When should a local series be preferred over a global series for approximating functions?

26 Eki 18:45

I’ve been reading about series expansions and noticed that many textbooks focus on global approaches like Taylor or Fourier series. However, in practice we often only need a good fit on a limited interval. How do local series methods work in general – e.g., using piecewise expansions or windowed transforms – and what are the main advantages or trade‑offs compared to global series? Are there typical scenarios where a local approach clearly outperforms a global one? Would love to hear your thoughts.