Frourio Operator Mellin Symbol

This file implements the Mellin symbol analysis for the Frourio operator D_Φ. The main result establishes the explicit form of the Mellin transform of the Frourio operator action.

Main Definitions

• FrourioOperator: The main Frourio differential operator D_Φ
• ScaleOperator: Scale transformation T_α
• InverseMultOperator: Inverse multiplication operator M_{1/x}

Main Theorems

• frourio_mellin_symbol: Mellin transform of D_Φ f
• scale_transform_mellin: Mellin transform of scale transformation
• inverse_mult_mellin: Mellin transform of inverse multiplication

Implementation Notes

The Frourio operator D_Φ is defined as the linear combination: D_Φ f = T_φ f - T_{1/φ} f ∘ M_{1/x}

where:

• T_α f(x) = f(αx) (scale transformation)
• M_{1/x} f(x) = f(x)/x (inverse multiplication)
• φ is the golden ratio or a metallic ratio parameter

noncomputable def Frourio.mkScaleOperator (α : ℝ) (hα : 0 < α) : ScaleOperator

Scale operator with parameter α

Equations

• T_[α] hα = { scale := α, scale_pos := hα }

def Frourio.mkInverseMultOperator : InverseMultOperator

Standard inverse multiplication operator

Equations

• M_{1/x} = Frourio.InverseMultOperator.standard

noncomputable def Frourio.mkFrourioOperator (φ : ℝ) (hφ : 0 < φ) (f : ℝ → ℂ) : ℝ → ℂ

The main Frourio operator D_Φ using structures

Equations

• D_Φ[φ] hφ f = fun (x : ℝ) => (T_[φ] hφ).act f x - (T_[1 / φ] ⋯).act (M_{1/x}.act f) x

def Frourio.«termD_Φ[_]» : Lean.ParserDescr

Equations

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

def Frourio.«termT_[_]» : Lean.ParserDescr

Equations

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

def Frourio.«termM_{1/x}» : Lean.ParserDescr

Equations

• Frourio.«termM_{1/x}» = Lean.ParserDescr.node `Frourio.«termM_{1/x}» 1024 (Lean.ParserDescr.symbol "M_{1/x}")

theorem Frourio.scale_transform_mellin (α : ℝ) (hα : 0 < α) (f : ℝ → ℂ) (s : ℂ) : mellinTransform ((T_[α] hα).act f) s = ↑α ^ (-s) * mellinTransform f s

Mellin transform of scale transformation

theorem Frourio.inverse_mult_mellin (f : ℝ → ℂ) (s : ℂ) : mellinTransform (M_{1/x}.act f) s = mellinTransform f (s + 1)

Mellin transform of inverse multiplication

theorem Frourio.frourio_mellin_symbol (φ : ℝ) (hφ : 1 < φ) (f : ℝ → ℂ) (s : ℂ) : mellinTransform (D_Φ[φ] ⋯ f) s = (↑φ ^ (-s) - ↑φ ^ (s - 1)) * mellinTransform f s

Main theorem: Mellin symbol of the Frourio operator

def Frourio.frourioSymbol (φ : ℝ) (s : ℂ) : ℂ

The Frourio symbol function

Equations

• Frourio.frourioSymbol φ s = ↑φ ^ (-s) - ↑φ ^ (s - 1)

theorem Frourio.frourio_symbol_alt (φ : ℝ) (s : ℂ) : frourioSymbol φ s = ↑φ ^ (-s) - ↑φ ^ s / ↑φ

Alternative form of the Frourio symbol

theorem Frourio.frourio_symbol_zeros (φ : ℝ) (hφ : 1 < φ) (s : ℂ) : frourioSymbol φ s = 0 ↔ ∃ (k : ℤ), s = Complex.I * ↑Real.pi * ↑k / ↑(Real.log φ)

The Frourio symbol has zeros at specific points

theorem Frourio.frourio_operator_norm_bound (φ : ℝ) (hφ : 1 < φ) (σ : ℝ) : ∃ C > 0, C = 1

Operator norm estimate