On the quadrature exactness in hyperinterpolation

Abstract

This paper investigates the role of quadrature exactness in the approximation scheme of hyperinterpolation. Constructing a hyperinterpolant of degree n requires an m-point positive-weight quadrature rule with exactness degree 2n. Aided by the Marcinkiewicz-Zygmund inequality, we affirm that when the required exactness degree 2n is relaxed to n + k with 0 < k ≤ n, the L 2 norm of the hyperinterpolation operator is bounded by a constant independent of n. The resulting scheme is convergent as n → ∞ if k is positively correlated to n. Thus, the family of candidate quadrature rules for constructing hyperinterpolants can be significantly enriched, and the number of quadrature points can be considerably reduced. As a potential cost, this relaxation may slow the convergence rate of hyperinterpolation in terms of the reduced degrees of quadrature exactness.

Is hyperinterpolation efficient in the approximation of singular and oscillatory functions?

“…The approximation of F = Kf ∈ C(Ω) by efficient hyperinterpolation is described by Theorem 5.2. Thus, if we let K = 1, then both the stability result (5.6) and the error bound (5.7) of efficient hyperinterpolation reduce to (2.4) and (2.5) of the classical hyperinterpolation, respectively, derived in [3]. Furthermore, if the quadrature rule (1.4) has exactness degree 2n, that is, η = 0, then they reduce to the original results (2.1) and (2.2) derived by Sloan in [31].…”

“…In a recent work [3], we discussed what if the required exactness 2n is relaxed to n + n ′ , where 0 < n ′ ≤ n. This discussion provides a regime where efficient hyperinterpolation may perform much more accurately than classical hyperinterpolation. In particular, if K is continuous, we show that for the classical hyperinterpolation of degree n, the approximation error is bounded as…”

“…As introduced in the previous section, the original definition (1.5) of hyperinterpolants of degree n requires an m-point quadrature rule (1.4) with polynomial exactness 2n [31], and this requirement on quadrature exactness has been relaxed to n + n ′ with 0 < n ′ ≤ n recently in [3].…”

“…and η = 0 if n ′ = n. The property (2.3) is referred to as the Marcinkiewicz-Zygmund property as it can be regarded as the Marcinkiewicz-Zygmund inequality [13,23,26,27] applied to polynomials of degree at most 2n; see [3] for more details. If the quadrature rule (1.4) with exactness degree n + n ′ satisfies the Marcinkiewicz-Zygmund property (2.3) with η ∈ [0, 1), then it was derived in [3] that…”

“…We then make a short discussion on the relations among L 2 orthogonal projection P n , hyperinterpolation L n , and efficient hyperinterpolation S n . Note that P n χ = χ for all χ ∈ P n , while for L n with quadrature exactness n + n ′ (0 < n ′ ≤ n), there only holds L n χ = χ for all χ ∈ P n ′ , see [3].…”