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Edit: I have found what I needed in Schwartz's Mathematics for the Physical Sciences. Will type up a reply when I have time.

  • Bourbaki does not explain the justifications behind operational calculus
  • The quantum book below does not justify the LT of the dirac distribution, nor the unbounded exponentials.
  • Schwartz's original: Theorie des distributions has partial explanations.

I am looking for an extremely rigorous treatment of the laplace transform.

Stating the usual formulas for convenience:

$$ \mathcal{L}: ?\to?\quad\mathcal{L}(f)(s) = \int_{\mathbb{R}^n}f(x)\operatorname{exp}(-\langle s,x\rangle)dx $$

Or as a map on the space of tempered distributions (is it even well defined?),

$$ \mathcal{L}: ? \to ?\quad\langle \mathcal{L}(F), \phi\rangle_{(\mathcal{S}', \mathcal{S})} = \langle F, \mathcal{L}(\phi)\rangle_{(\mathcal{S}', \mathcal{S})} $$

Starting from a measure-theoretic, functional-analytic standpoint, I would like to know the following:

  • Can we restrict the LT to some kind of isomorphism? What are the convergence/continuity properties? If its domain/codomain differs from that of the Fourier Transform (which is a toplinear isomorphism on the space of tempered distributions) - what topology are we using?

  • If the LT is defined on all tempered distributions, can we extend the domain from $\mathcal{S}'$ to a bigger space? Is the LT injective/surjective?

  • Is there a duality like in the Fourier Transform, regularity as a distribution gets converted to integrability?

  • With regards to solving differential equations, how is the recovery of a regular solution possible?

  • Is viewing the LT from a purely algebraic perspective, the way researchers today understand the LT?

The following is a common scenario in the applied sciences that I would want the proof (in the positive or negative) of. Let $X$ and $Y$ be 'function spaces in a real variable' (left imprecise), and $H: X\to Y$ be a linear map. If $H$ is shift-invariant, meaning

$$ H(\tau_a f) = \tau_a H(f)\quad \tau_af(x) = f(x-a)\quad\forall a,x\in\mathbb{R} $$

then we can apply the Laplace Transform. Somehow in this definition we have left out the usual a.e identifications/regularity assumptions that were present if the domain of $H$ were to include the point mass at the origin $\delta_0$: a tempered distribution defined by the duality pairing:

$$ \langle \delta_0, f\rangle_{(\mathcal{S}', \mathcal{S})} = f(0) $$

for every $f\in \mathcal{S}$ in the Schwartz space. It is often claimed (but not proven), that every linear, shift invariant 'operator' $H$ admits a rational 'transfer function' in the 'Frequency domain'

$$ \mathcal{L}(h)\mathcal{L}(f) = \mathcal{L}(H(f)) $$

Setting where $h = H(\delta_0)$ is commonly called the 'impulse response' of the 'system' $H$.

It is also often claimed that the LT does not work for functions that are not of 'exponential order',

$$ E_{order} = \biggl\{f: \mathbb{R}\to\mathbb{R},\: \exists C,k\in\mathbb{R},\: \vert f(x)\vert\leq C\operatorname{exp}(kx)\: \forall x\in\mathbb{R} \biggr\} $$

but somehow is compatible with extreme irregularity like $\delta_0$ and linear combinations of its derivatives. Smooth functions like $f(x) = \operatorname{exp}(\vert x\vert^2)$ are locally integrable, thus defines a distribution.

It is well known the space of test functions (we refer to $C_c^\infty$ as test functions, and $C^\infty$ as smooth functions), is dense in $\mathcal{S}'$. It is bizarre to rule out the functions that are above exponential order, if we were to take the 'distributional' approach to the LT.

Moreover, the domain of the LT has to extend beyond the Schwartz functions, because we allow for the 'causal' (or even 'eternal' exponentials, so $g(x)=e^{kx}$) exponentials $f(x)=e^{kx}$ for $x\geq 0$ and $f(x)=0$ for $x<0$, where $k\in\mathbb{R}$ to be 'transformed' into reciprocals

$$ \mathcal{L}(f)(s) = \dfrac{1}{s-k}\forall \operatorname{Re}(s)>k $$

In sum:

  • The LT clearly converges for every test function,
  • The point mass $\delta_0$ is in the domain of the LT,
  • The domain of the LT has to extend beyond the Schwartz functions,
  • No additonal regularity is imposed, because we also allow for the unit step $u(x)=0$ for $x<0$ and $u(x)=1$ for $x\geq 0$ to be LT-able
  • but we somehow exclude a subset of $L_{loc}^1$ from the domain?

Next, to my knowledge there is no agreed upon definition for the mathematical object $\mathcal{L}(f)(s)$ in the last equation. Should we extend $\mathcal{L}(f)$ for $\operatorname{Re}(s)<k$ by zero? Then we can identify $\mathcal{L}(f)$ with a distribution.

I have looked at the following posts but the answers (and the texts provided) are not satisfying.

along with countless (with respect) pseudo-explanations such as

  • the LT is 'just like' the FT, (no proof, no concrete definitions)
  • the LT converts differentiation, integration into pointwise multiplication (ok, but what function space? a.e class? duality pairing identification? what about $\delta_0$? weak derivatives?, we cannot integrate distributions and how about inversion?) the fourier transform does the same.

Some more oddities:

On a related note, what is the unilateral/one-sided/causal Laplace Transform? What is the rigour behind it?

Thank you for reading this long post.

Qqqq123123
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    The exponential order bit is obvious from the definition of $\mathcal L$. Multiplying such a function by $\exp(-st)$ for our choice of $s$ still cannot ensure even the limit of the integrand goes to zero as $t \to \infty$! – Snared Sep 17 '23 at 21:16
  • The point being, if we rule out the functions that are 'above' exponential order, such as $e^{|x|^2}$, which are well defined as locally-integrable functions; thus defines a distribution - we might have to rule out the possibility of defining the laplace transform of the point mass. Since $L_{loc}^1\subseteq \mathcal{D}'$, the space of distributions. – Qqqq123123 Sep 17 '23 at 21:22
  • Can always define a similar operator but with multiplication of $\exp^{\circ k}(\cdot)$ for some of the larger functions. – Snared Sep 17 '23 at 21:58
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    Fourier transform is defined on space of characters of locally compact abelian group, and Laplace is defined on space of characters of locally compact abelian monoid. Check Bourbaki, "Integration II", chapter IX. – dsh Sep 17 '23 at 22:14
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    The book by Laurent Schwartz on distributions has a chapter on LT but it's in French. The book by Barry Simon on $P(\Phi)_2$ quantum field theory has a quick intro to LT within the theory of distributions. – Abdelmalek Abdesselam Sep 18 '23 at 00:37

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