9.5 Discrete expansions and interpolation

Suppose one approximates a function f through its discrete truncated expansion,
∑N f(x) ≈ 𝒫dN[f](x) = ˆfdj pj(x). (9.76 ) j=0
That is, instead of considering the exact projection coefficients ˆ {fj}, these are approximated by discretizing the corresponding integrals using Gauss, Gauss–Lobatto or Gauss–Radau quadratures,
∫ b N fˆ = ⟨p ,f⟩ = f(x )p (x)ω (x)dx ≈ fˆd:= ∑ f(x )p (x )A (9.77 ) j j ω j j i j i i a i=0
with Ai given by Eq. (9.65View Equation) and {xi} any of the Gauss-type points. Putting the pieces together,
N∑ ( ∑N ) 𝒫d [f](x ) = f (xi) Ai pj(xi)pj(x ) . (9.78 ) N i=0 j=0
Since if f is a polynomial of degree smaller than or equal to N, the discrete truncated expansion 𝒫dN[f] = f is exact for Gauss or Gauss–Radau quadratures according to the results in Section 9.4, it follows that the above term inside the square parenthesis is exactly the i-th Lagrange interpolating polynomial,
∑N Ai pj(xi)pj(x) = ℓN(x ). (9.79 ) j=0 i
Therefore, we arrive at the remarkable result:

Theorem 22. Let ω be a weight function on the interval (a,b), as introduced in Eq. (9.17View Equation), and denote by 𝒫dN[f] the discrete truncated expansion of f corresponding to Gauss, Gauss–Lobatto or Gauss–Radau quadratures. Then,

N∑ 𝒫d [f](x ) = ℐ[f](x ) = f (x )ℓN (x). (9.80 ) N i i i=0
That is, the discrete truncated expansion in orthogonal polynomials of f is exactly equivalent to the interpolation of f at the Gauss, Gauss–Lobatto or Gauss–Radau points.

The above simple proof did not assume any special properties of the polynomial basis, but does not hold for the Gauss–Lobatto case (for which the associated quadrature is exact for polynomials of degree 2N − 1). However, the result still holds (at least for Jacobi polynomials); see, for example, [237Jump To The Next Citation Point].

Examples of Gauss-type nodal points {xi} are those given in Eq. (9.68View Equation) or Eq. (9.70View Equation). As we will see below, the identity (9.80View Equation) is very useful for spectral differentiation and collocation methods, among other things, since one can equivalently operate with the interpolant, which only requires knowledge of the function at the nodes.


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