Introduction

In a previous paper [1] a one-to-one correspondence between FC (Fourier Conjugate) pairs of DP (Definite Parity) Fourier series and inner analytic functions on the open unit disk was established. In a subsequent paper [2] the questions related to the convergence of such series were examined in the light of this correspondence. In those papers certain techniques were presented for the recovery of the real functions from the coefficients of their DP Fourier series, which work even if the series are divergent. This included a technique we called ``singularity factorization'' that from the (possibly divergent) DP Fourier series of a given real function leads to certain expressions involving alternative trigonometric series, with better convergence characteristics, that converge to that same real function. The reader is referred to those papers for many of the concepts and notations used in this paper.

In another previous paper [3] certain low-pass filters acting in the space of integrable real functions were introduced, and their use for the regularization of divergent Fourier series in boundary value problems was discussed. In the present paper we will show that these low-pass filters can be interpreted and realized within the open unit disk of the complex plane in a very simple way, in the context of the correspondence between FC pairs of DP Fourier series and inner analytic functions within that disk, which was discussed in the aforementioned earlier papers [1] and [2]. In line with our discussion in those papers, about ways of recovering the real functions from their Fourier coefficients when the Fourier series do not converge, or converge poorly, here we will show that the use of low-pass filters can be interpreted as one more such technique. However, unlike the previous ones it involves a certain type of approximation, and its application changes the series and functions in a specific way, that is small in a certain sense, as described in [3]. From a purely mathematical standpoint the discussion of these filters consists of the examination of the properties of a certain set of integral operators acting in the space of integrable real functions.

Let us review briefly the facts about the filters, when defined on a periodic interval. As given in [3], the first-order linear low-pass filter is defined in the following way, if we adopt as the domain of our real functions the periodic interval $[-\pi ,\pi ]$. Given a real function $f(\theta)$ of the real angular variable $\theta$ in that interval, of which we require no more than that it be integrable, we define from it a filtered function $f_{\epsilon}^{(1)}(\theta)$ as

$$f_{\epsilon}^{(1)}(\theta) = \frac{1}{2\epsilon} \int_{\... ...epsilon}^{\theta+\epsilon}d\theta'\, f\!\left(\theta'\right),$$ (1)

where the angular parameter $\epsilon\leq\pi$ is a strictly positive real parameter which we will refer to as the range of the filter. One can also define $f_{0}^{(1)}(\theta)$ by continuity, as the $\epsilon\to 0$ limit of this expression. The filter can be understood as a linear integral operator acting in the space of integrable real functions, as is done in [3]. It may be written as an integral over the whole periodic interval involving a kernel $K_{\epsilon}^{(1)}\!\left(\theta-\theta'\right)$ with compact support,

$$f_{\epsilon}^{(1)}(\theta) = \int_{-\pi}^{\pi}d\theta'\, ... ...}^{(1)}\!\left(\theta-\theta'\right) f\!\left(\theta'\right),$$

where the kernel is defined as $K_{\epsilon}^{(1)}\!\left(\theta-\theta'\right)=1/(2\epsilon)$ for $\left\vert\theta-\theta'\right\vert<\epsilon$, and as $K_{\epsilon}^{(1)}\!\left(\theta-\theta'\right)=0$ for $\left\vert\theta-\theta'\right\vert>\epsilon$. Since we are in the periodic interval, it should be noted that what we mean here by ``compact support'' is the fact that the kernel is different from zero only within an interval contained in the periodic interval. We may have at most that the two intervals coincide, with $\epsilon=\pi$, and in general we will assume that we have $\epsilon\leq\pi$. The most interesting case, however, is that in which we have $\epsilon\ll\pi$. We may say then that this kernel is a discontinuous even function of $\left (\theta -\theta '\right )$ that has unit integral and compact support. As shown in [3], it can be expressed in terms of a point-wise convergent Fourier series,

$$K_{\epsilon}^{(1)}\!\left(\theta-\theta'\right) = \frac{1... ...n)} \right] \cos\!\left[k\left(\theta-\theta'\right)\right].$$

The filter defined above has several interesting properties, which are the reasons for its usefulness, the most important and basic ones of which are listed and demonstrated in [3]. In this paper we will refer to and use these properties as the occasion arises. Also, as part of the demonstrations discussed in Section 3 we will have the opportunity to examine some more of these properties in Appendix B.

As discussed in [3], since the first-order filter defined here is a linear operator, one can construct higher-order filters by simply applying it multiple times to a given real function. This leads directly to the definition of higher-order filters, for example the second-order one, with range $2\epsilon$, and assuming that $\epsilon\leq\pi/2$,

$$f_{2\epsilon}^{(2)}(\theta) = \int_{-\infty}^{\infty}d\th... ...}^{(2)}\!\left(\theta-\theta'\right) f\!\left(\theta'\right),$$

where, as a consequence of the definition of the first-order filter, the second-order kernel with range $2\epsilon$ is given by the application of the first-order filter to the first-order kernel,

$$K_{2\epsilon}^{(2)}\!\left(\theta-\theta''\right) = \int_... ...ta'\right) K_{\epsilon}^{(1)}\!\left(\theta'-\theta''\right).$$

This second-order kernel is a continuous but non-differentiable even function of $\left (\theta -\theta '\right )$. Due to the properties of the first-order filter regarding its action on Fourier expansions [10,11], the second-order kernel is also given by the absolutely and uniformly convergent Fourier series

$$K_{2\epsilon}^{(2)}\!\left(\theta-\theta'\right) = \frac{... ... \right]^{2} \cos\!\left[k\left(\theta-\theta'\right)\right],$$

so long as $\epsilon\leq\pi/2$. Both the first and second-order kernels are even functions of $\left (\theta -\theta '\right )$ with unit integral and compact support. The range of the first-order filter is given by $\epsilon$, and if one just applies the filter twice as we did here, that range doubles do $2\epsilon$. However, one may compensate for this by simply applying twice the first-order filter with parameter $\epsilon/2$, thus resulting in a second-order filter with range $\epsilon$, given by the absolutely and uniformly convergent Fourier series

$$K_{\epsilon}^{(2)}\!\left(\theta-\theta'\right) = \frac{1... ... \right]^{2} \cos\!\left[k\left(\theta-\theta'\right)\right],$$

so long as $\epsilon\leq\pi$. This procedure can be iterated $N$ times to produce an order-$N$ filter with range $N\epsilon$. Given the properties of the first-order filter regarding its action on Fourier expansions [10,11], the Fourier representation of the order-$N$ kernel can easily be written explicitly,

$$K_{N\epsilon}^{(N)}\!\left(\theta-\theta'\right) = \frac{... ... \right]^{N} \cos\!\left[k\left(\theta-\theta'\right)\right],$$ (2)

so long as $\epsilon\leq\pi/N$. This definition can be extended down to the case of the order-zero kernel, with $N=0$, which is simply the Dirac delta ``function'', and which is in fact given, as shown in [2], by the divergent Fourier series

$$\begin{eqnarray*} \delta\!\left(\theta-\theta'\right) & = & K_{0}^{(0)}\!\le... ...{k=1}^{\infty} \cos\!\left[k\left(\theta-\theta'\right)\right]. \end{eqnarray*}$$

This can be understood as the kernel of an order-zero filter, which is the identity almost everywhere. If we simply exchange $\epsilon$ by $\epsilon/N$ in the expression in Equation (2) we get the order-$N$ filter with range $\epsilon$, written in terms of its Fourier expansion,

$$K_{\epsilon}^{(N)}\!\left(\theta-\theta'\right) = \frac{1... ... \right]^{N} \cos\!\left[k\left(\theta-\theta'\right)\right],$$

so long as $\epsilon\leq\pi$. Note that this series converges ever faster as $N$ increases, and that it can be differentiated $N-2$ times still resulting in absolutely and uniformly convergent series, and $N-1$ times still resulting in point-wise convergent series. The series for $K_{0}^{(0)}\!\left(\theta-\theta'\right)$ is the only one which is not convergent, and of the remaining ones that for $K_{\epsilon}^{(1)}\!\left(\theta-\theta'\right)$ is the only one which is not absolutely or uniformly convergent, although it is point-wise convergent. For $N\geq 2$ all the Fourier series of the kernels, regardless of range, are absolutely and uniformly convergent to functions which are $C^{N-2}$ everywhere. All these kernels, regardless of order or range, are even functions of $\left (\theta -\theta '\right )$ with unit integral and compact support, so long as $\epsilon\leq\pi$.

Therefore, one is led to think of the possibility that in the limit $N\to \infty$ this sequence of order-$N$ kernels with constant range $\epsilon$ could converge to a $C^{\infty}$ kernel function $K_{\epsilon}^{(\infty)}\!\left(\theta-\theta'\right)$ with compact support. The corresponding infinite-order filter would then map any merely integrable function to a $C^{\infty}$ function. Although it turns out to be possible to construct a infinite-order kernel with such a property, it is not to be obtained by the limit described here, as we will see later in Section 3.