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A spline knot sequence is a nondecreasing list of spline knots, including their multiplicities, that specifies the junctions and smoothness constraints of a spline space. An order-$k$ B-spline uses $k+1$ consecutive knot entries. Its support must have positive width, and repeated knots are interpreted by confluent divided differences or their limiting convention.

A spline is a piecewise polynomial function whose polynomial pieces obey specified smoothness conditions at their joins, called spline knots. For example, a cubic spline is cubic on each interval and commonly has two continuous derivatives across interior knots. Spline approximation and spline interpolation use these finite-dimensional function spaces to represent data or approximate functions.

For strictly increasing spline knots, $M_i(t)=k[t_i,\dots ,t_{i+k}](\cdot -t{)}_{+}^{k-1}$ is a nonnegative integral-normalized B-spline. Integrating the truncated power function over its support replaces its spline knot value by $(u-t_i{)}^k/k$. The order-$k$ divided difference of this degree-$k$ polynomial is its leading coefficient $1/k$. Multiplying by the prefactor $k$ in $M_i$ gives integral one. The corresponding partition-normalized B-spline is $N_i=(t_{i+k}-t_i)M_i/k$.