"Real numbers make possible limit operations with rational numbers, but they would be of little value if the corresponding limit operations carried out with them necessitated the introduction of some further kind of "unreal" numbers which would have to be fitted in between the real ones, and so on ad infinitum. Fortunately, the definition of real number is so comprehensive that no further extension of the number system is possible without discarding one of its essential properties." - Introduction to Calculus and Analysis: Volume I by Richard Courant and Fritz John
0 Introduction
Suppose we were provided a problem of trying to discover an optimal shape for the wheels of a vehicle traveling over terrain that is expected to be reasonably flat in an average sense.$^{[1]}$ Then, possibly from empirically informed intuition, we may suspect that the circle is most optimal in the sense that, for instance, every point on the circumference of a circle is equidistant from its center and so there may be a "continuous falling" phenomenon as a wheel rolls along reasonably flat surfaces.$^{[2]}$ Yet, there is no such thing as a "perfect circle", or, at the very least, according to the limits of modern physics as far as I am aware, there cannot be a perfect circle due to various phenomena as in the phenomenon of Heisenberg's uncertainty principle. So, at best, we would merely provide polygons, perhaps smoother and smoother, which approximate the circle in its abstract conception.
Then, a natural question arises. Could generalized approximation processes, algorithms, be provided that allow one to approximate the circle with greater and greater precision? In fact, can even more general conceptions be realized so as to extend beyond that of the circle? As a partial answer, one of the successful objects as utilized in the history of mathematics comes in the Cauchy sequences, and, in fact, equivalence classes$^{[3]}$ of Cauchy sequences have demonstrated their tremendous utilitarianism in certain more modern topics of analysis.
1 Historical Results in Approximations to Real Numbers
Recall that Cauchy sequences are sequences $\{ x_n \}_{n = 1}^{\infty}$ such that, for any arbitrarily small real $\epsilon > 0$, one can always find a sufficiently large $N$ such that for $n$ and $m$ greater than $N$, we have that $\left| x_n - x_m \right| < \epsilon$.$^{[4]}$ Intuitively, Cauchy sequences characterize the "squeezing" of convergent sequences without a priori knowledge of the limits of sequences. Regarding certain historical results in the appraisal of real numbers, it was always thought that, at least by certain mathematicians and scientists, corresponding to each point of the "real line" is a real number,$^{[5]}$ and that such real numbers can of course be appraised via sequences of rational numbers, or, more appropriately, appropriate Cauchy sequences of rational numbers, especially in light of the fact that irrational numbers can never be expressed completely if one were to, say, utilize the decimal number system in the sense that irrational numbers, in their decimal expansions, have periods that are infinite.
A natural approach then as taken by many mathematicians was of course the intuitive approach that allows the iterative intuitive appraisals of phenomena on the real line, in the sense of,
Theorem 1.1 (Nested Interval Theorem) If we were provided an infinite nested sequence of bounded closed intervals with real end-points in the sense of $\{ [a_i, b_i] \}_{i = 1}^{\infty}$ for $[a_m, b_m] \subset [a_n, b_n]$ whenever $n < m$ and such that $| b_i - a_i | \rightarrow 0$ for $i \rightarrow \infty$, then, we can always find some unique real number contained in the intersection of all such closed intervals.$^{[6]}$
PROOF.$^{[7]}$ First we demonstrate that such a real number must clearly be unique. If such a real number is not unique, then there exists, at least, $c$ and $c'$ for $c \neq c'$ in $[a_i, b_i]$ for all $i$ such that $|c - c'| > \epsilon$ for some sufficiently small real $\epsilon > 0$, but then, $b_i - a_i$ would not tend to 0 for $i \rightarrow \infty$, contradicting the provided hypothesis. Now, regarding the existence, we shall provide a proof from a certain conventional axiomatic foundation where the existence of suprema and infima are taken to exist.$^{[8]}$ So, for all $i$, we take the supremum of the set of numbers provided by $\{ a_i \}$ and the infimum of the set of numbers provided by $\{ b_i \}$ as denoted by $A$ and $B$ respectively, and such numbers certainly exist as the non-empty set $\{ a_i \}$ is bounded above by $b_1$, and, the non-empty set $\{ b_i \}$ is bounded below by $a_1$. Then, since $|b_i - a_i| \rightarrow 0$ for $i \rightarrow \infty$, we have that $A = B$ and, in fact, such a real number is the real number that is recovered in the provided intersection of the infinite nested sequence of bounded closed intervals with real end-points. $\square$
We comment, with the use of infinite nested sequences of bounded closed intervals with rational end-points rather than real end-points, it is actually possible to simply define real numbers in general, and, in particular, irrational numbers, by appropriate equivalence classes of nested intervals. If, in the limit, an infinite nested sequence of bounded closed intervals with rational end-points was to recover a rational number, then the rational number is provided. However, if, in the limit, no rational numbers are recovered, then we take the provided infinite sequence of nested intervals as, for all intents and purposes, representing a real number in general - that is, real numbers can be represented by their approximations as realized with infinite nested sequences of bounded closed intervals with rational end-points. Before we proceed to the next theorem, we remind the reader of the notion of "metric completion". A "space" is completed with respect to a metric if every Cauchy sequence converges to some object of the space in the metric. Such a definition is of course informal and various important subtleties have been missed, and we intend to touch on them very quickly before moving on, so consider two subtleties in,
I. Cauchy sequences are realized with the metrics defined. What is a completed space with respect to some metric may no longer be completed with respect to another metric.
II. In completed spaces, every Cauchy sequence is to converge to objects "of the same class", "of the same space". For example, there exist Cauchy sequences of rational numbers that do not converge to rational numbers, then, since there exists Cauchy sequences of rational numbers that do not converge to objects of the same class, the set, or class, of rational numbers, is not a completed set with respect to the conventional Archimedean metric, being, the "distance metric" of the real line.
Theorem 1.2 The real line $\mathbb{R}$ is a complete metric space with respect to the conventional Archimedean metric.$^{[9]}$
PROOF. Since every Cauchy sequence of real numbers allows one to provide nested sequences of closed intervals, we have then that every Cauchy sequence of real numbers converges to a real number and thus the completion. To see this, if we were given a Cauchy sequence, then, for any arbitrarily small real $\epsilon > 0$, one can always find some sufficiently large $N$ such that $|x_n - x_m| < \epsilon$ for $n$ and $m$ greater than $N$, that is, such a process could be iterated to provide a nested sequence of closed intervals. Thus, suppose we were given some $\epsilon_1$, then we find the smallest $N_1$ such that $|x_n - x_m| < \epsilon_1$ for all $n$ and $m$ greater than $N_1$. Then, we continue to $\epsilon_2 < \epsilon_1$, and we now find the smallest $N_2$ such that $|x_n - x_m| < \epsilon_1$ and $|x_n - x_m| < \epsilon_2$ for $n$ and $m$ greater than $N_2$. In general, for $\epsilon_{k} < \epsilon_{k - 1} < \dots < \epsilon_{1}$, we find the smallest $N_{k}$ such that $\left( |x_n - x_m| < \epsilon_1 \right) \land \dots \land \left( |x_n - x_m| < \epsilon_k \right)$ for $n$ and $m$ greater than $N_k$. We continue in such a fashion indefinitely, producing nested sequences of closed intervals with lengths $\epsilon_1 > \epsilon_2 > \dots$ for $N_1 < N_2 < \dots$, that is, with $\epsilon \rightarrow 0$ for $N \rightarrow \infty$. And so, an application of Theorem 1.1 completes the argument. $\square$
We emphasize again that real numbers in general can be characterized abstractly in the sense of their approximation schemes. Next, consider a theorem that existed even during the Middle Ages,
Theorem 1.3 Any real number can be represented decimally.
PROOF. In recalling that real numbers in general can be represented by approximation schemes that make use of rational numbers only, all that is required is to demonstrate that any rational number can be represented decimally, and that such a set of objects is dense in the set of reals. If so, then equivalence classes of Cauchy sequences are all that are required to realize the set of real numbers via decimal numbers. Indeed, without losing generality, we recall that a rational number is given by a ratio of integers $\frac{a}{b}$, and all one needs to do is to apply the Euclidean algorithm decimally in order to express such a rational number decimally. Then, an application of a variation of Theorem 1.1 completes the proof, where we use nested sequences of closed intervals with decimally represented rational end-points rather than real end-points.$^{[10]}$ $\square$
So, ultimately, in light of Theorem 1.1, Theorem 1.2, and Theorem 1.3, we can actually ascertain as a result that the set of real numbers actually suffices as a complete system of measurement in various contexts including in contexts of constructions of general approximation schemes that makes use of relatively simple integers (rational numbers can, of course, be given in terms of integers). Not merely is the set of real numbers completed, but we must repeat yet again an important point, that the set of real numbers can be realized by approximation schemes and thus a complete system of measurement has been constructed which allows the general appraisals of approximation schemes. To be more clear, Theorem 1.1 demonstrates that approximations can be provided always via systemic means that "locate" real numbers with greater and greater precision, Theorem 1.2 demonstrates completeness with respect to the extremely utilitarian Archimedean metric, and Theorem 1.3 demonstrates efficient computational feasibility in the possibility of decimal computations. So, it is unsurprising that real numbers and their decimal representations in particular have proliferated as the dominant form of computational objects.
2 A Primer in Elementary Analysis: Extensions of Certain Transcendental Functions to the Real Line
Suppose there were maps that map sets of real numbers to sets of real numbers, or, alternatively, the real line to the real line. Could we always be guaranteed that certain conventional definitions of such classes of real functions are unambiguous and well-defined as a result?$^{[11]}$ We now introduce two elementary transcendental functions, being $e^x$ and its inverse $\log(x)$ - despite their elementary nature when interpreted as functions that map from the set of rational numbers to the set of rational numbers, we would like to show how such functions, extended to functions that map from the real line to the real line, can be given as approximations in terms of their more primitive precursories that simply map from the set of rational numbers to the set of rational numbers.
Theorem 2.1 The exponential function $e^x$, initially defined on the rational numbers, can be extended to a continuous function which maps the real line to the positive real line.
PROOF. In making use of Theorem 1.1 as taken from the previous section, one can simply provide infinite sequences of nested intervals of the form $\left[ e^{a_i}, e^{b_i} \right]$ for $a_i$ and $b_i$ rational numbers,$^{[12]}$ where we recall that we have $|b_i - a_i| \rightarrow 0$ for $i \rightarrow \infty$. Then, ultimately, we are interested in demonstrating that, if $\{ x_n \}_{n = 1}^{\infty}$ is a Cauchy sequence of rational numbers, then $\{ e^{x_n} \}_{n = 1}^{\infty}$ is also a Cauchy sequence. Due to the elementary properties of the exponential function as defined on rational numbers,$^{[13]}$ we immediately have that $\left| e^{x_n} - e^{x_m} \right| = \left| e^{x_n} \right| \cdot \left| 1 - e^{x_m - x_n} \right|$. So, for arbitrarily small real $\epsilon > 0$, if we can always find some sufficiently large $N$ such that $|x_n - x_m| < \epsilon$ for $n$ and $m$ greater than $N$, then it is also clear that one can find some sufficiently large $N$ such that $\left| e^{x_n} \right| \cdot \left| 1 - e^{x_m - x_n} \right|$ is made arbitrarily small for $n$ and $m$ greater than $N$ since $\left| e^{x_n} \right|$ is bounded, and since $e^{x_m - x_n} \rightarrow 1$ for $N \rightarrow \infty$. $\square$
Theorem 2.2 The natural logarithm $\log(x)$, initially defined on rational numbers, can be extended to a continuous function which maps the positive real line to the real line.
PROOF. The argument as provided here is quite similar to that of the previous theorem. We simply realize that, given a Cauchy sequence of rational numbers $\{ x_n \}_{n = 1}^{\infty}$ and an arbitrarily small real $\epsilon > 0$, we find a sufficiently large $N$ such that, for $n$ and $m$ greater than $N$, we provide $\left| \log(x_n) - \log(x_m) \right| = \left| \log \left( \frac{x_n}{x_m} \right) \right|$, where $\left| \log \left( \frac{x_n}{x_m} \right) \right| \rightarrow 0$ for $N \rightarrow \infty$. And, as is clear, the natural logarithm being the inverse of the exponential function means that it maps the positive real line to the real line. $\square$
It is not entirely surprising that such elementary transcendental functions can be extended to the real line in making use of Cauchy sequences since, at the very least, initially, we have that these functions map from the set of rational numbers to the set of rational numbers, that is, we are merely "completing" two sets of rational numbers which are identified with one another via sufficiently well-behaved maps. In other words, we can notice that even here in topics of elementary real analysis, that elementary transcendental functions can be aptly and conveniently extended to continuous functions on the real line due simply to the fact that what were general approximation schemes can still be applied equally well to both the functional values and the domain of such functions (of course, we repeat again that these functions are sufficiently well-behaved and such is important, after all, not all functions are continuous).
And so, in this way, we notice that even functions, transcendental functions to be sure, can be extended to the set of real numbers in general via approximation schemes in the form of Cauchy sequences. We should perhaps comment very quickly prior to transitioning to the next section that such a point is not entirely trivial as transcendental functions cannot be expressed in terms of ratios of polynomials, or, as rational functions - it is obvious that ratios of polynomials as defined for the rational numbers can be easily extended continuously to the real line as all we would be doing is performing finite numbers of arithmetical operations involving real numbers. In light of the fact that rational functions can almost be trivially continuously extended to the real line (possibly excluding points of discontinuities), it no longer appears as straightforward that even the elementary transcendental functions $e^x$ and $\log(x)$ can be continuously extended to the real line as was demonstrated. After all, such functions cannot be evaluated via finite numbers of arithmetical operations involving the real numbers, and thus a non-trivial aspect of Theorem 2.1 and Theorem 2.2, their proofs made seemingly so simple with a conceptual approach that utilizes Cauchy sequences. To put the successes of Theorem 2.1 and Theorem 2.2 differently, such theorems had achieved a continuous correspondence between equivalence classes of Cauchy sequences of rational numbers, with equivalence classes of Cauchy sequences of rational numbers, where different functions will naturally provide different explicit correspondences, and different domains of definitions - that, both the exponential function and the natural logarithm provides a means of continuously mapping equivalence classes of Cauchy sequences of rational numbers to equivalence classes of Cauchy sequences of rational numbers, or, in other words, means of continuously mapping real numbers to real numbers.
We remind the reader yet again that, in this way, approximations have been provided for both $e^x$ and $\log(x)$ for the two functions as continuously extended to the real line in general, providing special cases of a realization as obtained in the previous section, that Cauchy sequences suffice completely to provide general systematic means of retrieving approximation schemes. Indeed, the fact that $\log(x_0)$ can always be approximated, for $x_0$ irrational, by a sequence $\{ \log(x_n) \}_{n = 1}^{\infty}$ corresponding to a Cauchy sequence of rational numbers $\{ x_n \}_{n = 1}^{\infty}$ is a property that was utilized by many mathematicians and scientists, and, in particular, by the astronomers. If $\log(x_0)$ was defined for some $x_0$ which fails to appear in the voluminous logarithmic tables of old, it was intuitively understood that all that was required was to provide approximations to $\log(x_0)$ for neighboring rational argument values that are reasonably close to $x_0$, those that actually appear in some logarithmic table.$^{[14]}$
Supplementarily, it is also the case that such kinds of arguments hold for the elementary transcendental trigonometric functions, and so the trigonometric functions can be continuously extended to the real line as well. In fact, the class of elementary trigonometric functions had also provided, historically, examples of functions that cannot be resolved into rational functions, and, that it was the class of elementary trigonometric functions that revealed the possibility of realizing real functions as defined on irrational numbers, albeit the algebraic irrational numbers, as realized on a geometric basis.$^{[15]}$
3 The Continuous Extension of the Fourier Transform to $\mathcal{L}^2$ Spaces
It is in this section where we can engage with potentially much less trivial topics in making use of Cauchy sequences, and, ultimately, what are essentially approximation schemes in the construction, and characterization, of objects in their furthered generalizations. In the theory of Fourier transforms, we recall that a Fourier transform $\mathcal{F}$ of a function $f(x)$ is defined to be $\mathcal{F} f = \int_{-\infty}^{\infty} f(x) e^{-2 \pi i x \xi} d \xi$. Yet, for those with sufficient training in analysis, one may immediately ask questions pertaining to the behavior of the Fourier transform, or, the integral in particular, since, if $f$ was not sufficiently well-behaved, how could one be confident of the convergence of the integral? One traditional approach is to simply define $f$ to be so sufficiently well-behaved that the definition of the Fourier transform would not result in particular issues. Then, one of the popularized classes of functions come in what is known as the class of "Schwartz functions" $\mathcal{S}$, consisting of functions $f$ that are infinitely differentiable and such that their derivatives always satisfy $\sup_{x \in \mathbb{R}} \left| \frac{\partial^k}{\left( \partial x \right)^k}f \right| \left| x \right|^n < \infty$ for the supremum taken over all $k$ and $n$.
But, given the discussions in the preceding sections, it becomes increasingly clear as to how one could attempt to generalize the Fourier transform so as to be defined on much broader classes of functions, and one of those classes is in fact provided by the equivalence class of square-Lebesgue integrable functions $\mathcal{L}^2$. Of the many approaches that are possible, here, we intend to provide a completion of $\mathcal{S}$ via equivalences classes of Cauchy sequences, and so we aim to demonstrate that the set of $\mathcal{L}^2$ functions is exactly that set which provides the completion of the class of Schwartz functions, but first, we require a classical theorem of real analysis and a particular lemma in,
Theorem 3.1 (The Lebesgue Dominated Convergence Theorem) Suppose we have a sequence of measurable functions $\{ f_n(x) \}_{n = 1}^{\infty}$ that converges almost everywhere to $f(x)$ and is bounded everywhere in absolute value by integrable $g$ on $E$ for $m \left( E \right) < \infty$, then we have that,
$$\lim_{n \rightarrow \infty} \int_E \left| f - f_n \right| dx = 0$$
And,
$$\lim_{n \rightarrow \infty} \int_E f_n dx = \int_E f dx$$
PROOF. Without losing generality, suppose that $f_n$ for all $n \in \mathbb{N}$, and $g$, are non-negative. By Egorov's theorem,$^{[16]}$ for any arbitrarily small real $\epsilon > 0$, one can always find some $E_{\epsilon} \subset E$ such that $\{ f_n \}_{n = 1}^{\infty}$ converges uniformly to $f$ on $E_{\epsilon}$ for $m \left( E \setminus E_{\epsilon} \right) <\epsilon$. And since, additionally, $g$ is integrable and we have that $f_n \leq g$ everywhere for all $n \in \mathbb{N}$,
\begin{align*} \int_E f_n dx &= \int_{E_{\epsilon}} f_n dx + \int_{E \setminus E_{\epsilon}} f_n dx^{[17]} \\ & \leq \int_{E_{\epsilon}} f_n dx + \int_{E \setminus E_{\epsilon}} g dx \end{align*}
But then, for $g$ integrable, we have that $\int_{E \setminus E_{\epsilon}} g dx$ can be made arbitrarily small as long as $m \left( E \setminus E_{\epsilon} \right)$ is sufficiently small, and since $m \left( E \setminus E_{\epsilon} \right)$ can be made arbitrarily small, we have instead then that uniform convergence on $E_{\epsilon}$ allows the interchanging of the integral with the limit on $E_{\epsilon}$. To be clear,
\begin{align*} \lim_{n \rightarrow \infty} \int_E f_n dx & \leq \lim_{n \rightarrow \infty} \int_{E_{\epsilon}} f_n dx + \lim_{n \rightarrow \infty} \int_{E \setminus E_{\epsilon}} g dx \\ &= \int_{E_{\epsilon}} \lim_{n \rightarrow \infty} f_n dx + \epsilon \end{align*}
And so, by Fatou's lemma, $\int f dx \leq \liminf_{n \rightarrow \infty} \int f_n dx$ implies $\lim_{n \rightarrow \infty}\int f_n dx = \int f dx$ since $\epsilon$ is arbitrary and since $f$ was assumed to be non-negative. Finally, we have also that,
$$\lim_{n \rightarrow \infty} \int_E \left| f - f_n \right| dx = 0$$
If $f$ is not non-negative, however, we merely need to consider a decomposition of $f$ into non-negative measurable parts in the sense of $f = f_1 - f_2$ for $f_1 = \max(f, 0)$ and $f_2 = \max(-f, 0)$. $\square$
Lemma 3.1 For $f \in \mathcal{L}^2$ a bounded function with compact support, and for $g$ a non-negative bounded infinitely differentiable function with compact support that also satisfies $\int g(x) dx = 1$, we have that, for $\epsilon \rightarrow 0$, $f$ converges to $f \ast \left( \frac{1}{\epsilon} g(\frac{x}{\epsilon}) \right)$ in $\mathcal{L}^2$. Additionally, we have that $f \ast \left( \frac{1}{\epsilon} g \left( \frac{x}{\epsilon} \right) \right)$ is infinitely differentiable with compact support for all $\epsilon$.
PROOF. First, we demonstrate that the family of functions as provided by $g_{\epsilon}(x) = \frac{1}{\epsilon} g(\frac{x}{\epsilon})$, as parameterized by $\epsilon$, provides what is an approximate identity family of kernels,$^{[18]}$ and since one of the properties is clearly satisfied already by hypothesis due to the property $\int g(x) dx = 1$, that $\int g_{\epsilon}(x) dx = 1$ is clear by opting for the substitution $u = \frac{x}{\epsilon}$, the only thing remaining is to demonstrate the other property of approximate identity kernels. The other property is, that $g_{\epsilon}(x)$ approaches 0 everywhere with the exception of the origin, which can be stated as being intuitively clear for the moment, that the mass of $\frac{1}{\epsilon}g \left( \frac{x}{\epsilon} \right)$ is increasingly concentrated at the origin for $\epsilon$ increasingly small. Indeed, as $g$ has compact support, and since we are working with the real line, its support is closed and bounded. And so, if we scale its support by $\frac{1}{\epsilon}$, for $\epsilon \rightarrow 0$, we have that we are concentrating its support increasingly around, and ultimately at, the origin. In obtaining an approximate identity family of kernels, we conclude $f$ converges to $f \ast \left( \frac{1}{\epsilon} g(\frac{x}{\epsilon}) \right)$ in $\mathcal{L}^2$.$^{[19]}$ Next, we make use of the Lebesgue dominated convergence theorem to allow differentiation under the integral sign as $f \ast g_{\epsilon}$ is uniformly bounded with compact support for all $\epsilon$, meaning,
\begin{align*} \left| f \ast g_{\epsilon} \right| & \leq \left| \int f(x - u) g_{\epsilon}(u) du \right| \\ & \leq \left| \sup f \right| \cdot \left| \int g_{\epsilon} du \right| \\ &= \left| \sup f \right|\end{align*}
We can then exploit the symmetric property of the convolutional operator in the sense of $\int f(x - u) g_{\epsilon}(u) du = \int f(u) g_{\epsilon}(x - u) du$ in order to relegate differentiation always to the differentiation of the infinitely differentiable function in $g$ under the integral sign in the sense of $\frac{d}{dx} \left( f \ast g_{\epsilon} \right) = \int f(u) \frac{d}{dx} g_{\epsilon}(x - u) du.$ $\square$
We comment that there is a subtlety in the above in that not merely are we interested in the convergence of $f \ast g_{\epsilon}$ to $f$ for $\epsilon \rightarrow 0$, but, we are interested in the infinite differentiability of $f \ast g_{\epsilon}$ for all $\epsilon$, that is, that one could always provide smooth approximations to $f \in \mathcal{L}^2$ given appropriate hypotheses. And, for the smooth approximations to be bounded and of compact support allows one to determine, almost in a straightforward fashion, that they are Schwartz functions. Next, we transition to the main theorem of this section in,
Theorem 3.2 The space of Schwartz functions as defined on the real line $\mathcal{S}(\mathbb{R})$ is dense in the equivalence class of square-Lebesgue integrable functions $\mathcal{L}^2(\mathbb{R})$ with respect to the $\mathcal{L}^2(\mathbb{R})$ norm.
PROOF. We intend to demonstrate that $f$ can be approximated by Schwartz functions for $f \in \mathcal{L}^2$, that, ultimately, for any $f \in \mathcal{L}^2$, there always exists some sequence of Schwartz functions that converges to $f$ with respect to the $\mathcal{L}^2(\mathbb{R})$ norm. So, consider a class of functions $F_n$ which is defined to be $F_n = f(x)$ for $\left( \left| f(x) \right| \leq n \right) \land \left( |x| \leq n \right)$ and 0 everywhere else. Such a function clearly converges to $f$ for $n \rightarrow \infty$, and it is clear that we can use $F_n \ast g_{\epsilon}$ to provide the sequences where $g_{\epsilon}$ is defined in Lemma 3.1. In fact, for all $n \in \mathbb{N}$, $F_n$ is bounded with compact support, and so an application of Lemma 3.1 allows us to provide a "regularization" in the sense that, we provide the result in $f = \left( \lim_{n \rightarrow \infty}\right) \left( \lim_{\epsilon \rightarrow 0}\right) F_n \ast g_{\epsilon}$. To be clear, $F_n \ast g_{\epsilon}$ belongs to the class of Schwartz functions and converges to $f$, in the $\mathcal{L}^2$ sense, for $n \rightarrow \infty$ and $\epsilon \rightarrow 0$. $\square$
For further supplementary remarks, the purpose of providing Lemma 3.1 above was simply due to the fact that even if we are able to provide sequences that converge to $f \in \mathcal{L}^2$, there was no guarantee that such sequences of functions are infinitely differentiable. Lemma 3.1 merely provides the necessary regularization to specifically guarantee that there always exists some sequence of Schwartz functions that converge to $f$ in Theorem 3.2, and, in fact, as is clear, that there always exists some relevant Cauchy sequence. Finally, the only task that remains is to continuously extend the Fourier transform to $\mathcal{L}^2$, and such a task can be finalized specifically by defining the Fourier transform more generally as $\mathcal{F}f = \lim_{n \rightarrow \infty} \mathcal{F} g_n$ for the $g_n$'s appropriate Schwartz functions which approximate $f$ (however, the "definition" is incomplete by itself as, for instance, well-definedness needs to be proven). Alternatively, if we had wanted to explicitly make use of Cauchy sequences, we can then instead investigate whether $\| \mathcal{F}f_n - \mathcal{F}f_m \|_{\mathcal{L}^2}$ can be made arbitrarily small for $f_n$ and $f_m$ belonging to the same Cauchy sequence. Fortunately, such is actually the case due in part to Plancherel's theorem, where we state an elementary variation in,
Theorem 3.3 (Plancherel's Theorem) For $f \in \mathcal{S}(\mathbb{C})$, we always have that $\| \mathcal{F}f \| = \| f \|$.
PROOF. Consider $F(x) = f(x) \ast \overline{f(-x)} = \int_{-\infty}^{\infty} f(x - u) \cdot \overline{f(-u)} du$, then, we have that.
$$\mathcal{F}F = \mathcal{F} \left( f (x) \ast \overline{f(-x)} \right)$$
$$\hspace{1.3cm} = \mathcal{F}(f(x)) \cdot \mathcal{F}(\overline{f(-x)})$$
$$\hspace{-1.6cm} = \left| \mathcal{F}f \right|^2$$
Additionally, we also have that,
$$F(0) = \int_{-\infty}^{\infty} f(-u) \cdot \overline{f(-u)} du$$
$$\hspace{-1.05cm} = \int_{-\infty}^{\infty} |f|^2 du$$
Now, we apply the inverse Fourier transform to $F(x)$ as evaluated on $x = 0$ to give,
$$\int_{-\infty}^{\infty} \mathcal{F}F d \xi = F(0)$$
Substituting $\mathcal{F}F = \left| \mathcal{F}f \right|^2$ and $F(0) = \int_{-\infty}^{\infty} |f|^2 du$ into the above then finalizes the argument. $\square$
The previous theorem however does not suffice in the context of Fourier transforms as defined on $\mathcal{L}^2$ spaces. So we consider a more general version in,
Theorem 3.4 (Plancherel's Theorem for $\mathcal{L}^2$ Spaces) Given a Fourier transform $\mathcal{F}$ that maps from $\mathcal{L}^2$ to $\mathcal{L}^2$, we have that $\| \mathcal{F}f \|_{\mathcal{L}^2} = \| \mathcal{f} \|_{\mathcal{L}^2}$.
PROOF. First, we state the inequality $\| \mathcal{F}(f) \|_{\mathcal{L}^2} \leq c \| f \|_{\mathcal{L}^2}$ for some real $c$ since $f \in \mathcal{L}^2$. Then, we consider the set of Schwartz functions $\mathcal{S}$ which is everywhere dense in $\mathcal{L}^2$, where we recall that $\mathcal{F}$ as defined on $\mathcal{S}$ is completely unambigious. So, given a Cauchy sequence of Schwartz functions $\{ f_n \}_{n = 1}^{\infty}$, we make use of the inequality $\| \mathcal{F}(f) \|_{\mathcal{L}^2} \leq c \| f \|_{\mathcal{L}^2}$ to give,
$$\| \mathcal{F}(f_n) - \mathcal{F}(f_m) \|_{\mathcal{L}^2} \leq c \| f_n - f_m \|_{\mathcal{L}^2}$$
Since $\mathcal{L}^2$, being a Hilbert space, is a complete metric space, we conclude that Cauchy sequences of objects $\mathcal{F}(f_n)$, given Cauchy sequences of Schwartz functions $\{ f_n \}_{n = 1}^{\infty}$, always converge to some object of $\mathcal{L}_2$. But then, due to the more prototypic Theorem 3.3 as provided previously, in combination with the continuous extension of $\mathcal{F}$ from $\mathcal{S}$ to the entirety of $\mathcal{L}^2$, the argument is finalized. For further clarity, since, for any $f_0 \in \mathcal{L}^2$, one can always find sequences of Schwartz functions $\{ f_n \}_{n = 1}^{\infty}$ that converge to $f_0$ and such that $\| \mathcal{F}f_n \| = \| f_n \|$, the continuity of $\mathcal{F}$ on the $f_n$'s gives $\left( \lim_{n \rightarrow \infty} \| \mathcal{F}f_n \| = \lim_{n \rightarrow \infty} \| f_n \| \right) \Rightarrow \left( \| \mathcal{F} \lim_{n \rightarrow \infty}f_n \| = \| \lim_{n \rightarrow \infty}f_n \| \right) \Rightarrow \left( \| \mathcal{F}f_0 \| = \| f_0 \| \right)$. $\square$
What have we actually demonstrated? Recall that, from the section titled 0 Introduction, it was suggested that perhaps one could look for "general" schemes of approximations, and such a scheme was found, and utilized, throughout sections 1, 2, and 3 in the Cauchy sequences. Then, the utilitarian aspect of the Cauchy sequence was made use of so as to persistently characterize much wider classes of mathematical objects in general, whether they be numbers, functions, or functionals as defined on functions. It did not matter that the step from the set of rational numbers to the set of real numbers was possibly too dramatic, or, that the step from transcendental functions as defined on rational numbers, to becoming defined on the set of real numbers, was possibly too dramatic. It did not even matter that the Fourier transform would be extended, possibly so dramatically, to much wider classes of functions that may threaten its convergence. As, ultimately, all these wider classes of mathematical objects can always be realized via the equivalence classes of approximations to them, as given in terms of much more fundamental and well-behaved mathematical objects. Thus, entire subfields of mathematics and mathematical analysis in particular, whether in such fields as Fourier and Harmonic analysis or the theory of generalized functions and topological vector spaces have been founded on such early conceptions, that one must not forget that one of the seeds was provided in these general approximation schemes of old.
Incidentally, regarding the extension of the Fourier transform to the $\mathcal{L}^2$ space as initially defined on the class of Schwartz functions, such an approach was realized, intuitively, by even the physicists and engineers of the 19th century since the defining of the Fourier transform on less restrictive classes of functions was considered to be so practical in "reality", that certain figures had not even waited for the rigorous development of the theory of generalized functions that would be provided by the likes of Schwartz or Sobolev.$^{[20]}$
In remembering the discussion regarding the circle as provided in the section titled 0 Introduction, we will return to the discussion of such topics and approach them from the perspective of optimization for certain problems of physics, the sciences in general, and engineering. We comment supplementarily that there, of course, exist much more general forms of completions of metric spaces via Cauchy sequences, but such general variations will be discussed in future blog posts.
FOOTNOTES
[1] Average in what sense? For the moment, many simple definitions are possible, as in definitions that may involve integral averages of curvatures of surfaces. But, for the purpose of this blog post, we do not intend to discuss differential geometric phenomena.
[2] If surfaces were not reasonably flat in a reasonable average sense, then other shapes may become optimal. For instance, consider square wheels over certain surfaces with periodic indentations. In fact, for a certain surface consisting of an appropriate kind of periodic indentations, the corners of the square wheels trace out inverted catenaries as they roll along such a surface.
[3] In many contexts, what matters is not so much the particular Cauchy sequences themselves, but the equivalence classes of them. In an analogous fashion, in many contexts, it is not the particular representations of rational numbers that are of interest, but the equivalence classes of rational numbers, that, for instance, $\frac{1}{2}$ belongs to the same equivalence class as $\frac{2}{4}$, $\frac{4}{8}$, and so on.
[4] We note that such is not the most general definition of Cauchy sequences. For instance, as is well known, Cauchy sequences could be defined with the more general notion of a metric.
[5] Consider the various illustrations of approximations of real numbers via sequences of rational numbers with the use of continued fractions, or infinite series, in such historical documents as Newton's The Method of Fluxions and Infinite Series: Applications to the Geometry of Curve-Lines.
[6] We require that the intervals be closed, since, if the intervals were open, then it may just so happen that the infinite intersection of all such open intervals produces the empty set. Incidentally, such a result is one of the classic results of analysis that allows one to recall, for instance, that countable intersections of closed sets produce closed sets, where, in such a context of nested intervals, we have that the set as produced is a closed singleton set.
[7] In truth, many such kinds of "proofs" can be somewhat ambiguous if the axiomatic system is not agreed upon. Indeed, in our provided proof, notice that, by supposing that one can always find some sufficiently small real $\epsilon$ such that $|c - c'| > \epsilon$ for $c \neq c'$, one is assuming that there is no such thing as a "number" that is smaller than any arbitrarily small real number, in the sense that there does not exist some kind of an "infinitesimal" $\epsilon$ such that $c + \epsilon = c'$ for $c \neq c'$. Incidentally, such assumptions had provided the axiomatic foundations of an early style of infinitesimal calculus, a style that we do not adopt. For more information, please see a fascinating annotated historical exposition in L'Hôpital's Analyse des infiniments petits: An Annotated Translation with Source Material by Johann Bernoulli - indeed, compare with Courant's quote as cited at the beginning of this blog post. Additionally, many subtleties were not explicitly appraised, as in the subtlety of the fact that such sequences of nested intervals are obviously indexed by the countable set of natural numbers - recall that, for instance, according to the axioms of general topology, an arbitrary intersection of closed sets produces a closed set.
[8] Actually, a conventional axiomatic system in analysis is the ZFC set-theoretic foundation. But, for the purpose of our proof, the Peano axioms will even suffice - although the Peano axioms do not engage with certain subtleties of general topology and the uncountable, our proof makes the transition to the countably infinite rather than the uncountably infinite. For slightly more clarity, we may sketch some suggestions here. The Peano axioms provide the natural numbers, then, with the introduction of an additional "minus" arithmetic map $-$, we transition to the integers as appropriate equivalence classes of natural numbers. After, we provide rational numbers via appropriate equivalence classes of integers, and, transition to the real numbers via appropriate equivalence classes of rational numbers, where the existence of suprema is realized - indeed, a popular approach is to realize equivalence classes of rational numbers in using "Dedekind cuts" with no explicit references to Cauchy sequences.
[9] In identifying such a notion as an "Archimedean metric", we acknowledge that there may exist non-Achimedean metrics where attempts of metric completions will produce different kinds of mathematical spaces. An example is of course provided by the set of p-adics, although, according to Ostrowski's theorem, up to an isomorphism, the set of p-adics is the only alternative to the set of reals as a metric completion of the set of rational numbers.
[10] Despite the feigned simplicity of the supposed "proof", many subtleties have been missed, culminating in millenniums of mathematical developments in such topics as arithmetic, algebra, the theory of continued fractions, and so on. For instance, Joseph's The Crest of the Peacock: The Non-European Roots of Mathematics documents various algorithms of iterated decompositions of numbers that had existed during the Middle Ages.
[11] In our current context, well-definedness requires that $[x = x'] \rightarrow [f(x) = f(x')]$. In real analysis, a standard approach to achieve well-definedness is to make use of the uniqueness of the limit, that, limiting operations are utilized as a device to achieve well-definedness. In the context of Theorem 2.1 and Theorem 2.2, the limiting operations are provided by relevant Cauchy sequences of rational numbers. In other subfields as those of topics of abstract algebra where there are no topological bases, it may be the case that there are no limiting operations, and well-definedness is typically achieved differently.
[12] Although Theorem 1.1 is stated specifically for real end-points, we remind outselves that rational numbers are also real numbers, that, all rational numbers are real numbers, but not all real numbers are rational numbers.
[13] For instance, see page 117 of Zorich's Mathematical Analysis I. In fact, it is also the "elementary properties" of the exponential function, or, more generally, of $a^x$ for $a > 0$ real and $x$ rational, that allows us to conclude that $e^x$ only maps to the positive real line rather than the entire real line. The elementary properties of $\log(x)$ as defined on the set of rationals can be realized similarly.
[14] It was apparently the case that, historically, in various academic subcultures, trigonometric and logarithmic tables would become very voluminous due to millennia of astronomical data. Although I would love to provide some interesting historical references, including material related to the House of Wisdom, I feel that there is a lack of rigor at the moment. Possibly for future reference, consider The Verified Tables, consisting of astronomical data as compiled by the House of Wisdom during the 9th century - note that the House of Wisdom could simply refer to an academic subculture that had existed, rather than actual physical infrastructure.
[15] Consider the fascinating history surrounding the number $\sqrt{2}$.
[16] See for instance Theorem 1.2 of the blog post titled Some Desirable Properties of Measurable Sets & Functions in Real Analysis: The Littlewood Heuristics & Applications.
[17] A few subtleties are not discussed here. For example, such a decomposition is made possible by the fact that we are dealing with Lebesgue integrals as defined on measurable $E_{\epsilon}$ and $E \setminus E_{\epsilon}$. For more information, please see the blog post titled Some Desirable Properties of Measurable Sets & Functions in Real Analysis: The Littlewood Heuristics & Applications.
[18] For more information on "approximate identity family of kernels", please see the blog post titled Approximations via "Filters" in Convolutional Operators & Some Explicit Results.
[19] For example, see Theorem 2.1 of Approximations via "Filters" in Convolutional Operators & Some Explicit Results. But actually, that theorem is not sufficient as it assumes more than we had here. For those who are interested in further rigor, a generalization of Theorem 2.1 of Approximations via "Filters" in Convolutional Operators & Some Explicit Results can be found in 3.11.12. Corollary of Bogachev & Smolyanov's Real and Functional Analysis, page 121, where all that was assumed was $\mathcal{L}^p$ integrability for $\mathcal{L}^p$ convergence. Additionally, Theorem 2.1 of Stein & Shakarchi's Real Analysis: Measure Theory, Hilbert Spaces, & Integration, page 112, provides a variation for $\mathcal{L}^1$. Although one feels perhaps just a little lazy in not providing further material, these are blog posts, not a mathematical exposition.
[20] Although we could provide some references here, we briefly mention the well-known stories of the mathematical engineer Heaviside, and the mathematical physicist Dirac.
REFERENCES
Bogachev, V. I., & Smolyanov, O. G. (2020). Real and Functional Analysis. Springer.
Courant, R., & Fritz, J. (1989). Introduction to Calculus and Analysis: Volume I. Springer New York. (Original work published 1965)
Stein, E. M., & Shakarchi, R. (2003). Fourier Analysis: An Introduction. Princeton University Press.
Stein, E. M., & Shakarchi, R. (2005). Real Analysis: Measure Theory, Hilbert Spaces, & Integration. Princeton University Press.
Zorich, V. A. (2016). Mathematical Analysis I (2nd ed.). Springer Berlin.
Zorich, V. A. (2016). Mathematical Analysis II (2nd ed.). Springer Berlin.
