An operation is defined on polynomials. How do I generalize it to other classes of functions?

Question

I am currently researching divergent integrals.

1. Definition. An extended number is an expression of the form $\int_a^b f(x)\,dx$, where $a,b\in \overline{\mathbb{R}}$ and function $f(x)$ is defined almost everywhere at $(a,b)$. Generally (when Riemann or Lebesgue sum converges or when the equivalence follows from the rules expressed below), an extended number can be equal to a real or complex number.

2. There are four simple equivalence rules based on linearity:

$$\int_a^c f(x) \,dx=\int_a^b f(x)\,dx+\int_b^c f(x)\,dx$$

$$\int_a^b (f(x)+g(x))\,dx=\int_a^b f(x)\,dx+\int_a^b g(x)\,dx$$

$$\int_a^b c f(x) \, dx =c \int_a^b f(x) \, dx$$

$$\int_{-\infty}^{-a} f(x) \, dx=\int_a^\infty f(-x) \, dx$$

3. There is one complicated Laplace-transform based rule:

$$\int_0^\infty f(x)\,dx=\int_0^\infty\mathcal{L}_t[t f(t)](x) \, dx=\int_0^\infty\frac1x\mathcal{L}^{-1}_t[ f(t)](x)\,dx$$

4. Divergent integrals, summable by Cesaro, Abel, Cauchy mean or other averaging technique are considered equal to their regularized values.

5. The changes-of-variables $\int_0^\infty f(a x)dx=\frac1a\int_0^\infty f(x)dx$ and $\int_a^\infty f(x)\,dx = \int_{a-b}^\infty f(x+b)\,dx $ are permitted only when $\lim_{x\to\infty}\overline{f(x)}=0$, where $\overline{f(x)}$ is the average value (by Cesaro, Abel, etc).

6. There is a rule that allows to represent divergent integrals of polynomials via the most basic divergent integral $\tau=\int_0^\infty dx$:

$$\int_0^\infty x^n \, dx=\frac{\left(\tau +\frac{1}{2}\right)^{n+2}-\left(\tau -\frac{1}{2}\right)^{n+2}}{(n+1)(n+2)}$$

7. Following Laplace transform, there is a similar rule (for $n>1$):

$$\int_0^\infty \frac1{x^n} \, dx = \frac1{(n-1)!}\int_0^\infty x^{n-2} \, dx = \frac{\left(\tau +\frac{1}{2}\right)^n-\left(\tau -\frac{1}{2}\right)^n}{(n-1)n!}$$

8. There is the opposite rule, converting in the opposite direction:

$$\tau^n=B_n(1/2)+n\int_0^\infty B_{n-1}(x+1/2)\,dx$$

---

That said, one can use these rules to multiply divergent integrals of polynomials.

Example.

\begin{align} & \int_0^\infty (2x^3-3x^2+x-4) \, dx \cdot \int_0^\infty (2x^2-3x+1) \, dx \\[8pt] = {} & \left(\frac{\tau ^4}{2}-\tau^3+\frac{3 \tau^2}{4}-\frac{17 \tau}{4}+\frac{23}{480} \right)\left(\frac{2 \tau^3}{3}-\frac{3 \tau ^2}{2} + \frac{7 \tau }{6}-\frac{1}{8}\right) \\[8pt] = {} & \frac{\tau^7}{3}-\frac{17 \tau^6}{12} + \frac{31 \tau^5}{12}-\frac{83 \tau^4}{16}+\frac{5333 \tau^3}{720}-\frac{4919 \tau^2}{960}+\frac{1691 \tau }{2880}-\frac{23}{3840} \\[8pt] = {} & \int_0^\infty \left(\frac{7 x^6}{3}-\frac{17 x^5}{2} + 10 x^4-\frac{41 x^3}{3}+\frac{1007x^2}{60}-\frac{63 x}{10}-\frac{113}{120}\right) \, dx+\frac{127}{420} \end{align}

---

Now, I have the following questions:

1. Can one obtain the formula for multiplication of integrals of polynomials in more simple or general form, for instance, involving matrices or binomial coefficients?

2. Can I in natural way generalize the formula to a larger class of the functions, particularly, rational functions?

Answer 1

Answering my own question. SEE THE UPDATE BELOW

The operator looks like this:

$\int_0^\infty f(x)dx\cdot\int_0^\infty g(x)dx=\int_0^\infty D^2 \Delta^{-1} \left(\Delta D^{-2}f(x)\cdot\Delta D^{-2}g(x)\right) dx +D \Delta^{-1} \left(\Delta D^{-2}f(x)\cdot\Delta D^{-2}g(x)\right)|_{x=0}$

Here $\Delta D^{-2}f(x)=\int_x^{x+1}\int_0^s f(t)\,dt\,ds$.

The formula is unambiguous. Here is the Mathematica code:

f[x_] := x
g[x_] := x

Func := D[
    D[Sum[(DifferenceDelta[Integrate[Integrate[f[t], {t, 0, x}], x], 
         x]*DifferenceDelta[Integrate[Integrate[g[t], {t, 0, x}], x], 
         x]), x], x], x]
Const := (D[
     Sum[(DifferenceDelta[Integrate[Integrate[f[t], {t, 0, x}], x], 
         x]*DifferenceDelta[Integrate[Integrate[g[t], {t, 0, x}], x], 
         x]), x], x] /. x -> 0) 
Inactivate[
 Integrate[f[x], {x, 0, Infinity}]\[CenterDot]Integrate[
   g[x], {x, 0, Infinity}], Integrate] == FullSimplify[
Const + Integrate[Func, {x, 0, Infinity}]] // Expand // Quiet // TraditionalForm

Out:=$\int _0^{\infty }xdx\cdot \int _0^{\infty }xdx=\int_0^{\infty } \left(x^3-\frac{x}{6}\right) \, dx+\frac{1}{180}$

The code works only with polynomials.

With other types of functions, it seems, it does not ($f(x)=\exp(-x), g(x)=\exp(x)$ gives a wrong constant term).

Thus, the question on how to generalize this still remains.

UPDATE At my current unde3rstanding, this definition of multiplication is not the natural one (does not coincide with analogues in Levi-Civita and Hardy fields). It is better termed "umbral".

A more simple and natural one would be given by the following code (providing two methods):

f[x_] := -4 - 3 x^2 + x + 2 x^3
g[x_] := 1 - 3 x + 2 x^2
prod1[x_] := 
 Evaluate[Refine[Integrate[f[x], x] Integrate[g[x], x], x > 0]]
prod2[x_] := 
 Evaluate[Refine[
   Integrate[f[x], {x, 0, x}] Integrate[g[x], {x, 0, x}], x > 0]]
prod2[x]
Inactivate[
    Integrate[f[x], {x, 0, Infinity}]\[CenterDot]Integrate[
      g[x], {x, 0, Infinity}], Integrate] == 
   FullSimplify[
     prod1[0] + 
      Distribute[
       Integrate[
        ExpandAll[FullSimplify[D[prod2[x], x]]], {x, 
         0, \[Infinity]}]]] - 
    Limit[Sum[D[prod2[s x], x], {x, 1, Infinity}, 
       Regularization -> "Dirichlet"] // FullSimplify, s -> 0] // 
  ExpandAll // Quiet
Inactivate[
  Reg[Integrate[f[x], {x, 0, Infinity}]\[CenterDot]Integrate[
     g[x], {x, 0, Infinity}]], Integrate] == FullSimplify[prod1[0]]

Or this one:

f[x_] := x
g[x_] := x
ni := 1/4 (LaplaceTransform[f[t], t, v] - 
       LaplaceTransform[f[-t], t, v]) (LaplaceTransform[g[t], t, v] - 
       LaplaceTransform[g[-t], t, v]) /. v -> 0  /. 
   ComplexInfinity -> 0 // Quiet
prod2[x_] := 
 Evaluate[Refine[
    Integrate[f[x], {x, 0, x}] Integrate[g[x], {x, 0, x}], x > 0] // 
   FullSimplify]
div := Distribute[
  Integrate[
   ExpandAll[FullSimplify[D[prod2[x], x]]], {x, 0, \[Infinity]}]] 
term1 := FullSimplify[
       Limit[(LaplaceTransform[Evaluate[D[prod2[t], t]], t, o] + 
           LaplaceTransform[Evaluate[D[prod2[t], t]], t, -o])/2, 
        o -> 0]] /. \[Infinity] -> 0 /. -\[Infinity] -> 0 /. 
    ComplexInfinity -> c /. Indeterminate -> c // Evaluate
term2 := Limit[
     Sum[D[prod2[s x], x], {x, 1, Infinity}, 
       Regularization -> "Dirichlet"] // FullSimplify, 
     s -> 0] /. \[Infinity] -> 0 /. -\[Infinity] -> 0 // Evaluate
Print["Method 1:"]
Inactivate[
    Integrate[f[x], {x, 0, Infinity}]\[CenterDot]Integrate[
      g[x], {x, 0, Infinity}], Integrate] == 
   FullSimplify[ni + div - term1] // ExpandAll // Quiet
Print["Method 2:"]
Inactivate[
    Integrate[f[x], {x, 0, Infinity}]\[CenterDot]Integrate[
      g[x], {x, 0, Infinity}], Integrate] == 
   FullSimplify[ni + div - term2] // ExpandAll // Quiet
Inactivate[
  Reg[Integrate[f[x], {x, 0, Infinity}]\[CenterDot]Integrate[
     g[x], {x, 0, Infinity}]], Integrate] == FullSimplify[ni]

According to this definition (based on multiplication of germs at infinity), the following identities hold:

$\int _0^{\infty }xdx\cdot \int _0^{\infty }xdx=\int_0^{\infty } x^3 \, dx,$

$\int _0^{\infty }1dx\cdot \int _0^{\infty }1dx=\int_0^{\infty } 2 x \, dx$

and

$\int _0^{\infty }\left(2 x^3-3 x^2+x-4\right)dx\cdot \int _0^{\infty }\left(2 x^2-3 x+1\right)dx=\int_0^{\infty } \frac{7 x^6}{3} \, dx-\int_0^{\infty } \frac{17}{2} x^5 \, dx+\int_0^{\infty } \frac{35 x^4}{3} \, dx-\int_0^{\infty } \frac{53}{3} x^3 \, dx+\int_0^{\infty } \frac{39 x^2}{2} \, dx-\int_0^{\infty } 8 x \, dx$

Nevertheless, umbral multiplication remains of interest as it provides non-trivial results.