engleski

Algebraic Proof of the B-spline Derivative Formula

We prove a well known formula for the generalized derivatives of Chebyshev B--splines: \begin{; ; ; eqnarray*}; ; ; L_1B_i^k(x) & = & \frac{; ; ; B_i^{; ; ; k-1}; ; ; (x)}; ; ; {; ; ; C_{; ; ; k-1}; ; ; (i)}; ; ; - \frac{; ; ; B_{; ; ; i+1}; ; ; ^{; ; ; k-1}; ; ; (x)}; ; ; {; ; ; C_{; ; ; k-1}; ; ; (i+1)}; ; ; , \end{; ; ; eqnarray*}; ; ; where \begin{; ; ; eqnarray}; ; ; C_{; ; ; k-1}; ; ; (i) & = & \int_{; ; ; t_{; ; ; i}; ; ; }; ; ; ^{; ; ; t_{; ; ; i+k-1}; ; ; }; ; ; B_i^{; ; ; k-1}; ; ; (x) d\sigma, \end{; ; ; eqnarray}; ; ; in a purely algebraic fashion, and thus show that it holds for the most general spaces of splines. The integration is performed with respect to a certain measure associated in a natural way to the underlying Chebyshev system of functions. Next, we discuss the implications of the formula for some special spline spaces, with an emphasis on those that are not ssociated with ECC-systems.

Chebyshev splines; Divided differences

nije evidentirano