@casperyc Okay, I understand your question now. No, there is no predefined way to do that. It seems very complicated, akin to the identify command.

@casperyc Sorry, I don't understand that any better than the original. Can you give an actual example rather than an explanation?

@Mac Dude Preben's solution does not strictly require eval but it does require evaluation before passing to inttrans[fourier]. This is because assuming does not work with variables in functions. So, using evaluation, this works:

i[beam]:= phi-> exp(-(phi-phi0)^2/2/sigma^2):
ex:= i[beam](phi):
i[beamspec]:= inttrans[fourier](ex, phi, omega) assuming sigma>0;

Please see my Answer below, which you seem to have overlooked.

@Kitonum The animation quality can be improved dramatically by including spacestep= .02 in the call to pdsolve. Note that setting higher values of numpoints does not change the discretization error. Using spacestep=.02 gives 200 points on the interval x= 0..4.

@Markiyan Hirnyk Axel's code works for me. I get the same numeric answer in 10 seconds total (mostly for the integration). I am one of the upvoters.

@Carl Love Using Preben's code, I now understand why f(1) is -x (actually x-> -x) and f(x) is -1. What was confusing me was that I thought that P(x) was always 1+x+x^2.

@Alejandro Jakubi While we cannot construct a patterned matrix of unspecified (abstract) size in Maple, all the OP wants is a procedure that constructs it for specified sizes. This is quite doable in Maple.

@xcyborg You can change the options on this plot, of course. I just made some stuff up.

plot(
     [zip(`[]`, W, M), zip(`[]`, W, B)],
     style= point, symbol= [diamond, box],
     symbolsize= 15, color= [brown, green]
);

@mahmood180 

Replace the G2 loops by the single statement:

G2:= Matrix(r, (i,j)-> `if`(i+j <= r, z[i+j+r+1], 0));

When you use a[k]^`*`, do you mean the complex conjugate of a[k]? Or is it just a notation to indicate something different from a[k]?

You say that the expected result of f(1) is -x and that the expected result of f(x) is -1. Neither of those make sense to me. Could you explain why you expect those results? Then maybe I can help you with the rest. Writing a function within a function is easy.

I just need a mathematical explanation; no knowledge of or use of Maple is needed.

@ecterrab Can you provide an explicit example showing the difference between PDEtools:-Solve and solve to which you are referring? I get identical RootOf results from both PDEtools:-Solve(Theta=F(s), s) and solve(Theta=F(s), s).

@mahmood180

You need to correct the Error, Vector index out of range. Vector z has 2*r+1 elements, but the code z[i+j+r+1] is trying to access z[2*r+2] (because i = r-1 and j = 2).

@Markiyan Hirnyk Your Maple output from the latex command is different from mine. Mine does not have any internal square brackets, and mine uses underscores to indicate subscripting.

Please give a more-detailed example.

@casperyc add is the correct thing to use; sum is incorrect, or at least suspicious. Quoting from ?sum :

To add a finite sequence of values, rather than compute a formula, use the add command.  For example, add(k, k=0..9) returns 45.  Although the sum command can often be used to compute explicit sums, it is strongly recommended that the add command be used in programs if an explicit sum is needed, in particular, when summing over all elements of a list, Array, Matrix, or similar data structure.