. This proves the unit-integral normalization of a B-spline. Admissible repeated knots follow by coalescing knots, with the usual one-sided endpoint conventions.

Solution

 0  0

Write

N_{i, k}

for the order-

k

B-spline. At order one,

N_{i, 1} = 1_{[t_{i}, t_{i + 1})}

, and the finite sum is one on its basic knot interval. For larger order, the Cox-de Boor recurrence is

N_{i, k} (t) = \frac{t - t _{i}}{t _{i + k - 1} - t _{i}} N_{i, k - 1} (t) + \frac{t _{i + k} - t}{t _{i + k} - t _{i + 1}} N_{i + 1, k - 1} (t),

(1)

with a zero-denominator term defined as zero. Sum over

i = 1, \dots, n

. For each interior lower-order index

j = 2, \dots, n

, the combined coefficient is

\frac{t - t _{j}}{t _{j + k - 1} - t _{j}} + \frac{t _{j + k - 1} - t}{t _{j + k - 1} - t _{j}} = 1.

(2)

The two unmatched lower-order terms have supports

[t_{1}, t_{k}]

and

[t_{n + 1}, t_{n + k}]

, respectively, and vanish on the open basic interval

(t_{k}, t_{n + 1})

. Induction therefore gives

i = 1 \sum n N_{i, k} (t) = 1, t_{k} \leq t \leq t_{n + 1} .

(3)

For simple knots and

k \geq 2

, continuity extends the identity to both endpoints. In the order-one case and at admissible repeated endpoint knots, use the one-sided value from the basic interval. Extending the knot sequence in both directions also proves the subpartition of unity for B-splines,