Hard m-term retention is optimal for an orthonormal coefficient basis

Since $U^{T}U=I_N$, $x=U^{T}c$. Orthogonality gives $\mathrm{\parallel }x-U^T{P}_{S}c{\mathrm{\parallel }}_2^2=\mathrm{\parallel }{U}^T(c-P_{S}c){\mathrm{\parallel }}_2^2=\mathrm{\parallel }c-P_{S}c{\mathrm{\parallel }}_2^2={\sum }_{i\mathrm{\notin }S}\mathrm{\mid }{c}_i{\mathrm{\mid }}^2\mathrm{.}$ This proves the exact discarded-tail identity.

Minimizing the tail is equivalent to maximizing ${\sum }_{i\in S}\mathrm{\mid }{c}_i{\mathrm{\mid }}^2$ among sets with at most $m$ indices. A maximizing set can include no zero coefficient while omitting a nonzero one; and if it includes $j$ while omitting $i$ with $\mathrm{\mid }{c}_i\mathrm{\mid }>\mathrm{\mid }{c}_j\mathrm{\mid }$, replacing $j$ by $i$ strictly increases that sum. Since the index set is finite, a maximizing set exists, and every set of $m$ largest absolute coefficients is maximizing; when fewer than $m$ coefficients are nonzero, adding zero coordinates changes no error. By [F2], $H_m$ is one such set, proving optimality including ties and the endpoints $m=0,N$.