"The theorem guarantees convergence by averaging out whatever disturbing oscillations might occur in the ordinary Fourier approximation." - Introduction to Calculus and Analysis: Volume I by Richard Courant and Fritz John
3 Some Applications in Fourier Analysis
Fourier analysis classically constituted the theory of approximations via trigonometric polynomials, where Fourier himself had originally described certain tools of Fourier analysis as the resolution of certain functions into infinite sums of "cosines of multiple arcs".$^{[13]}$ And, although such interpretations can be subsumed under various generalized frameworks, as in the functional analytic framework of orthogonal projections in inner-product spaces, there is another framework that could subsume the theory of Fourier analysis, and given the context of this series, perhaps it is not entirely too surprising that the direction that we are interested in is the framework as provided by the theory of convolutions. Before we proceed, we provide a slight altering of property I. of families of approximate identity kernels in (for the properties, see Convolutional Approximations II: A Systematic Theory with Identity Approximating Kernels),
I. We have that, for $n \in \mathbb{N}$,
$$\frac{1}{2 \pi} \int_{-\pi}^{\pi} g_n(x) dx = 1$$
And, for the other properties II. and III., we intend to integrate over the interval $[-\pi, \pi]$ instead. Such is simply due to the fact that we would like to accommodate the $2\pi$-periodic properties of Fourier series as conventionally stated, and, also, that we would like to ultimately be able to recover the approximation to the identity on $[-\pi, \pi]$. So, consider a first class of kernels that is classical in,
Definition 3.1 (The Dirichlet Kernels) The Dirichlet kernels are defined to be, for $n \in \mathbb{N}$,
$$D_n(x) = \frac{\sin \left[ \left( n + \frac{1}{2} \right)x \right]}{\sin \left( \frac{1}{2}x \right)}$$
We also take it as known, and so by definition, that,$^{[14]}$
$$D_N(x) = \sum_{n = -N}^{N} e^{inx}$$
Definition 3.2 (The Fejér Kernels) The Fejér kernels are defined to be, for $n \in \mathbb{N}$,
$$F_n(x) = \frac{1}{n} \left( \frac{\sin \left( \frac{1}{2}nx \right)}{\sin \left( \frac{1}{2}x \right)} \right)^2$$
Then, in pursuing certain convergence results of Fourier analysis, in recalling the treatment as provided in the previous parts of this sequel, we can then proceed in the following manner. If Fourier series can be provided in general in terms of convolutions with families of kernels, and if only such families of kernels are approximate identity kernels, then we can guarantee certain convergence results. A natural question is then, are the family of kernels as defined in Definition 3.1 and Definition 3.2 families of approximate identity kernels?
Theorem 3.1 The family of Dirichlet kernels is not a family of approximate identity kernels.
Intuitively, we first comment that it is not entirely clear that the family of Dirichlet kernels is not a family of approximate identity kernels, where we state now that it is in fact the second property in II. which is violated. Such is not entirely clear as families of Dirichlet kernels provide iterated oscillations of positive and negative amplitudes, that is, it is not so straightforward as to whether, for $n \rightarrow \infty$, that the cancellation effect between the positive amplitudes and the negative amplitudes is sufficient to provide convergence. We thus require a quantitative estimate at least and so we consider,
Lemma 3.1 We have that the Dirichlet kernel satisfy $\frac{1}{2 \pi} \int_{-\pi}^{\pi} \left| D_n(x) \right| dx > c \log(n)$ for some real $c$.
PROOF. Since $\sin(x) \leq x$, we have that $\frac{1}{x} \leq \frac{1}{\sin(x)}$, and so we first immediately obtain,
$$\begin{align} \int_{-\pi}^{\pi} \left| D_n(x) \right| dx & \geq \int_{-\pi}^{\pi} \frac{\left| \sin \left[ \left( n + \frac{1}{2} \right)x \right] \right|}{\left| x \right|} dx \\ &= \int_{-\left( n + \frac{1}{2} \right) \pi}^{\left( n + \frac{1}{2} \right) \pi} \frac{\left| \sin(u) \right|}{\left| u \right|} du \\ &> \int_{\pi}^{n \pi} \frac{\left| \sin(u) \right|}{\left| u \right|} du \\ &= \sum_{k = 1}^{n - 1} \int_{k \pi}^{(k + 1)\pi} \frac{\left| \sin(u) \right|}{\left| u \right|} du \\ & \geq \frac{1}{\pi} \sum_{k = 1}^{n - 1} \frac{1}{k + 1} \int_{k \pi}^{(k + 1)\pi} \left| \sin(u) \right| du \end{align}$$
Then, finally, since $\sum_{k = 1}^{n} \frac{1}{k} > c' \log(n)$ for some appropriate $c'$, we obtain the estimate $\frac{1}{2 \pi} \int_{-\pi}^{\pi} \left| D_n(x) \right| dx > c \log(n)$.$^{[15]}$ $\square$
The proof of Theorem 3.1 is then an immediate consequence of Lemma 3.1. Given such an estimate, a quantitative bound is then provided which allows us to realize that the family of Dirichlet kernels consists of exactly those kernels that are unable to control their corresponding iterated oscillations, in absolute value, for $n \rightarrow \infty$. Since Fourier series can typically be formulated in terms of convolutions with respect to Dirichlet kernels, what Theorem 3.1 allows us to conclude then essentially is that there exist Fourier series that are not convergent, a particularly simple result, a special case of a general theory in the theory of convolutions.$^{[16]}$
However, there exist other classes of kernels where such iterated oscillations can be controlled. Indeed, the family of Fejér kernels resolves, in a certain sense, exactly the weaknesses of the family of Dirichlet kernels in that what is provided is an appropriate averaging phenomenon so as to control the iterated oscillations of the family of Dirichlet kernels. And so, consider the following formulation below,
Theorem 3.2 The Fejér kernels can be provided by Cesàro sums of families of Dirichlet kernels in the sense of,
$$F_n(x) = \frac{D_0(x) + D_1(x) + \dots + D_{n - 1}(x)}{n}$$
PROOF. First, recall that since the Dirichlet kernels can be given by $D_n(x) = \frac{\sin \left[ \left( n + \frac{1}{2} \right)x \right]}{\sin \left( \frac{1}{2}x \right)}$, we can rewrite the Dirichlet kernels as,
$$\begin{align} \frac{\sin \left[ \left( n + \frac{1}{2} \right)x \right]}{\sin \left( \frac{1}{2}x \right)} &= \frac{\cos \left( \frac{1}{2}x \right) \cos \left[ \left( n + \frac{1}{2} \right)x \right] - \cos \left[ \left( n + 1 \right)x \right]}{\left[ \sin \left( \frac{1}{2}x \right) \right]^2} \\ &= 2 \frac{\cos \left( \frac{1}{2}x \right) \cos \left[ \left( n + \frac{1}{2} \right)x \right] - \cos \left[ \left( n + 1 \right)x \right]}{1 - \cos(x)} \end{align}$$
Where, in the first equality, we have used the trigonometric addition formula in $\sin(\frac{1}{2}x) \sin \left[ \left( n + \frac{1}{2} \right)x \right] = \cos(\frac{1}{2}x) \cos \left[ \left( n + \frac{1}{2} \right)x \right] - \cos \left[ \left( n+ \frac{1}{2} \right)x + \frac{1}{2}x \right]$, and, in the second equality, the half-angle formula in $\left[ \sin \left( \frac{x}{2} \right) \right]^2 = \frac{1 - \cos(x)}{2}$. And, since,
$$\begin{align} \cos(nx) &= \cos \left[ \left( n + \frac{1}{2} \right)x - \frac{1}{2}x \right] \\ &= \cos \left[ \left( n + \frac{1}{2} \right)x \right] \cos \left( \frac{1}{2}x \right) + \sin \left[ \left( n + \frac{1}{2} \right)x \right] \sin \left( \frac{1}{2}x \right) \end{align}$$
$$\begin{align} \cos \left[ (n + 1)x \right] &= \cos \left[ \left( n + \frac{1}{2} \right)x + \frac{1}{2}x \right] \\ &= \cos \left[ \left( n + \frac{1}{2} \right)x \right] \cos \left( \frac{1}{2}x \right) - \sin \left[ \left( n + \frac{1}{2} \right)x \right] \sin \left( \frac{1}{2}x \right) \end{align}$$
We proceed to,
$$\begin{align} \frac{\sin \left[ \left( n + \frac{1}{2} \right)x \right]}{\sin \left( \frac{1}{2}x \right)} &= \frac{\cos(nx) - \cos \left[ (n + 1)x \right]}{1 - \cos(x)} \end{align}$$
The point of the above being, that such an expression then allows us to provide a telescoping sum so as to retrieve the Fejér kernels,
$$\begin{align} \frac{D_0(x) + \dots + D_{n - 1}(x)}{n} &= \frac{1}{n} \left( \frac{\cos(0x) - \cos \left[ \left( 0 + 1 \right)x \right] + \cos(1x) - \cos \left[ \left( 1 + 1 \right)x \right] + \dots - \cos (nx)}{1 - \cos(x)} \right) \\ &= \frac{1}{n} \left( \frac{1 - \cos(nx)}{1 - \cos(x)} \right) \\ &= \frac{1}{n} \left( \frac{\sin \left( \frac{1}{2}nx \right)}{\sin \left ( \frac{1}{2}x \right)} \right)^2 \end{align}$$
$\square$
Notice that in the above Cesàro sums in the providing of Fejér kernels, all that was provided is a certain arithmetic averaging phenomenon. Although it is not entirely clear that such an approach can actually resolve the iterated oscillations of the Dirichlet kernels, it turns out that, in fact, such an averaging technique is sufficient for our current purposes, as formalized in the following,
Theorem 3.3 The family of Fejér kernels is a family of approximate identity kernels.
PROOF. First, since Dirichlet kernels satisfy property I. of approximate identity kernels as,
$$\begin{align} \frac{1}{2 \pi} \sum_{n = -N}^{N} \int_{-\pi}^{\pi} e^{inx} dx &= \frac{1}{2\pi} \int_{-\pi}^{\pi} e^{i \cdot 0 \cdot x} dx + \frac{1}{2 \pi} \sum_{n = -N}^{-1} \int_{-\pi}^{\pi} \left[ \cos(nx) + i \sin(nx) \right] dx + \frac{1}{2 \pi} \sum_{n = 1}^{N} \int_{-\pi}^{\pi} \left[ \cos(nx) + i \sin(nx) \right] dx \\ &= \frac{2 \pi}{2 \pi} \\ &= 1 \end{align}$$
It is clear that the Fejér kernels, being kernels that can be given in terms of Cesàro sums of Dirichlet kernels, also satisfy property I.. Property II. is realized almost immediately since Fejér kernels are non-negative and since property I. is satisfied. Next, we demonstrate property III. of approximate identity kernels, which can be realized by noticing that $\left( \frac{\sin \left( \frac{1}{2}nx \right)}{\sin \left( \frac{1}{2}x \right)} \right)^2$ is bounded for $x \neq 0$, and so, away from $x = 0$, the Fejér kernels are of the same order of magnitude as $\frac{1}{n}$. $\square$
In other words, Fourier series are Cesàro summable in the sense that Cesàro sums of partial Fourier sums always provide results that converge. Or, alternatively, for entire classes of functions, even if their Fourier series do not converge, at the very least, we can always provide trigonometric polynomials that converge to them in the limit, and these trigonometric polynomials are exactly those Cesàro sums of partial Fourier sums, seemingly quite a powerful and profound result since such demonstrates that even the class of elementary trigonometric sinusoids are sufficient so as to provide approximations for surprisingly wide classes of functions in general.
Below, an animation that shows how incredibly controlled the family of Fejér kernels is, as opposed to the family of Dirichlet kernels, where the integral of the absolute value of the functions grows (developed using Python).
For further commentary, notice that all the kernels as provided in this blog post are kernels that in some sense converge closer and closer to functions with graphs that spike at the origin, with masses that are mostly concentrated at the origin, or, in other words, these kernels appear to resemble, more and more, the delta function at the origin for $n \rightarrow \infty$. However, for the Dirichlet kernels, they do not mimic the delta function sufficiently well in the limit in that property II. of approximate identity kernels is not satisfied. Alternatively, approximate identity kernels are kernels that are able to aptly mimic the delta function for $n \rightarrow \infty$.
Footnotes
[13] "Regarding the researches of d'Alember and Euler could one not add that if they knew this expansion, they made but a very imperfect use of it. They were both persuaded that an arbitrary and discontinuous function could never be resolved in series of this kind, and it does not even seem that anyone had developed a constant in consines of multiple arcs, the first problem which I had to solve in the theory of heat." - Joseph Fourier
[14] See for instance page 37 of Stein and Shakarchi's Fourier Analysis: An Introduction.
[15] See for instance page 505 of Courant and John's Introduction to Calculus and Analysis: Volume I.
[16] Actually, we have not proven that convolutions with families of kernels that are not approximations to the identity could diverge, but, for the moment, we take it as known that there exist Fourier series' that do not converge to appropriate limit functions.
References
Courant, R., & Hilbert, D. (1989). Methods of Mathematical Physics: Volume I. Wiley-VCH Verlag GmbH & Co. KGaA. (Original work published 1953)
Courant, R., & John, F. (1989). Introduction to Calculus and Analysis: Volume I. Springer New York. (Original work published 1965). https://doi.org/10.1007/978-1-4613-8955-2
Jahnke, H. N. (2003). A History of Analysis. American Mathematical Society; London Mathematical Society. (Original work published 1999). https://doi.org/10.1090/hmath/024
Lützen, J. (1979). Heaviside's operational calculus and the attempts to rigorise it. Archive for History of Exact Sciences, 21, 161-200. https://doi.org/10.1007/BF00330405
Stein, E. M., & Shakarchi, R. (2003). Fourier Analysis: An Introduction. Princeton University Press.
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