Order of Growth
The order of growth of Dirichlet's kernel is logarithmic, i.e.
See Big O notation for the notation O(1). It should be noted that the actual value is both difficult to calculate (see Zygmund 8.3) and of almost no use. The fact that for some constant c we have
is quite clear when one examines the graph of Dirichlet's kernel. The integral over the n-th peak is bigger than c/n and therefore the estimate for the harmonic sum gives the logarithmic estimate.
This estimate entails quantitative versions of some of the previous results. For any continuous function f and any t one has
However, for any order of growth ω(n) smaller than log, this no longer holds and it is possible to find a continuous function f such that for some t,
The equivalent problem for divergence everywhere is open. Sergei Konyagin managed to construct an integrable function such that for every t one has
It is not known whether this example is best possible. The only bound from the other direction known is log n.
Read more about this topic: Convergence Of Fourier Series
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