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The integral form of the Taylor theorem about the left endpoint iswhere $q_{k-1}\in {\mathcal{P}}_{k-1}$. Apply the order-$k$ divided difference at $x_i,\dots ,x_{i+k}$. The polynomial term vanishes, and linearity permits interchange with the integral:By the definition $M_i(t)=k[x_i,\dots ,x_{i+k}](\cdot -t{)}_{+}^{k-1}$,This is the Peano kernel theorem for the divided-difference functional.

The explicit divided difference formula makes $M_i(t)$ a finite linear combination of $(t_j-t{)}_{+}^{k-1}$. It is therefore polynomial of degree at most $k-1$ between knots and globally $C^{k-2}$. For $t<t_i$, the nodal data come from a polynomial of degree $k-1$, whose order-$k$ divided difference vanishes; for $t\ge t_{i+k}$ all truncated powers vanish. Thusand normalization does not change the degree, smoothness, knots, or support.

For distinct increasing knots and $k\ge 2$, the explicit divided difference formula writes an order-$k$ B-spline as a finite sum of truncated power functions of degree $k-1$. It is a piecewise polynomial function and is globally $C^{k-2}$. Below its first knot the order-$k$ divided difference annihilates a degree-$k-1$ polynomial; above its last knot all values vanish. Nonzero first and last pieces give the exact closed support. Order one gives interval indicators instead of continuous splines.

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.

An order-$k$ B-spline is supported on $[t_i,t_{i+k}]$. Below this interval its truncated-power knot data agree with a degree-$k-1$ polynomial, annihilated by the order-$k$ divided difference; above it all data vanish. Therefore its ordinary or mixed Gram matrix has entries zero when $|i-j|\ge k$. The half-bandwidth is $k-1$, and for strictly increasing knots this bound is sharp.

For integer $r\ge 1$, the truncated power is zero for $x\le t$ and equals $(x-t{)}^r$ for $x>t$. It is globally $C^{r-1}$ and its $r$th derivative has a jump at $t$. Order zero is the step-function convention $1_{\{x>t\}}$. Knot divided differences of these functions produce B-splines.