Return to computing page for the first course APMA0330
Return to computing page for the second course APMA0340
Return to computing page for the fourth course APMA0360
Return to Mathematica tutorial for the first course APMA0330
Return to Mathematica tutorial for the second course APMA0340
Return to Mathematica tutorial for the fourth course APMA0360
Return to the main page for the course APMA0330
Return to the main page for the course APMA0340
Return to the main page for the course APMA0360
Introduction to Linear Algebra with Mathematica
Theorem 2:
If f, g ∈ 𝔏¹(ℝ), then their convolution f☆g is another function in 𝔏¹(ℝ), defined by
\[
\left( f \star g \right) (x) = \int_{-\infty}^{+\infty} f(x-y)\,g(y)\,{\text d} x = \left( g \star f \right) (x) .
\]
Their Fourier transform is the product of the Fourier transforms:
\[
ℱ_{x\to \xi}\left[ f \star g \right] = \hat{f}(\xi )\cdot \hat{g}(\xi ) .
\]
Theorem 6:
Example 7:
■
End of Example 7
Theorem 3:
Let f,g ∈ 𝔏¹(ℝ). For almost all x, f(x − y) g(y) is integrable (as a function of y) and if we denote by
\[
h(x) = \int_{\mathbb{R}} f(x-y)\,g(y)\,{\text d}y = f\ast g
\]
their convolution, then h ∈ 𝔏¹(ℝ) and
\[
\| h \| = \| f \ast g | \leqslant \| f \| \, \| g \| .
\]
Moreover,
\[
\hat{h}(\xi ) = 𝔉\left[ f \ast g \right] (\xi ) = \hat{f} (\xi ) \cdot \hat{g} (\xi ) \qquad \mbox{for all } \xi .
\]
It is not hard to show that convolution f ✶ g of two functions f and g is commutative, associatjve, and distributive.
Example 4:
■
End of Example 4
Theorem 4:
Let f,h ∈ 𝔏¹(ℝ) and
\[
h(x) = \mbox{V.P.} \,\frac{1}{2\pi} \int H(\xi )\, e^{{\bf j}\xi x} {\text d}\xi
\]
with integrable H(ξ). Then
\[
\left( h \ast f \right) (x) = \frac{1}{2\pi} \int H(\xi )\,\hat{f}(\xi ) \, e^{{\bf j}\xi x} {\text d}\xi .
\]
The product of two functions H(ξ) f(y is integrable in (ξ, y) ∈ ℝ×ℝ; hence, by Fubini's theorem,
\begin{align*}
\left( h \ast f \right) &= \int h(x-y)\, f(y)\,{\text d}y = \frac{1}{2\pi} \iint H(\xi )\, e^{{\bf j}\xi x} e^{-{\bf j}\xi x} f(y)\,{\text d}\xi{\text d}y
\\
&= \frac{1}{2\pi} \int H(\xi )\, e^{{\bf j}\xi x} \int e^{-{\bf j}\xi x} f(y)\,{\text d}y {\text d}\xi = \frac{1}{2\pi} \int H(\xi )\,\hat{f} (\xi )\, e^{{\bf j}\xi x} {\text d}\xi .
\end{align*}
Example 5:
■
End of Example 5
Return to Mathematica page
Return to the main page (APMA0340)
Return to the Part 1 Basic Concepts
Return to the Part 2 Fourier Series
Return to the Part 3 Integral Transformations
Return to the Part 4 Parabolic PDEs
Return to the Part 5 Hyperbolic PDEs
Return to the Part 6 Elliptic PDEs
Return to the Part 6P Potential Theory
Return to the Part 7 Numerical Methods