[pool noodle]
I want to glue the ends together to make a hoop. I don't know what kind
of plastic foam it is made of, but it is one of the closed cell slippery
plastic foams. I have not been successful finding an adhesive that
works. I expect that joining the ends will require a fusing or welding
operation.

What is the intended application?

Cut a deep 'V' in one end, and a matching wedge
in the other end, and then ream a hole in the
middle of each so you can insert a pipe-insulation
"dowel". Use a GOOD rubber cement all over
the dowel and the wedges. Then, if your application
permits, stitch through the whole thing with a
thick, soft cord, and whip the outside.

You can heat a nail to red-hot and draw it
across the seams to get a fairly strong weld,
but that will leave hard crunchy bits every
where that you melt the plastic.