The question asks to construct a piecewise quadratic polynomial defined on the interval $\mathbb{R}$ of the form $$B_0= \begin{cases} p(x)=x^2,\qquad\qquad\quad\; 0\leq x<1,\\ q(x)=ax^2+bx+c,\quad\, 1\leq x<2,\\ r(x)=(3-x)^2,\qquad\quad 2\leq x<3,\\ 0\qquad\qquad\qquad\qquad\quad\,\text{otherwise} \end{cases}$$

I need to find the coefficients for $q(x)$. However, I'm having trouble with figuring out which method to use. We're dealing with uniform quadratic B-splines. Would I need to use the Gram-Schmidt method?

Thanks.

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Hint B-splines have the property that they are continuous. Higher order splines have continuity of derivatives, how many depends on the order of the approximation. – Daryl Nov 5 '12 at 5:50
Do you mean I would have to derive q(x)? – Alti Nov 5 '12 at 5:55
Possibly, it depends on the way that they are defined. You need to have $p(1) = q(1)$ and $q(2)=r(2)$ for continuity. Do these uniquely describe $q(x)$ or is derivative information needed? – Daryl Nov 5 '12 at 5:58
I'm sorry, i'm not sure what you mean by your last statement. The only other information given is to determine a,b,c so that $B_0$ is in $C^1$(ℝ). – Alti Nov 5 '12 at 6:01
The two conditions that I give don't uniquely determine $q(x)$. You also require that $p'(1)=q'(1)$ and $q'(2)=r'(2)$. This is also needed for it to be in $C^1$. – Daryl Nov 5 '12 at 6:04

From my comments above, you require the spline to satisfy $p(1)=q(1)$, $q(2)=r(2)$ for function continuity and $p'(1)=q'(1)$ and $q'(2)=r'(2)$ for derivative continuity.