Cesaro-Fourier mean

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Cesaro-Fourier mean

Let $f \in L^{2} ([- π, π])$ . The $N$ th Cesaro-Fourier mean of $f$ is

σ_{N} f (x) = \frac{1}{N + 1} \sum_{k = 0}^{N} S_{k} f (x)

Where $S_{k} f (x)$ is the $k$ th partial Fourier sum.

Note

If the Fourier coefficients are all zero, then the function is zero
Want to show that if the partial Fourier sums converge, then it must be a sum of zeros. But doing this with the $S_{N}$ directly is the original question.
The "averaged" version (the Cesaro-Fourier mean) is easier to work with, so we will try and show convergence here instead

Note

We know from real analysis that the Cesaro means of a sequence behave better but do not lose any information.

Example

Sequences like ${- 1, 1, - 1, 1, \dots}$ do not converge, but their Cersaro means do

File	Last Modified
Cesaro-Fourier mean	2025-07-15
Fourier series	2025-07-15
Functional Analysis Lecture 15	2025-07-15

Cesaro-Fourier mean

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