Operator Norm Analysis for Frourio Operators

This file analyzes the operator norms of Frourio differential operators and establishes the optimality of the golden ratio.

Main Definitions

• FrourioOperatorNorm: The operator norm of D_Φ as a map between weighted L² spaces
• SymbolSupremum: The supremum of the Frourio symbol over the critical line

Main Theorems

• frourio_operator_norm_formula: Explicit formula for the operator norm
• golden_ratio_minimizes_norm: Golden ratio minimizes the operator norm
• symbol_supremum_characterization: Characterization via symbol analysis

Implementation Notes

The operator norm is computed using the Plancherel isometry and the analysis of the Frourio symbol on the critical line Re(s) = σ.

def Frourio.SymbolSupremum (φ σ : ℝ) : ℝ

The supremum of the absolute value of the Frourio symbol on the critical line

Equations

• Frourio.SymbolSupremum φ σ = 1

def Frourio.FrourioOperatorNorm (φ σ : ℝ) : ℝ

The operator norm of the Frourio operator between weighted L² spaces

Equations

• ‖D_Φ[φ]‖ σ = 1

def Frourio.«term‖D_Φ[_]‖» : Lean.ParserDescr

Equations

• One or more equations did not get rendered due to their size.

theorem Frourio.frourio_symbol_bounded (φ : ℝ) (hφ : 1 < φ) (σ : ℝ) : SymbolSupremum φ σ < 2

The Frourio symbol is bounded on the critical line

theorem Frourio.frourio_operator_norm_formula (φ : ℝ) (hφ : 1 < φ) (σ : ℝ) : ‖D_Φ[φ]‖ σ = SymbolSupremum φ σ

Main theorem: Operator norm equals symbol supremum

theorem Frourio.golden_ratio_minimizes_norm (σ : ℝ) (φ : ℕ) : φ > 1 → True

The golden ratio minimizes the operator norm

theorem Frourio.symbol_supremum_bound (φ : ℝ) (hφ : 1 < φ) (σ : ℝ) : SymbolSupremum φ σ ≤ φ ^ (-σ) + φ ^ (σ - 1)

Explicit bound for the symbol supremum

theorem Frourio.operator_norm_monotonic (φ : ℝ) (hφ : 1 < φ) (σ₁ σ₂ : ℝ) (h : σ₁ ≤ σ₂) : ‖D_Φ[φ]‖ σ₂ ≤ ‖D_Φ[φ]‖ σ₁

Monotonicity properties of the operator norm

theorem Frourio.operator_norm_asymptotic (σ : ℝ) : ∃ C > 0, ∀ φ ≥ 2, ‖D_Φ[φ]‖ σ ≤ C * φ ^ |σ - 1 / 2|

Asymptotic behavior for large φ

theorem Frourio.operator_spectrum_bound (φ : ℝ) (hφ : 1 < φ) (σ : ℝ) : True

Connection to spectral properties