Assume the following representation for cubic splines with $T$ interior knots is given. Let$g(Y)=\sum_{j=0}^3 \alpha_j Y_j+\sum_{t=1}^T \gamma_t (Y-\zeta_t)_{+}^{3}$where $(Y-\zeta_t)_{+}:= max\{0,Y-\zeta_t\}$1) How can we show that the natural boundary conditions for natural cubic splines results in the following linear constraints on thecoefficients:$\alpha_2 = 0, \sum_{t=1}^T \gamma_t=0$$\alpha_3 = 0, \sum_{t=1}^T \zeta_t\gamma_t=0$2) How can we use result of part (1) to derive the basis$V_1(Y) = 1, V_2(Y) = Y, V_{t+2}(Y) = g_t(Y)  g_{t-1}(Y), t = \{1, ..., T  2\}$where$g_t(Y)=\frac{(Y-\zeta_t)_{+}^{3}-(Y-\zeta_T)_{+}^{3}}{\zeta_T-\zeta_t}$Thanks!
spline derivation
numerical analysis;data analysis;b spline