Convergence of Fourier Series - Norm Convergence

Norm Convergence

The simplest case is that of L2, which is a direct transcription of general Hilbert space results. According to the Riesz–Fischer theorem, if ƒ is square-integrable then

\lim_{N\rightarrow\infty}\int_0^{2\pi}\left|f(x)-S_N(f) \right|^2\,dx=0

i.e.,  converges to ƒ in the norm of L2. It is easy to see that the converse is also true: if the limit above is zero, ƒ must be in L2. So this is an if and only if condition.

If 2 in the exponents above is replaced with some p, the question becomes much harder. It turns out that the convergence still holds if 1 < p < ∞. In other words, for ƒ in Lp,  converges to ƒ in the Lp norm. The original proof uses properties of holomorphic functions and Hardy spaces, and another proof, due to Salomon Bochner relies upon the Riesz–Thorin interpolation theorem. For p = 1 and infinity, the result is not true. The construction of an example of divergence in L1 was first done by Andrey Kolmogorov (see below). For infinity, the result is a more or less trivial corollary of the uniform boundedness principle.

If the partial summation operator SN is replaced by a suitable summability kernel (for example the Fejér sum obtained by convolution with the Fejér kernel), basic functional analytic techniques can be applied to show that norm convergence holds for 1 ≤ p < ∞.

Read more about this topic:  Convergence Of Fourier Series

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