# Approximate a constant function with sequence of spline functions

Suppose that for a constant $c \in \mathbb{R}$ $$\sup_{t \in [a,b]}\Big|c- \sum_{l=1}^{m}a_{l}\ B_{l}(t;q)\Big|< \epsilon.$$

The $B_l$ form a B-spline basis of degree $q$ on the interval $[a,b]$ partitioned by $m+1-q$ (equidistant) knots. I am convinced that $|c-a_l|< M \epsilon$ for some constant $M$ which does not depend on the B-spline basis used here, that is if we increase $m$ the constant $M$ does not change.

The problem is I cannot prove it. Can someone help me with the above. If something is vague just ask.

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