Re: [Phys-l] Convolution (was Sun's image ...)

[Ken Caviness <caviness@southern.edu>]

Brian or anyone who wants to chip in,

Am I misunderstanding something here?  It seems to me that you are not talking about convolution at all, just ordinary function multiplication, showcasing integer multiplication as a special case of polynomial multiplication (with x=10) and then putting in the carried digits (the sense of this also escapes me).

Can you tie this in for me with the only way I've used the idea of convolution, namely in regard to Fourier and Laplace transforms?  For example, http://mathworld.wolfram.com/ConvolutionTheorem.html reminds us that F(f*g) = F(f) F(g), where ordinary multiplication of the Fourier transforms of the functions leads to the same result as the transform of the convolution of the original functions:  '*' represents function convolution, defined as

	f * g = Integral[g(t')f(t-t')dt',{t,-Infinity,Infinity}].

Feeling lost,

Ken

Ken Caviness
Physics
Southern Adventist University

> -----Original Message-----
> From: phys-l-bounces@carnot.physics.buffalo.edu [mailto:phys-l-
> bounces@carnot.physics.buffalo.edu] On Behalf Of Brian Whatcott
> Sent: Friday, May 01, 2009 1:18 PM
> To: Forum for Physics Educators
> Subject: Re: [Phys-l] Convolution (was Sun's image ...)
>
> Now hang on a darn' minute!
>  Every single one of your students has been convolving three and four
> term
> polynomials since third grade, and then some!
>
> Look at this polynomial a(x) = x^3 + 2x^2 + 3x + 4
> and    b(x) = x^3 + 3x^2 + x + 3
>
> The convolution is 1   5  10  18  21  13  12
>
>
> Now if I mention that your students typically use x = 10
> And they express  two polynomials such as....
> a(x) = 10^3 + 2X 10^2 + 3X 10 + 4             as 1234
> and
> b(x) as 10^3 + 3 X 10^2 +  10 + 3 X 10^0     as 1313
>
>
> They convolve these polynomials.
>
> In other words, they not only derive the convolution of these two
> polynomials, but they THEN
> carry  values greater than the base to the next column:
> so  (from the lowest power)
> 12                 = 2                 carry 1
> 13 plus 1 = 14 = 4                 carry 1
> 21 plus 1 = 22 = 2                carry 2
> 18 plus 2 = 20 = 0               carry 2
> 10 plus 2 = 12 = 2                carry 1
> 5 plus 1          = 6
>                         1
>
> They then write
>  1 6 2 0 2 4 2 =  1234 X 1313
>
> :-)
>
> Who knew?
>
> Brian W
>
>
> Anthony Lapinski wrote:
> > Interesting. Thanks! I know there are numerous applications, but this
> is
> > too advanced for my students in trying to explain the solar image.
> >
> >
> > > Take  two polynomial equations, say
> > > x^3 +2x^2 + 3x + 4   and  x^3 + 4x^2 + 9x + 16
> > >
> > > Just as we can multiply two scalars 3  X 4 = 12,
> > > we can multiply these two polynomial functions in a similar way.
>
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