Re: Function-type arguments in function definition

*To*: mathgroup at smc.vnet.net*Subject*: [mg44351] Re: Function-type arguments in function definition*From*: Paul Abbott <paul at physics.uwa.edu.au>*Date*: Wed, 5 Nov 2003 10:02:20 -0500 (EST)*Organization*: The University of Western Australia*References*: <bo7o3k$aep$1@smc.vnet.net>*Sender*: owner-wri-mathgroup at wolfram.com

In article <bo7o3k$aep$1 at smc.vnet.net>, "Carsten Reckord" <news at reckord.de> wrote: > I'm pretty new to Mathematica so please excuse me if this is kind of a silly > question (though I couldn't find any answer after a full day of searching). > I'm trying to define functions that take other functions as arguments and > need those functions' arguments in their own definition. An example would be > the definition of convolution: > > h(x)=f(x)*g(x) is defined as Integral over f(y)g(x-y) with respect to y. > > As you can see it is important in the definition of convolution to treat the > arguments f and g as functions because the definition makes use of their > arguments. > I've seen this done in Mathematica as > > convolute[f_,g_,x_]:=Integrate[f[y]*g[x-y],{y,-inf,inf}] > > but that's not exactly what I'm looking for because I can only use function > names as arguments to convolute[...], not arbitrary expressions in x. So I > can't for example use it for the convolution f(s(x))*g(t(x)) without > defining intermediate functions for f(s(x)) and g(t(x))... > > So, my question is if there is any way to define such a function that can > make use of its arguments being functions and yet supports arbitrary > expressions as its arguments? I would do this as follows. Define your convolution operation for pure functions f and g with variable x as Star[f_Function, g_Function][x_] := Integrate[f[y] g[x - y], {y, -Infinity, Infinity}] If you now add a rule that converts f and g to pure functions (assumed to be functions of x), Star[f_,g_][x_] := Star[Function[x,f], Function[x,g]][x] then you can compute convolutions of the type you require as Star[f[s[x]], g[t[x]]][x] (An aside: Mathematica already has an infix notation suitable for convolution -- the Star operator. Above I've used the InputForm, Star, which adds rules to Star, so that you can enter input expressions as (f[s[x]] \[Star] g[t[x]])[x] where \[Star] is converted to the appropriate special character on input.) Cheers, Paul -- Paul Abbott Phone: +61 8 9380 2734 School of Physics, M013 Fax: +61 8 9380 1014 The University of Western Australia (CRICOS Provider No 00126G) 35 Stirling Highway Crawley WA 6009 mailto:paul at physics.uwa.edu.au AUSTRALIA http://physics.uwa.edu.au/~paul