How do you blur out one end of an object more than the other end?

[Delia5E4B]

I am busy creating shadows for a 3d glass box that I am working on. I want to create the shadows that the pillars make on the ground however, I seem to be having trouble trying to figure out how to blur one end of the shadow more than the other end. My shadow is a vector shape that I have drawn with the pen tool. I do not want to blur the whole shape evenly, I want to blur one end more than the other end. Kindly see my screenshot below. Any help would be appreciated.

 

[Monika Gause]

Draw the shadow with a freeform gradient or a gradient mesh.

Or try applying an envelope after applying the blur.

[Anna Lander]

What about to draw the forms in Illustrator and then copy/paste them and blur in Potoshop? There you will have smart filters or Blur Gallery which allow you to control the blurriness much more free and precise.