       Convolution and Integration of Top Hat Functions

• To: mathgroup at smc.vnet.net
• Subject: [mg33196] Convolution and Integration of Top Hat Functions
• From: Roger Jones <rmj at SLAC.Stanford.EDU>
• Date: Thu, 7 Mar 2002 02:24:08 -0500 (EST)
• Sender: owner-wri-mathgroup at wolfram.com

```The first convolution of two Top Hat  functions is the triangle
function.  This does not come out explicitly in Mathematica 4.1 as can
be seen from the below example.  The first part of the IF statement is
indeed the correct solution but the second option to the IF is not
required.   Obviously this can be coded to force only the first part of
the IF condition to be displayed but I would have expected this to be
automatic.

I also, note that in Mathematica 3.0 this was not the case as only the
first part of the if conditional was given as the solution.

Thanks!

-Roger Jones

Clear[II];
(* Define "Top Hat" function *)
Clear[II];
II[xx_] = UnitStep[1\/2 + xx] - UnitStep[xx - 1\/2];

(* Convolve *)
f[yy_]=Integrate[II[xx]  II[yy-xx],{xx,-Infinity,Infinity}]

1                            1
If[- - yy > -Infinity && yy < -(-) &&
2                            2

1
yy < -, (-1 + yy) UnitStep[-1 + yy] -
2

2 yy UnitStep[yy] +

(1 + yy) UnitStep[1 + yy],

1
Integrate[(-UnitStep[-(-) + xx] +
2

1
UnitStep[- + xx])
2

1
(-UnitStep[-(-) - xx + yy] +
2

1
UnitStep[- - xx + yy]),
2

{xx, -Infinity, Infinity}]]

```

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