Uniform approximation by partial sums of one power series

DefinitionThe claim of Scottish Book Problem 159#✓fully proved

There is a sequence of real numbers \(a_0, a_1, a_2, \dots\) such that for every \(\varepsilon > 0\) there is a natural number \(N\) with the following property: for every function \(f : \mathbb{R} \to \mathbb{R}\) that is continuous on \([0, 1]\) and satisfies \(f(0) = 0\) and \(0 \le f(x) \le 1\) for all \(x \in [0, 1]\), there is a natural number \(n \le N\) with \(|f(x) - P_n(x)| < \varepsilon\) for all \(x \in [0, 1]\), where \(P_n(x) = \sum_{k=0}^{n} a_k x^k\) is the \(n\)-th partial sum of the power series \(\sum a_k x^k\).

def ScottishBookProblem159Claim : Prop :=
  ∃ a : ℕ → ℝ, ∀ ε : ℝ, 0 < ε → ∃ N : ℕ, ∀ f : ℝ → ℝ,
    ContinuousOn f (Set.Icc 0 1) → f 0 = 0 →
      (∀ x ∈ Set.Icc (0 : ℝ) 1, f x ∈ Set.Icc 0 1) →
      ∃ n : ℕ, n ≤ N ∧ ∀ x ∈ Set.Icc (0 : ℝ) 1,
        dist (f x) (∑ k ∈ Finset.range (n + 1), a k * x ^ k) < ε

Used by 1

Used byScottish Book Problem 159 has a negative answer

TheoremScottish Book Problem 159 has a negative answer#●ready to prove

There is no sequence of real numbers \(a_0, a_1, a_2, \dots\) with the following property: for every \(\varepsilon > 0\) there is a natural number \(N\) such that for every function \(f : \mathbb{R} \to \mathbb{R}\) that is continuous on \([0, 1]\) and satisfies \(f(0) = 0\) and \(0 \le f(x) \le 1\) for all \(x \in [0, 1]\), there is a natural number \(n \le N\) with \(\left|f(x) - \sum_{k=0}^{n} a_k x^k\right| < \varepsilon\) for all \(x \in [0, 1]\).

Sternbach's argument in the addendum Ad 159 uses the functions \(\sin^2(2^n \pi x)\). Mazur showed that for an arbitrary family \(\Phi\) of functions of this kind, a power series with this property exists exactly when the functions of \(\Phi\) are equicontinuous.

theorem_wanted problem159Refuted : ¬ ScottishBookProblem159Claim

Sources

• S. Ruziewicz, Problem 159 of the Scottish Book; Ad 159 in the typescript (Sternbach's negative argument; Mazur's equicontinuity criterion).

Uses 1

Statement usesThe claim of Scottish Book Problem 159

1 fully proved1 ready to prove

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