Considering an approximation $f$ for a set of $N$ data points $(x,y)$ using, for example, $M$ radial basis functions at arbitrary sites in the domain
$f_i = \sum_{j=1} ^M c_j\phi(||x_i-x_j||)$
where $\phi(||\cdot||)$ is a basis function, one would normally find the fitting coefficients $c$ by solving the matrix expression
$\mathbf{c} = \mathbf{\Phi}^{-1} \mathbf{y}$
where $\mathbf{y}$ is the column vector of responses from the data and $\mathbf{\Phi}$ is an interpolation matrix whose elements are given by $\Phi_{i,j} = \phi(||x_i-x_j||)$.
Is it possible to estimate the range of values for the fitting coefficients $c$ without calculating them as above? If so, can we say anything with certainty about those estimates, e.g., that they are upper/lower bounds on the values of $c$?