abs_iteratedDeriv_finitePowerSeries_le_mass

Plain-English statement

- On the full correlation interval, every iterated derivative is controlled by its weighted absolute coefficient mass. No sign condition on the coefficients is required.

Scope guard. This card records a compiled local declaration. Its mathematical scope is exactly the Lean statement below; the Registry note and source correspondence may describe motivation but do not strengthen it.

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Lean statement

theorem abs_iteratedDeriv_finitePowerSeries_le_mass
    (order : ℕ) (support : Finset ℕ) (a : ℕ → ℝ) {u : ℝ}
    (hu : |u| ≤ 1) :
    |iteratedDeriv order (finitePowerSeries support a) u| ≤
      finitePowerSeriesDerivativeMass order support a := by
  rw [iteratedDeriv_finitePowerSeries]
  unfold finitePowerSeriesDerivative finitePowerSeriesDerivativeMass
  calc
    |∑ j ∈ support,
        (j.descFactorial order : ℝ) * a j * u ^ (j - order)| ≤
        ∑ j ∈ support,
          |(j.descFactorial order : ℝ) * a j * u ^ (j - order)| :=
      Finset.abs_sum_le_sum_abs _ _
    _ = ∑ j ∈ support,
        (j.descFactorial order : ℝ) * |a j| * |u| ^ (j - order) := by
      apply Finset.sum_congr rfl
      intro j hj
      rw [abs_mul, abs_mul, abs_pow,
        abs_of_nonneg (show 0 ≤ (j.descFactorial order : ℝ) by positivity)]
    _ ≤ ∑ j ∈ support, (j.descFactorial order : ℝ) * |a j| := by
      apply Finset.sum_le_sum
      intro j hj
      exact mul_le_of_le_one_right
        (mul_nonneg (by positivity) (abs_nonneg (a j)))
        (pow_le_one₀ (abs_nonneg u) hu)

/-- Termwise control of the scalar fourth-order chain-rule expression.

For smooth scalar functions `k` and `u`, the left side is the absolute value
of the formal expression for `(k ∘ u)⁽⁴⁾`.  The numerical coefficients
`1, 6, 3, 4, 1` are the exact one-dimensional Faà di Bruno coefficients. -/

Open AutoSamplingTheory/TechnicalLemmas/Analysis/FinitePowerSeries.lean:66published source at ed709a7f320d

Proof architecture

Reusable signed finite-polynomial derivative budget for smooth kernels, moment features, and local truncation estimates.

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Pointwise support, topological support, and compact support retain distinct meanings.