Let $n \in \mathbb{N}$, for an integrable function $f$ we define $$S_{n}(f) = \sum_{j = 0}^{n} \sum_{k=0}^{2^{j}-1}<f,h_{j,k}>h_{j,k}.$$ Prove that if $f \in C([0,1])$, then $S_{n}(f)$ converges uniformly to $f$.
Could anyone give me a hint?
2026-03-27 00:03:01.1774569781
Haar functions question and sequence of partial sums.
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