Theorems · Theorem · harmonic analysis
has_pointwise_sum_fourier_series_of_summable
∀ {T : ℝ} [hT : Fact (0 < T)] {f : C(AddCircle T, ℂ)},
Summable (fourierCoeff ⇑f) → ∀ (x : AddCircle T), HasSum (fun i => fourierCoeff (⇑f) i • (fourier i) x) (f x)If the sequence of Fourier coefficients of f is summable, then the Fourier series of f
converges everywhere pointwise to f.
- Defined in
- Mathlib.Analysis.Fourier.AddCircle
- Cited by
- 2 results in Mathlib
- Foundations
- Depth 274 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- Fact
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites18
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
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- SummationFilterproof · cited by 607
- HasSumstatement and proof · cited by 518
- AddSubgroup.zmultiplesstatement · cited by 493
Cited by2
Results whose statement or proof uses this declaration.
- hasSum_one_div_pow_mul_fourier_mul_bernoulliFunproof · cited by 1
- Real.tsum_eq_tsum_fourierproof · cited by 1