The condition number (CN) of a matrix (in $L_2$ norm) is the square root of of the ratio of the largest eigenvalue to the smallest eigenvalue. We see here that CN is related to the extreme (largest and smallest) eigenvalues. It looks like we can vary the CN by varying the eigenvalues.
Convolution matrix is a Toeplitz matrix. There are some special Toeplitz matrices such as tri-diagonal and penta-diagonal matrices whose eigenvalues in terms of matrix entries are well-known [1-2]. In these case, by varying the matrix entries we can vary the eigenvalues and in turn the CN.
 S. Noschese, L. Pasquini, and L. Reichel, "Tridiagonal Toeplitz
matrices: properties and novel applications," Numer. Linear Algebra
Appl., 20 (2013), pp. 302-326
 G. D. Smith, Numerical Solution of Partial Diﬀerential Equations,
2nd ed., Clarendon Press, Oxford, 1978.