Outstanding text treats numerical analysis with mathematical rigor, but relatively few theorems and proofs. Oriented toward computer solutions of problems, it stresses errors in methods and computational efficiency. Problems — some strictly mathematical, others requiring a computer — appear at the end of each chapter.
It incorporates concepts from Functional Analysis well, in useful ways to describe approximation processes. The proofs are easy to follow if you don't mind dulling a pencil out about the details of the equations. It's great for engineering; it looks like it has good coverage of the Fourier Transform. Looking forward to reading the rest of it in closer detail.