Hi Philip -
I haven’t seen samples of gradients printed from FAA, but if you are working in Photoshop, it will help to make sure your gradients are dithered (this is an option in the gradient editor, and it’s not on by default; click the little square to turn it on). Dithering helps prevent banding in gradients.
Also, it looks like the colors you are using in this piece may be out of gamut, meaning those highly saturated bright magentas and blues are likely not available in the printed inks and thus the print will turn out somewhat darker and more muted color than you are seeing on the screen. When you are selecting colors in Photoshop, be sure not to select those that show an exclamation mark in the color picker. The exclamation mark indicates the color is out of gamut and will not print like you see it on the screen.