P_{x} = T A_{x}^{- 1} R

, where

R

samples at the interpolation sites and

T

synthesizes the normalized B-spline expansion. Sampling has operator norm at most one, and nonnegative B-splines with sum at most one give

∥ T ∥ \leq 1

. Hence

∥ P_{x} g ∥_{\infty} \leq ∥ A_{x}^{- 1} R g ∥_{ℓ^{\infty}} \leq ∥ A_{x}^{- 1} ∥_{ℓ^{\infty}} ∥ g ∥_{\infty}, ∥ P_{x} ∥ \leq ∥ A_{x}^{- 1} ∥_{ℓ^{\infty}} .

(1)

Here the matrix norm is the maximum absolute row sum. Existence of the inverse matrix requires the usual ordered distinct interpolation sites and the Schoenberg–Whitney theorem support conditions, with endpoint values interpreted by their one-sided limits. If the displayed diagonal condition is read without the implicit ordering and distinctness, it is insufficient: all three sites equal to

1/2

in the quadratic example give positive diagonal entries but three identical rows. The estimate applies whenever the interpolating linear map in the question is defined uniquely.

For the quadratic Bernstein basis,

N_{1} = (1 - x)^{2}

N_{2} = 2 x (1 - x)

and

N_{3} = x^{2}

. Sampling and inverting give

A = 1 1/4 0 0 1/2 0 0 1/4 1, A^{- 1} = 1 - 1/2 0 020 0 - 1/2 1, ∥ A^{- 1} ∥_{ℓ^{\infty}} = 3.

(2)

The three cardinal Lagrange interpolation polynomials are

ℓ_{0} (x) = 2 x^{2} - 3 x + 1, ℓ_{1} (x) = 4 x (1 - x), ℓ_{2} (x) = 2 x^{2} - x .

(3)

The Lebesgue constant of interpolation gives the exact operator norm: the upper bound follows from

∣ P g (x) ∣ \leq ∥ g ∥_{\infty} \sum_{i} ∣ ℓ_{i} (x) ∣

. To attain it at a maximizing point, prescribe at the three sites the signs of the corresponding cardinal polynomials and extend those values by a continuous piecewise linear function of norm one.