Hyperinterpolation beyond exact cubature: a spectral multiplier approach

Abstract: We study hyperinterpolation and its spectral multiplier variants on the sphere under weak cubature assumptions formulated through Sobolev discrepancy estimates. In contrast with classical hyperinterpolation theory, our framework does not require exact polynomial cubature formulas or Marcinkiewicz--Zygmund inequalities. The main idea is to interpret the discretization error as the action of a spectral multiplier operator on the cubature discrepancy measure. This viewpoint separates approximation properties of the underlying spectral operator from geometric properties of the sampling measure, leading to stable Sobolev approximation estimates under weak cubature assumptions. The resulting theory applies to a broad class of spectral approximation operators, including sharp spectral projections, compactly supported smooth filters, Bessel potential operators, and heat kernel operators. For sufficiently localized spectral multipliers, we additionally obtain uniform $L<sup>\infty$-stability of the corresponding discrete approximation operators. The results establish a direct connection between hyperinterpolation, Sobolev discrepancy, and quasi-Monte Carlo (QMC) designs, showing that stable approximation from scattered data can be achieved without exact polynomial reproduction.

• The paper develops spectral multiplier hyperinterpolation on the sphere using only an H⁻ʳ discrepancy bound, replacing exact cubature and MZ inequalities with distributional error analysis.
• Balancing approximation and discretization errors gives the Sobolev rate δₘ^{r(s−q)/(s+r)}, including L² rate δₘ^{rs/(s+r)}, for sufficiently smooth targets.
• Localized filtered, Bessel, and heat multipliers provide uniform L∞ stability under quasi-uniform sampling, while sharp projections retain approximation guarantees but lack stable Lebesgue constants.

Overview and motivation

The paper develops a theory of hyperinterpolation on the unit sphere $\mathbb S^d$ that dispenses with the two pillars of classical analysis: exact polynomial cubature formulas (e.g., spherical $t$-designs) and Marcinkiewicz–Zygmund (MZ) inequalities. Instead, the author assumes only a weak cubature condition expressed as an $H^{-r}$ Sobolev discrepancy bound on the sampling measure, $\|\nu_m\|_{H^{-r}} \lesssim \delta_m^r$, where $\nu_m = \mu_m - \sigma$ is the difference between the discrete measure $\mu_m = \sum_j w_j \delta_{x_j}$ and normalized surface measure, and $\delta_m$ is the mesh norm. This condition is equivalent to controlling worst-case integration error over $H^r(\mathbb S^d)$ and is naturally satisfied by quasi-uniform sequences of QMC designs for $H^r$, which achieve $\|\nu_m\|_{H^{-r}} \le C m^{-r/d}$.

The unifying device is a family of spectral multiplier operators $t$0 with kernels $t$1, discretized by cubature to yield the spectral multiplier hyperinterpolation operator $t$2. The choice of multiplier recovers sharp projection hyperinterpolation ($t$3), filtered hyperinterpolation with compactly supported smooth filters, Bessel potential operators $t$4, and the heat semigroup $t$5.

Discretization error as distributional perturbation

The central structural observation is the identity

$t$6

where $t$7 is extended to distributions by duality against the kernel. Since $t$8 ensures $t$9, and since $H^{-r}$0 is a multiplier algebra on $H^{-r}$1 whenever $H^{-r}$2 (Lemma on Sobolev multiplication), the product $H^{-r}$3 is well defined. The analysis thus decomposes into three independent ingredients: approximation properties of the continuous operator, smoothing estimates of the multiplier, and discrepancy estimates of the sampling measure. No algebraic cancellation or discrete $H^{-r}$4-projection structure is invoked; all discretization error is controlled analytically through smoothing acting on the discrepancy distribution.

Main approximation result

Under assumptions (A1) (weak cubature), (A2) ($H^{-r}$5), and (A3) ($H^{-r}$6), the main theorem gives

$H^{-r}$7

for $H^{-r}$8 with $H^{-r}$9. Balancing the two terms yields the bandwidth choice $\|\nu_m\|_{H^{-r}} \lesssim \delta_m^r$0 and the balanced rate

$\|\nu_m\|_{H^{-r}} \lesssim \delta_m^r$1

with corresponding $\|\nu_m\|_{H^{-r}} \lesssim \delta_m^r$2 rate $\|\nu_m\|_{H^{-r}} \lesssim \delta_m^r$3 at $\|\nu_m\|_{H^{-r}} \lesssim \delta_m^r$4 and an $\|\nu_m\|_{H^{-r}} \lesssim \delta_m^r$5 bound of the same order when additionally $\|\nu_m\|_{H^{-r}} \lesssim \delta_m^r$6. The author explicitly notes that these rates differ from classical hyperinterpolation estimates precisely because exact polynomial reproduction is absent: the discretization error cannot be removed algebraically and must be absorbed through smoothing. In the equal-weight QMC setting, substituting $\|\nu_m\|_{H^{-r}} \lesssim \delta_m^r$7 gives rates in terms of $\|\nu_m\|_{H^{-r}} \lesssim \delta_m^r$8 alone, consistent with prior $\|\nu_m\|_{H^{-r}} \lesssim \delta_m^r$9 analysis of QMC-based hyperinterpolation.

Uniform $\nu_m = \mu_m - \sigma$0 stability via localization

For multipliers whose kernels satisfy a localization estimate $\nu_m = \mu_m - \sigma$1 with $\nu_m = \mu_m - \sigma$2, and under quasi-uniform sampling with weights $\nu_m = \mu_m - \sigma$3 and the bandlimiting constraint $\nu_m = \mu_m - \sigma$4, the paper proves uniform operator norm bounds:

$\nu_m = \mu_m - \sigma$5

with constants independent of both $\nu_m = \mu_m - \sigma$6 and $\nu_m = \mu_m - \sigma$7. The proof uses an annular decomposition of the sampling points around each evaluation point, together with quasi-uniform counting bounds. Notably, the balancing choice $\nu_m = \mu_m - \sigma$8 automatically satisfies the stability constraint since $\nu_m = \mu_m - \sigma$9. The result extends the Sloan–Womersley filtered hyperinterpolation stability theorem in two directions: no exact cubature is required, and the class of admissible multipliers is broader than compactly supported smooth filters. The remark that this mechanism fails for sharp projections—whose Lebesgue constants grow with $\mu_m = \sum_j w_j \delta_{x_j}$0 due to poor spatial localization—is a genuine limitation of the framework rather than a technical artifact.

Verification for specific multipliers

The paper verifies (A2), (A3), and localization for four classes:

Multiplier	Approximation order	Smoothing	Localization	Caveat
Sharp projection $\mu_m = \sum_j w_j \delta_{x_j}$1	$\mu_m = \sum_j w_j \delta_{x_j}$2	holds	none	Lebesgue constants grow
Filter $\mu_m = \sum_j w_j \delta_{x_j}$3, $\mu_m = \sum_j w_j \delta_{x_j}$4, $\mu_m = \sum_j w_j \delta_{x_j}$5	$\mu_m = \sum_j w_j \delta_{x_j}$6	holds	any $\mu_m = \sum_j w_j \delta_{x_j}$7	—
Bessel $\mu_m = \sum_j w_j \delta_{x_j}$8	$\mu_m = \sum_j w_j \delta_{x_j}$9, only if $\delta_m$0	requires $\delta_m$1	for large $\delta_m$2	saturation at $\delta_m$3
Heat kernel $\delta_m$4	$\delta_m$5, only if $\delta_m$6	holds	Gaussian, arbitrary $\delta_m$7	saturation at $\delta_m$8

Two observations deserve emphasis. First, the Bessel and heat multipliers exhibit saturation: because $\delta_m$9 and $H^r(\mathbb S^d)$0 near $H^r(\mathbb S^d)$1, increasing $H^r(\mathbb S^d)$2 improves smoothing and kernel decay but cannot push the consistency error beyond second order. The paper states plainly that stronger localization does not imply higher approximation order—the latter is governed solely by low-frequency behavior of the multiplier. Second, the sharp projection satisfies the full Sobolev theory but not uniform $H^r(\mathbb S^d)$3 stability, so the two desiderata (approximation under weak cubature and uniform stability) are achieved by different mechanisms and are simultaneously available only for localized multipliers.

Limitations and open questions

Several restrictions are intrinsic to the framework. The regularity requirement $H^r(\mathbb S^d)$4 couples smoothness of $H^r(\mathbb S^d)$5 to the discrepancy exponent $H^r(\mathbb S^d)$6, so rougher targets demand better cubature quality. The Bessel and heat examples are limited to $H^r(\mathbb S^d)$7 by their first-order behavior at the origin; higher-order analogues would require multipliers vanishing to higher order at $H^r(\mathbb S^d)$8. Uniform $H^r(\mathbb S^d)$9 stability requires quasi-uniformity, weight bounds $H^r$0, and the bandlimiting condition $H^r$1; whether stability persists for non-quasi-uniform or adaptively weighted designs is not addressed. Finally, the paper establishes rates but provides no numerical experiments, leaving open the practical constants and the empirical sharpness of the balanced rate $H^r$2 relative to MZ-based theories.

Conclusion

The paper reframes hyperinterpolation error analysis as the action of a spectral multiplier operator on the cubature discrepancy measure, yielding stable Sobolev approximation rates from scattered data under a weak, discrepancy-type cubature assumption alone. It unifies sharp, filtered, Bessel, and heat-kernel variants within one set of hypotheses, identifies spatial localization as the mechanism behind uniform $H^r$3 stability, and connects the resulting theory directly to spherical QMC design guarantees. The trade-offs it exposes—saturation of diffusive multipliers versus instability of sharp projections—delimit precisely which combinations of accuracy and stability are attainable without exact polynomial reproduction.