L_{n} : C [0, 1] \to C [0, 1]

converge uniformly to the identity on every continuous function if they do so on the three test functions

1, t, t^{2}

b

Solution

Let

e_{m}

denote the

m

t_{i + 1}, \dots, t_{i + k - 1}

. Expanding both sides of the Marsden identity in powers of

x

and equating the coefficient of

x^{k - 1 - m}

gives

a_{m, i} = \frac{e _{m} ( t _{i + 1} , \dots , t _{i + k - 1} )}{( m k - 1 )}, 0 \leq m \leq k - 1.

Thus

a_{0, i} = 1

, while

a_{1, i}

and

a_{2, i}

are respectively the means of the interior knots and of their pairwise products.

V_{n} 1 = 1

. If

ξ_{i} = a_{1, i}

, then both

τ_{i}

and

ξ_{i}

lie in

[t_{i}, t_{i + k}]

, so

∥ V_{n} t - t ∥_{\infty} \leq k ∣ Δ_{n} ∣ \to 0.

Because

a_{2, i}

averages products of knots in the same interval and all points lie in

[0, 1]

∥ V_{n} t^{2} - t^{2} ∥_{\infty} \leq 2 k ∣ Δ_{n} ∣ \to 0.

∥ V_{n} (f) - f ∥_{C [0, 1]} \to 0

for every

f \in C [0, 1]

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