I read in some lecture notes that if a function is uniformly Lipschitz, then its Fourier series is also uniformly converging. I was wondering is this true, and if so how can it be proven?
I understand that a function being Lipschitz means that its Fourier summation is pointwise convergent, and I understand that if the Fourier summations are collectively Lipschitz as well then the convergence is uniform like shown here:
Given sequence of $L-$Lipschitz functions which converges pointwise, prove uniform convergence
and I understand that there is a bound on the Fourier coefficients for the function being Lipschitz as shown here:
A bound on the Fourier coefficients of an $\alpha$-Lipschitz function
I was hoping these facts might help with constructing the proof of what I want. I was also hoping that the resulting proof would be simple, since the notes I read from said that a proof contained in "Fourier Series and Orthogonal Polynomials" by Jackson (1941) was very simple, but I can't seem to find the book available online.
