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By Brown E. W.

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P (t) The Poisson kernel. 9 If f ∈ Lp [−π, π], 1 ≤ p < +∞, or p = +∞ and f is continuous with f (π) = f (−π), we have limr→1− Pr ∗ f − f p = 0. Now we consider f ∈ L1 [−π, π] and ask about the a. e. convergence of Fn ∗f (x) or Pr ∗f (x) to f (x). Obviously there exist some subsequences Fnk ∗f and Prk ∗ f that converge a. e. to f . u(reiθ ) = Pr ∗ f (θ) is a harmonic function on the unit disc.

A. de Reyna: LNM 1785, pp. 47–49, 2002. c Springer-Verlag Berlin Heidelberg 2002 1 1 − x − t x − tJ dt, 48 where tJ is the center of J. All these terms can be bounded by weak local norms f (n,J) = j∈Z 1 c 2 1 + j |J| f (t) exp −2πi n + J j t 3 |J| dt . Hence, given α = (n, J), f α is a mean value of absolute values of (generalized) Fourier coeﬃcients of f . Observe now that the ﬁrst term is of the type with which we started, only that the new integral is more simple because it corresponds to fewer cycles, but a non integer number of cycles.

5 Summability As we have said, Du Bois Reymond constructed a continuous function whose Fourier series diverges at some point. Lipot Fej´er proved, when he was 19 years old, that in spite of this we can recover a continuous function from its Fourier series. Recall that if a sequence converges, there converges also, and to the same limit, the series formed by the arithmetic means of his terms. Fej´er considered the mean values of the partial sums 1 σn (f, x) = n+1 n Sn (f, x). j=0 We have an integral expression for these mean values σn (f, x) = 1 2π π −π Fn (x − t)f (t) dt where Fn is Fej´er kernel: 1 Fn (t) = n+1 j=n n 1− Dj (t) = j=0 j=−n |j| eijt .