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Robert Baillie
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By using computers to do experimental manipulations on Fourier series, we construct additional series with interesting properties. For example, we construct several series whose sums remain unchanged when the nth term is multiplied by sin(n)/n. One series with this property is this classic series for pi/4: pi/4 = 1 - 1/3 + 1/5 ... = 1*(sin(1)/1) - (1/3)*(sin(3)/3) + (1/5)*(sin(5)/5).... Another example is sum (n = 1 to infinity) of (sin(n)/n) = sum (n = 1 to infinity) of (sin(n)/n)^2 = (pi - 1)/2. This material should be accessible to undergraduates. This paper also includes a Mathematica package that makes it easy to calculate and graph the Fourier series of many types of functions.
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For 1<p<infty and for weight w in A_p, we show that the r-variation of the Fourier sums of any function in L^p(w) is finite a.e. for r larger than a finite constant depending on w and p. The fact that the variation exponent depends on w is necessary. This strengthens previous work of Hunt-Young and is a weighted extension of a variational Carleson theorem of Oberlin-Seeger-Tao-Thiele-Wright. The proof uses weighted adaptation of phase plane analysis and a weighted extension of a variational inequality of Lepingle.
Let $fin L_{2pi}$ be a real-valued even function with its Fourier series $ frac{a_{0}}{2}+sum_{n=1}^{infty}a_{n}cos nx,$ and let $S_{n}(f,x), ngeq 1,$ be the $n$-th partial sum of the Fourier series. It is well-known that if the nonnegative sequence ${a_{n}}$ is decreasing and $limlimits_{nto infty}a_{n}=0$, then $$ limlimits_{nto infty}Vert f-S_{n}(f)Vert_{L}=0 {if and only if} limlimits_{nto infty}a_{n}log n=0. $$ We weaken the monotone condition in this classical result to the so-called mean value bounded variation ($MVBV$) condition. The generalization of the above classical result in real-valued function space is presented as a special case of the main result in this paper which gives the $L^{1}$% -convergence of a function $fin L_{2pi}$ in complex space. We also give results on $L^{1}$-approximation of a function $fin L_{2pi}$ under the $% MVBV$ condition.
We consider the class of multiple Fourier series associated with functions in the Dirichlet space of the polydisc. We prove that every such series is summable with respect to unrestricted rectangular partial sums, everywhere except for a set of zero multi-parametric logarithmic capacity. Conversely, given a compact set in the torus of zero capacity, we construct a Fourier series in the class which diverges on this set, in the sense of Pringsheim. We also prove that the multi-parametric logarithmic capacity characterizes the exceptional sets for the radial variation and radial limits of Dirichlet space functions. As a by-product of the methods of proof, the results also hold in the vector-valued setting.
Every set $Lambdasubset R$ such that the sum of $delta$-measures sitting at the points of $Lambda$ is a Fourier quasicrystal, is the zero set of an exponential polynomial with imaginary frequencies.
An abstract theory of Fourier series in locally convex topological vector spaces is developed. An analog of Fej{e}rs theorem is proved for these series. The theory is applied to distributional solutions of Cauchy-Riemann equations to recover basic results of complex analysis. Some classical results of function theory are also shown to be consequences of the series expansion.
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