Here's my problem- I'm not an expert but have a fair grasp of his main work in solid state physics. Fourier analysis is fundamental (literally, on Day 1 of any intro course in solid state, you cover representations in k space) and he clearly understands it on a professional level.
Which leads me to conclude that he absolutely realizes this and is deliberately misleading his audience. It's not ignorance, it's dishonesty. I find this absolutely reprehensible.