a function space.
Note that there are a few more details involved, such as the existence
of an identity function and an additive inverse. Similar to the additive/multiplicative
proofs, these follow from the properties of the vector space. One interesting question to note
is what if there is no function in the space that maps an element of
Function transformations
If function spaces behave like vector spaces, this begs the question: what’s the
equivalent of linear transformations for function spaces? The equivalent is called
a function transformation, and it converts a function
Some examples of function transformations include
Integral transformations
Integral transformations are just one type of functional transform, but they’re the most used type of transform due to their usefulness. A general integral transform takes the form
- The integral transform exists only if the integral converges.
- All integral transforms are linear: this stems from the linearity property of the integral itself. Therefore, for a given function space, applying an integral transform on every member of that function space would also give us a function space.
The Laplace transform
We finally come to the Laplace transform: for the laplace transform, the kernel
is
Another bit of intuition regarding the laplace series comes from thinking about it as a continuous version of a power series. If we consider the power series
The continuous analogue of this power series is the integral
For this integral to converge,
Therefore, the laplace transform is like a continuous analogue of the power series.
Further reading
I haven’t covered the several properties of the laplace transform: those specific details can be found on the Wikipedia page. This was mainly to give an overview of the mathematical backbone that goes into transformations such as the laplace and fourier transforms.
For further reading on function spaces, see Vipul Naik’s notes and Terry Tao’s notes on the same. More on laplace transforms (including problems) are given in George Simmons’ Differential Equations with Applications and Historical Notes, chapter 9.
1No, it wouldn’t. This is analogous to having