Math 2.0

Integral transforms, operational calculus Mikusinski calculus and other operational calculi Proposition

Integration operator as convolution with the unit step

Builds on Convolution on the half-lineHeaviside step function

Statement

Let ff be a locally integrable function on [0,∞)[0, \infty). Let hh be the constant function h(t)=1h(t) = 1 on [0,∞)[0, \infty), the restriction of the Heaviside step function. Write f∗gf * g for the convolution on the half-line, (f∗g)(t)=∫0tf(τ) g(t−τ) dτ(f * g)(t) = \int_0^t f(\tau) \, g(t - \tau) \, d\tau. Then:

  1. (h∗f)(t)=∫0tf(τ) dτ(h * f)(t) = \int_0^t f(\tau) \, d\tau for all t≥0t \ge 0.
  2. For every integer n≥1n \ge 1, the nn-fold convolution power of hh is h∗n(t)=tn−1(n−1)!h^{*n}(t) = \dfrac{t^{n-1}}{(n-1)!}.
  3. For every integer n≥1n \ge 1, (h∗n∗f)(t)=∫0t(t−τ)n−1(n−1)!f(τ) dτ(h^{*n} * f)(t) = \displaystyle \int_0^t \frac{(t - \tau)^{n-1}}{(n-1)!} f(\tau) \, d\tau, and this is the nn-fold iterated integral of ff from 00 to tt.

Remarks

Clause 1 is the definition of convolution with h(t−τ)=1h(t - \tau) = 1 for 0≤τ≤t0 \le \tau \le t. Clause 2 follows by induction, since ∫0tτn−1/(n−1)! dτ=tn/n!\int_0^t \tau^{n-1}/(n-1)! \, d\tau = t^n / n!. Clause 3 follows from clauses 1 and 2 and the associativity of convolution; it is the Cauchy formula for repeated integration. For example, with f(t)=cos⁡tf(t) = \cos t, one gets (h∗f)(t)=sin⁡t(h * f)(t) = \sin t and (h∗h∗f)(t)=1−cos⁡t(h * h * f)(t) = 1 - \cos t. In the Mikusiński operational calculus this result lets one treat integration as multiplication by the element hh of a commutative ring.

Going deeper

For a student

This is the starting point of Mikusiński's calculus. The convolution ring of functions on [0,∞)[0, \infty) has no zero divisors by the Titchmarsh convolution theorem, so it has a field of fractions. In that field hh has an inverse ss, which plays the role of differentiation. Under the Laplace transform, hh corresponds to 1/s1/s, in line with the rule that the transform of ∫0tf\int_0^t f is F(s)/sF(s)/s.

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Where this leads

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Further reading

  1. Web https://en.wikipedia.org/wiki/Operational_calculus
About this page
TagBDBDJSP6DF
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Kindstatement, proposition
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MSC44A40, 44A35
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AboutMikusiński operational calculus, Convolution on the half-line
UsesConvolution on the half-line, Heaviside step function

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