In Shumaker's book (Spline Functions: Basic Theory), we know that the $l^\infty$-norm of B-spline coefficients is bounded above and below by the $L^\infty$-norm of the spline itself. Are there similar results about other norms? Such as $l^1$-norm of the coefficients? 



Norms of B-spline coefficients
This question is closely related to the so-called "condition number" of the B-spline basis. Basically, for a spline $f$ of some degree $p$ with a coefficient vector $c=(c\_i)$, you generally have for any $q \in [1,\infty]$ that
$$
 A\_{p,q} \|c\|\_{\ell\_q} \le \| f \|\_{L\_q} \le B\_{p,q} \|c\|\_{\ell\_q},
$$
and the smallest possible ratio $\kappa\_{p,q} = B\_{p,q} / A\_{p,q}$ is called the condition number of the basis.


Unfortunately, although this type of estimate seems to be "folklore" in the spline community, it's surprisingly hard to find a good reference for it. I will point you back to Schumaker's book, but to Theorem 9.27, equation (9.68), which states essentially this. The caveat is that this equation occurs in a chapter on Chebysheffian splines, and I don't yet fully understand if it fully applies to the standard B-spline situation. I have seen more knowledgeable people cite this equation for this purpose, though.


**Update:** I have since found a more palatable reference, namely De Boor, [Splines as linear combinations of B-splines. A Survey (1976)](http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.34.8204). Theorem 5.2 states what you need. I also forgot to mention above that, for $q \ne \infty$, the knot spacing enters the inequality, but with the same order for the lower and upper bound.


