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Rich Farmbrough (talk | contribs) m Import from Planet Math |
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==Definition== |
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The '''scale convolution''' of two functions <math>s(t)</math> and <math>r(t)</math>, also known as their '''logarithmic convolution''' is defined as the function<br> |
The '''scale convolution''' of two functions <math>s(t)</math> and <math>r(t)</math>, also known as their '''logarithmic convolution''' is defined as the function<br> |
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<math> s * |
:<math> s *_l r(t) = r *_l s(t) = \int_0^\infty s\left(\frac{t}{a}\right)r(a) \frac{da}{a}</math> |
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when this quantity exists. |
when this quantity exists. |
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The logarithmic convolution can be related to the ordinary convolution by changing the variable from <math>t</math> to <math>v = \log t</math>: |
The logarithmic convolution can be related to the ordinary convolution by changing the variable from <math>t</math> to <math>v = \log t</math>: |
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<math> s * |
: <math> s *_l r(t) = \int_0^\infty s\left(\frac{t}{a}\right)r(a) \frac{da}{a} = |
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\int_{-\infty}^\infty s(\frac{t}{e^u}) r(e^u) du </math> |
\int_{-\infty}^\infty s\left(\frac{t}{e^u}\right) r(e^u) du </math> |
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:<math> = \int_{-\infty}^\infty s(e^{\log t - u})r(e^u) du</math> |
:<math> = \int_{-\infty}^\infty s\left(e^{\log t - u}\right)r(e^u) du.</math> |
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Define <math>f(v) = s(e^v)</math> and <math>g(v) = r(e^v)</math> and let <math>v = \log t</math>, then |
Define <math>f(v) = s(e^v)</math> and <math>g(v) = r(e^v)</math> and let <math>v = \log t</math>, then |
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< |
:<math> s *_l r(v) = f * g(v) = g * f(v) = r *_l s(v).\, </math> |
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{{planetmath|id=5995|title=logarithmic convolution}} |
{{planetmath|id=5995|title=logarithmic convolution}} |
Revision as of 20:10, 12 August 2006
The scale convolution of two functions and , also known as their logarithmic convolution is defined as the function
when this quantity exists.
Results
The logarithmic convolution can be related to the ordinary convolution by changing the variable from to :
Define and and let , then