DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge pass (gpt-6-luna)verified 2026-10-22 (gpt-6.1-sol)

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Distributional TV as an extended functional on L²

Definition

Let $Ω \subseteq R^{2}$ be open and let $X = L^{2} (Ω; R)$ . Every $u \in X$ is integrable on compact subsets by Cauchy–Schwarz. Define its extended distributional total variation by $T V_{2} (u; Ω) : = \sup {\int_{Ω} u div φ d x : φ \in C_{c}^{\infty} (Ω; R^{2}), \sup_{x \in Ω} ∣ φ (x) ∣ \leq 1} \in [0, + \infty],$ where $C_{c}^{\infty} (Ω; R^{2})$ denotes smooth compactly supported vector fields, $∣ \cdot ∣$ is the Euclidean norm, and $div φ = \partial_{1} φ_{1} + \partial_{2} φ_{2}$ . This definition treats $L^{2}$ functions as almost-everywhere classes and does not require $u \in L^{1} (Ω)$ .

Definition references

For globally integrable functions, spatial TV is the mass of the distributional derivative measure.

The real L² space is defined from Lebesgue measure and almost-everywhere classes.

Cauchy–Schwarz gives $\int_{K} ∣ u ∣ \leq ∣ K ∣^{1 / 2} ∥ u ∥_{L^{2} (K)}$ on every compact $K$ .

Iglesias, Mercier, and Scherzer, A note on convergence of solutions of total variation regularized linear inverse problems, §1.1

Terra, Introduction to Geometric Measure Theory, §7.4

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Sources

José A. Iglesias, Gwenael Mercier, and Otmar Scherzer, A note on convergence of solutions of total variation regularized linear inverse problems, §1.1 (standard reference, not scraped)
Gláucio Terra, Introduction to Geometric Measure Theory — Lecture Notes, version 1.6, §7.4 (standard reference, not scraped)