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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?

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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). 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.

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