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The Lebesgue constant for uniform approximation of differential forms

Ludovico Bruni Bruno, Fwderico Piazzon

TL;DR

The paper develops a unified framework for uniform approximation of differential forms from weak, current-based data by projecting onto finite-dimensional form spaces via interpolation or least-squares fitting. It proves that the operator norm of the projection equals the Lebesgue constant under mild disjointness assumptions on sampling supports and extends this equivalence to countable sampling sets, utilizing reproducing kernels and Riesz representers. It further analyzes how the Lebesgue constant and the projection behave under domain mappings, providing explicit bounds under diffeomorphisms (with a sharp affine-case factor) and establishing domain-independence results in certain cases. These results bridge geometric measure theory with approximation theory and finite element exterior calculus, offering stability insights and practical guidance for geometry-preserving discretizations of differential forms.

Abstract

In this work we address the problem of uniform approximation of differential forms starting from weak data defined by integration on rectifiable sets. We study approximation schemes defined by the projection operator L given by either generalized weighted least squares or interpolation. We show that, under a natural measure theoretic condition, the norm of such operator equals the Lebesgue constant of the problem. We finally estimate how the Lebesgue constant varies under the action of smooth mappings from the reference domain to a physical one, as is customarily done e.g. in finite element method.

The Lebesgue constant for uniform approximation of differential forms

TL;DR

The paper develops a unified framework for uniform approximation of differential forms from weak, current-based data by projecting onto finite-dimensional form spaces via interpolation or least-squares fitting. It proves that the operator norm of the projection equals the Lebesgue constant under mild disjointness assumptions on sampling supports and extends this equivalence to countable sampling sets, utilizing reproducing kernels and Riesz representers. It further analyzes how the Lebesgue constant and the projection behave under domain mappings, providing explicit bounds under diffeomorphisms (with a sharp affine-case factor) and establishing domain-independence results in certain cases. These results bridge geometric measure theory with approximation theory and finite element exterior calculus, offering stability insights and practical guidance for geometry-preserving discretizations of differential forms.

Abstract

In this work we address the problem of uniform approximation of differential forms starting from weak data defined by integration on rectifiable sets. We study approximation schemes defined by the projection operator L given by either generalized weighted least squares or interpolation. We show that, under a natural measure theoretic condition, the norm of such operator equals the Lebesgue constant of the problem. We finally estimate how the Lebesgue constant varies under the action of smooth mappings from the reference domain to a physical one, as is customarily done e.g. in finite element method.

Paper Structure

This paper contains 15 sections, 13 theorems, 141 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Vectors, covectors and norms
  4. Rectifiable sets and integration
  5. The zero norm, a “ uniform” norm for differential forms
  6. Sampling differential forms by currents of order zero
  7. The interpolation operator $\Pi$ and the fitting operator $P$
  8. Riesz representers and reproducing kernels
  9. Equivalent definitions of the Lebesgue constant
  10. Pushforwarding currents and projection operators
  11. Operator norm and Lebesgue constant
  12. Proof of Theorem \ref{['thm1']}
  13. Continuity properties of $\mathscr{L}$
  14. Interplay between Hilbert and Banach structures
  15. Dependence of $\mathscr{L}$ on the reference domain: proof of Theorem \ref{['thm2']}

Key Result

Theorem 1

Let $\mathcal{T}=\{T_1,T_2,\dots\}$ be a countable $\mathscr{V}^k(E)$-determining subset of $\mathscr{A}^k(E)$ and $w\in \ell^1(\mathbb{N})$. If $\mathcal{H}^k(\mathop{\mathrm{supp}}\limits T_i \cap \mathop{\mathrm{supp}}\limits T_j)=0$ for $i\neq j$, then the discrete least squares projection $P$ s Inequality $\Vert P \Vert_{\mathrm{op}}\leq\mathscr{L}(\mathcal{T},\mathscr{V}^k (E))$ still holds

Theorems & Definitions (44)

  • Theorem 1: Characterization of the norm of the projection operator
  • Corollary 1
  • Theorem 2: Dependence of $\mathscr{L}$ on $E$
  • Definition 1: Orientation of a $k$-rectifiable set
  • Remark 1
  • Remark 2
  • Remark 3: Orientability of $k$-rectifiable set of locally finite measure
  • Example 1
  • Example 2
  • Definition 2: Test forms
  • ...and 34 more