Meir and Sharma (see SharmaMeir66) were the first to make use of the diagonal dominance of the standard equations, for the slopes or the second derivatives of the interpolating cubic spline at knots, to obtain bounds, on the first or the second derivative, of the error at knots, in terms of the max-norm of some derivative of the interpoland. %
Hall (see Hall68) refined this approach (without explicit credit to Meir and Sharma concerning the above-mentioned use of the standard equations and their diagonal dominance) by splitting the error into two parts, the error in local Hermite interpolation, and the difference, locally, between the local Hermite interpolant and the spline interpolant. %
Simultaneously, B. Swartz (see Swartz68) derived error bounds for spline interpolation, but I have to look up the paper; it may well not belong into this section. %
Hall and Meyer (see HallMeier76) improved upon this by showing that Hall's constants and/or those of Hall and Meyer are optimal, in the sense that they are taken on in the cardinal case. %
Howell and Varma (see HowellVarma89) apply the same idea to $C^2$-quartic interpolation in which, in addition to interpolation at knots (and complete end conditions), one also interpolates at the midpoint between knots. They also prove sharpness. %
Already SharmaMeir66 also applied their technique to $C^3$-quintic interpolation at knots, but failed to get sharp results (here or in the cubic case). For that, Hall's idea of bringing local Hermite interpolation to bear, together with the sharp error bounds for it from BirkhoffPriver67, is needed, as is the fact, first observed by HallMeyer76, that both the bound on the error-derivative at knots and the local Hermite error bound are extremized simultaneously by a function whose next derivative is absolutely constant and changes sign across breaks. In hindsight, this is obvious, here and in the cubic case since the tridiagonal matrix is nonnegative, hence it inverse is alternating, while the Peano kernel changes sign at the break interior to its support. This, by the way, is also the case in C^2-quartic interpolation, except that there the tridiagonal matrix has negative diagonal, and positive next-to-diagonal entries, hence its inverse is nonpositive, and, correspondingly, the Peano kernel for the rhs of error equation has no sign change. Such sharpness is certainly also attainable, for the same reason as in the cubic case, in $C^1$-quartic interpolation in which one matches, in each break interval, function values at three points interior to that interval (and matches value as end conditions). %
HallMeyer76 consider also $C^m$ (2m-1)st degree spline interpolation, in which one matches, at each breakpoint, $D^j g$, for $j=0,...,m-2$ (plus endpoint conditions) and state, without proof, a sharp error bound in this case, the latter obtained by computing the dominant error term in the cardinal case. %
4 jul 95 \bye