at t, then the Fourier series of ƒ cannot converge to a value different from ƒ(t). It may either converge to ƒ(t) or diverge. This is because, if converges to some value x, it is also summable to it, so from the first summability property above, x = ƒ(t). Order of growth The order of growth of Dirichlet's kernel is logarithmic, i.e. See Big O notation for the notation O(1). 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. Multiple dimensions Upon examining the equivalent problem in more than one dimension, it is necessary to specify the precise order of summation one uses. For example, in two dimensions, one may define which are known as "square partial sums". Replacing the sum above with lead to "circular partial sums". The difference between these two definitions is quite notable. For example, the norm of the corresponding Dirichlet kernel for square partial sums is of the order of while for circular partial sums it is of the order of . Many of the results true for one dimension are wrong or unknown in multiple dimensions. In particular, the equivalent of Carleson's theorem is still open for circular partial sums. Almost everywhere convergence of "square partial sums" (as well as more general polygonal partial sums) in multiple dimensions was established around 1970 by Charles Fefferman. Notes References Textbooks Dunham Jackson The theory of Approximation, AMS Colloquium Publication Volume XI, New York 1930. Nina K. Bary, A treatise on trigonometric series, Vols. I, II. Authorized translation by Margaret F. Mullins. A Pergamon