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The system $$\sum_{k=0}^n p_kx_i^k=P(x_i),$$ for $i=0,\cdots n$ is linear in the $a_k$ and its matrix is of the Vandermonde form. As is well known, the determinant of a Vandermonde matrix is $$\Delta=\prod_{0\le i<j\le n}(x_i-x_j),$$ which is nonzero when all $x$'s are different. Hence the solution of the system is unique.


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Clearly $s$ is continuous and linear for $x\in(x_i,x_{i+1})$. The slope of $s$ in the $i$'th subinterval is equal to the $i$'th coordinate of: $$\begin{pmatrix} -1&-1&-1&\ldots&-1\\ 1&-1&-1&\ldots&-1\\ 1&1&-1&\ldots&-1\\ \vdots&\vdots&\vdots&\vdots&\vdots&\\ 1&\ldots&1&1&-1\\ ...


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You are really constructing the function from 3 branches, the only interpolation in your example is done on $(a,b)$. As for the forms to use, the most basic is linear interpolation, which guarantees continuity but unfortunately does not make the result differentiable. To achieve differentiability (assuming $g'(a)$ and $h'(b)$ are well-defined) you can ...


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Given a power function $$ y=bx^m $$ graphed on a log-log scale, we have \begin{eqnarray} \log y&=&m\log x+\log b\\\ Y&=&mX+B \end{eqnarray} where $Y=\log y$, $X=\log x$, and $B=\log b$. So given two points $(x_0,y_0)$ and $(x_1,y_1)$ on a log-log graph we have $$ m=\frac{Y_1-Y_0}{X_1-X_0}=\frac{\log(y_1/y_0)}{\log(x_1/x_0)}\tag{1} $$ ...


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