Representability by the Fourier Coefficients

Again we start with an arbitrary integrable DP real function $f(\theta)$ that has zero average value, with $\theta\in[-\pi,\pi]$. Let us now assume that this function is such that the corresponding Fourier coefficients $a_{k}$ satisfy the condition that

$$\lim_{k\to\infty} \vert a_{k}\vert\,{\rm e}^{-Ck} = 0,$$ (3)

for all real $C>0$. What this means is that $a_{k}$ may or may not go to zero as $k\to\infty$, may approach a non-zero real number, and may even diverge to infinity as $k\to\infty$, so long as it does not do so exponentially fast. This includes therefore not only the sequences of Fourier coefficients corresponding to all possible convergent Fourier series, but many sequences that correspond to Fourier series that diverge almost everywhere. In fact, it even includes sequences of coefficients that cannot be obtained at all from a real function, such as the sequence $a_{k}=1/\pi$ for all $k\geq 1$, which is associated to the Dirac delta ``function'' $\delta(\theta)$, as shown in [1] and as will be discussed in Section 5 of this paper. It is therefore a very weak condition indeed.

Before anything else, let us establish a preliminary result, namely that the condition in Equation (3) implies that we also have

$$\lim_{k\to\infty} \vert a_{k}\vert k^{p}\,{\rm e}^{-Ck} = 0,$$ (4)

for all real powers $p>0$. This is just a formalization of the well-known fact that the negative-exponent real exponential function goes to zero faster than any positive power goes to infinity, as $k\to\infty$. We may write the function of $k$ on the left-hand side as

$$\vert a_{k}\vert k^{p}\,{\rm e}^{-Ck} = \vert a_{k}\vert\,{\rm e}^{p\ln(k)}\,{\rm e}^{-Ck}.$$

Recalling the properties of the logarithm, we now observe that, given an arbitrary real number $A>0$, there is always a sufficiently large value $k_{m}$ of $k$ above which $\ln(k)<Ak$. A simple proof can be found in Appendix A. Due to this we may write, for all $k>k_{m}$,

$$\vert a_{k}\vert k^{p}\,{\rm e}^{-Ck} < \vert a_{k}\vert\,{\rm e}^{pAk}\,{\rm e}^{-Ck},$$

since the exponential is a monotonically increasing function. If we choose $A=C/(2p)$, which is positive and not zero, we get that, for all $k>k_{m}$,

$$\begin{eqnarray*} \vert a_{k}\vert k^{p}\,{\rm e}^{-Ck} & < & \vert a_{k}\ver... ...}\,{\rm e}^{-Ck} \\ & = & \vert a_{k}\vert\,{\rm e}^{-Ck/2}. \end{eqnarray*}$$

According to our hypothesis about the coefficients $a_{k}$, the $k\to\infty$ limit of the expression in the right-hand side is zero for any strictly positive value of $C'=C/2$, so that taking the $k\to\infty$ limit we establish our preliminary result,

$$\lim_{k\to\infty} \vert a_{k}\vert k^{p}\,{\rm e}^{-Ck} = 0,$$

for all real $C>0$ and all real $p>0$. If we now construct the complex power series $S_{z}$ as before, using the coefficients $a_{k}$, we are in a position to show that it is absolutely convergent inside the open unit disk. In order to do this we consider the real power series $\overline{S}_{z}$ of the absolute values of the terms of the series $S_{z}$, which we write as

$$\begin{eqnarray*} \overline{S}_{z} & = & \sum_{k=1}^{\infty} \vert a_{k}\ver... ... & \sum_{k=1}^{\infty} \vert a_{k}\vert\,{\rm e}^{k\ln(\rho)}. \end{eqnarray*}$$

Since $\rho<1$ inside the open unit disk, the logarithm shown is strictly negative, and we may put $\ln(\rho)=-C$ with real $C>0$. We can now see that, according to our hypothesis about the coefficients $a_{k}$, the terms of this series go to zero exponentially fast as $k\to\infty$. This already suffices to establish its convergence, but we may easily make this more explicit, writing

$$\begin{eqnarray*} \overline{S}_{z} & = & \sum_{k=1}^{\infty} \vert a_{k}\ver... ...1}^{\infty} \frac{k^{2}\vert a_{k}\vert\,{\rm e}^{-Ck}}{k^{2}}. \end{eqnarray*}$$

According to our preliminary result in Equation (4) the numerator shown goes to zero as $k\to\infty$, and therefore above a sufficiently large value $k_{m}$ of $k$ it is less that one, so we may write that

$$\begin{eqnarray*} \overline{S}_{z} & = & \sum_{k=1}^{k_{m}} \vert a_{k}\vert... ...t\,{\rm e}^{-Ck} + \sum_{k=k_{m}+1}^{\infty} \frac{1}{k^{2}}. \end{eqnarray*}$$

The first term on the right-hand side is a finite sum and therefore is finite, and the second term can be bounded from above by a convergent asymptotic integral on $k$, so that we have

$$\begin{eqnarray*} \overline{S}_{z} & < & \sum_{k=1}^{k_{m}} \vert a_{k}\vert... ...1}^{k_{m}} \vert a_{k}\vert\,{\rm e}^{-Ck} + \frac{1}{k_{m}}. \end{eqnarray*}$$

It follows that $\overline{S}_{z}$, which is a real sum of positive terms, so that its partial sums form a monotonically increasing sequence, is bounded from above and is therefore convergent. It then follows that $S_{z}$ is absolutely convergent and therefore convergent. Since this is valid for all $\rho<1$, we may conclude that $S_{z}$ converges on the open unit disk. We may now recover $f(\theta)$ as the $\rho\to 1$ limit of the real or imaginary part of $w(z)$, as the case may be, almost everywhere on the unit circle, as was mentioned before and shown in [1].

This provides therefore a very general condition on the Fourier coefficients of the real functions that ensures that the correspondence established in [1] holds. This then ensures that the real functions can be recovered from, and therefore can be represented by, their Fourier coefficients. Note that the condition in Equation (3) can be considered as an even weaker form of the condition discussed in the previous section. We are therefore ready to state our first important conclusion:

$\parbox{\textwidth}{\bf\boldmath If the... ...n unit disk, from which the real function can be recovered almost everywhere.}$

Note that, although we formulated this condition in terms of the Fourier coefficients $a_{k}$ of a given DP real function, the fact that $a_{k}$ are the Fourier coefficients of the function has in fact not been used at all. Therefore, the conclusion is valid for any sequence of coefficients that satisfies Equation (3), regardless of whether or not they can be obtained as the Fourier coefficients of some real function.