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I am having a seemingly easy question as what follows. Suppose I am estimating a smooth function $f(x_i)$, with $i=1,...,n$ and $n$ is the sample size. Now, if I have the restriction on $f(\cdot)$ such that $f(x=0)=f(0)=0$ (i.e., passing through the origin), then a B-spline approximation of $f(x=0)$ is given by $f(0)\approx \phi(0)'\beta$, where $\phi(\cdot)=[\phi_1(\cdot),...,\phi_J(\cdot)]'$ and $J$ is the number of the basis function that goes to infinity when $n$ goes to infinity.

Question

Since $f(x=0)=f(0)=0$, then $f(0)\approx \phi(0)'\beta=0$, implying that $\phi(0)'=0$ for all J. However, the basis function of B-spline does not give zero when evaluating at zero, i.e., $\phi(0)\neq 0$. Is there any way I can center the basis function around zero such that $\phi(0)'=0$ for all J?

Thank you all in advance.

Rico Wang
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  • "implying that $\phi(0)'=0$ for all J" - if that is true, then $x=0$ must not be in the interior of the support of the B-Spline, because otherwise it's not a B-Spline. I don't think it's true though, because $f(0) = 0$ holds as long as the dot product between the basis functions and their coefficients is $0$ at $x=0$. – Coolwater Aug 05 '19 at 14:26
  • @Coolwater Thank you for the reply. If I understand your explanation correctly, then it is impossible to have $\phi(0)'\beta=0$, unless $x=0$ is not in the interior support of B-spline? – Rico Wang Aug 06 '19 at 01:51

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