A continuous Fourier series supported on a Sidon set has ell-one coefficients

The $N$-th Fejer mean is the finite polynomial [given, algebra] ${\sigma }_{N}f={\sum }_{\mathrm{\mid }k\mathrm{\mid }\le N}(1-\frac{\mathrm{\mid }k\mathrm{\mid }}{N+1})\widehat{f}(k)e_k\mathrm{.}$ Its spectrum lies in $E$. Positivity and mass one of the Fejer kernel give $\mathrm{\parallel }{\sigma }_{N}f{\mathrm{\parallel }}_{\mathrm{\infty }}\le \mathrm{\parallel }f{\mathrm{\parallel }}_{\mathrm{\infty }}$.

Apply the Sidon inequality to this polynomial: [step 1.1, algebra] ${\sum }_{\begin{array}{c}k\in E\\ \mathrm{\mid }k\mathrm{\mid }\le N\end{array}}(1-\frac{\mathrm{\mid }k\mathrm{\mid }}{N+1})\mathrm{\mid }\widehat{f}(k)\mathrm{\mid }\le C_E\mathrm{\parallel }f{\mathrm{\parallel }}_{\mathrm{\infty }}\mathrm{.}$ For every fixed finite subset of $E$, the displayed weights tend monotonically to $1$.

Letting $N\to \mathrm{\infty }$ first for each finite subset and then taking the [step 2.1, algebra] supremum over finite subsets gives ${\sum }_{k\in E}\mathrm{\mid }\widehat{f}(k)\mathrm{\mid }\le C_E\mathrm{\parallel }f{\mathrm{\parallel }}_{\mathrm{\infty }}$. Uniform Fejer convergence identifies the same continuous function with these means, so no separate representative is introduced.