A proposal to characterize and quantify superoscillations

We present a formal definition of superoscillating function. We discuss the limitations of previously proposed definitions and illustrate that they do not cover the full gamut of superoscillatory behaviors. We demonstrate the suitability of the new proposal with several examples of well-known superoscillating functions that were not encompassed by previous definitions.


I. INTRODUCTION
The concept of superoscillations originated in [1] (although it was anticipated in [2][3][4][5]), and its mathematical foundations were first laid out in [6].A great part of their appeal resides on their paradoxical nature: a bandlimited function (i.e., a function with a bounded Fourier spectrum) can, in a bounded interval, oscillate faster than its fastest Fourier component.Even more strikingly, this "superoscillatory behaviour" can be arbitrarily fast, and the intervals over which this phenomenon happens can be arbitrarily long (see, e.g., [7]).
The properties of superoscillations have been the object of extensive study, both from a mathematical [52][53][54][55][56][57][58][59][60][61][62][63] and a physical [64][65][66][67][68][69][70][71][72][73][74][75][76][77] perspective.However, even with all the progress that has been made so far, a clear-cut, rigorous definition of what it means for a function to display superoscillations is still arguably missing.One research line that has approached the problem of studying the mathematical properties of superoscillations (in a systematic, mathematically rigorous way) resorts to defining the concept of superoscillating sequences (see, e.g., [54,78]), rather than defining what it means for a given, single function to display superoscillations.However, we will argue that this way of approaching superoscillations can be too restrictive.Most of the other lines of study, as far as we are aware of, have traditionally either used the most compelling and graphically self-evident examples as guiding principles (such as [13,56,69]), or used a notion of 'local wavenumber' that leads to a definition [52,53,67] that suffers from limitations and ambiguities, and does not identify superoscillations in some simple examples, as we will see in this paper.
Here, we propose a formal definition of what it means for a real-valued bandlimited function to display superoscillatory behaviour in a particular interval.This definition can be applied unambiguously to decide if any given function is superoscillating in any specified interval.We will also argue that the definition we introduce captures the physically relevant notion of superoscillations.
The paper is organized as follows: in Sec.II, we review the two main descriptions of superoscillations that have been previously used in the literature, and discuss the problems they exhibit.In Sec.III, we present two criteria to identify superoscillations, and we then use them to formulate a rigorous definition of superoscillating function in an interval.Sec.IV is dedicated to illustrating how the proposed definition identifies and quantifies superoscillations by testing it against several examples.Finally, we present our conclusions in Sec.V.

II. THE CHALLENGE OF DEFINING SUPEROSCILLATIONS
Perhaps surprisingly, after over 30 years of study, there is still no consensus on a rigorous definition of superoscillating function.In fact, the concept of superoscillations is usually introduced in the literature in a predominantly qualitative way.The reason for this is probably that it is actually very easy to provide an intuitive notion of superoscillation: a superoscillating function is a bandlimited function (i.e., with bounded Fourier transform) that exhibits a region (the superoscillating interval or set) where it oscillates with a frequency outside of its bandwidth.More colloquially, we would say that there are intervals in the graph of the function where it oscillates "faster than its fastest Fourier component" [7].This intuitive notion is often complemented with the most preeminent example of superoscillating function [1,7,66]: As can be seen explicitly from Eq. ( 2), g is bandlimited to the interval [−1, 1].However, from Eq. ( 1) it is also easy to see that near the origin g(x, a, N ) ≈ e iax .Thus, for |a| > 1, f will exhibit superoscillations in a neighbourhood of x = 0, which is usually estimated to be the region |x| < √ N (see, e.g., [7,66]).The superoscillatory behaviour of this family of functions is illustrated in Fig. 1, where Re{g(x, 2, 1000)} and cos(2x) are shown to oscillate synchronically around x = 0.
One particular feature of the family of examples given by g(x, a, N ) is the fact that lim This convergence is, of course, pointwise in the whole real line, but it is also uniform when we restrict ourselves to a fixed compact interval.This property motivated the introduction of the concept of superoscillating sequence, which paved the way for a significant amount of the past and ongoing investigation of superoscillations from the mathematical point of view (see, e.g., [54,78] for reviews): following [78], a generalized Fourier sequence {Y n } is a sequence of functions of the form where n ∈ N, x ∈ R, a ∈ R + , and C j and k j are realvalued functions, respectively.Such a sequence is then considered a superoscillating sequence in a compact set This definition, while formal, is still not a definition of superoscillating function.It defines specific classes of functions with a special property, rather than a property that a given function may or may not have.Furthermore, even within the study of sequences of functions, the condition given by Eq. ( 5) is somewhat restrictive1 , since it does not allow the sequence to converge to any oscillating function, but specifically a monochromatic one.This, as we will see, excludes from the definition some functions that are arguably superoscillating from the qualitative point of view.
On the other hand, a popular attempt to characterize superoscillations for a generic function that relies on the qualitative notion of superoscillations was first considered in [6], and has been further used either as a definition or as a measure in later references, such as, e.g., [52,53,66,67,74]: in these references, a complex function u is defined to be superoscillating if its local gradient, or local wavenumber, is higher than the frequency of its fastest Fourier component.|k(x)| is then suggested as a measure of the degree of superoscillatory behaviour: the larger |k(x)|, the more significant the superoscillation.This definition has the clear advantage of its simplicity, and that it may, at first sight, seem to capture the intuition of what superoscillations ought to look like.However, this definition has three critical issues.The first issue is that the local wavenumber defined in Eq. ( 6) is identically zero when evaluated for real-valued functions, regardless of their behaviour.Yet, we should be able to talk about a real-valued function being superoscillating, just in the same way we say that the real part of g(x, a, N ) is superoscillating (cf.Fig. 1).For physical superoscillations in, e.g., optics, the definition has been used by expressing the oscillating function as the real part of a complex-valued function, and then applying this definition to the latter (see, e.g., [67]).However, this procedure is ambiguous, since a real-valued function can have more than one complex-valued function of which it is the real part, and each of these complex functions can yield a different value of the local wavenumber: consider, for example [56], for m, n ∈ Z.Then, we have that and This illustrates the ambiguity of this way of defining the local wavenumber for real-valued functions (and therefore of this definition of superoscillatory behaviour), also casting doubts on the ability to use |k(x)| as a measure of the degree of superoscillation.
The previous issue could still be addressed by fixing a specific method to construct a complex-valued function from the real signal: for instance, it is standard in signal theory to obtain the local phase of a real function using the Hilbert transform [79].Specifically, given an analytic real function v(x), we can prescribe the complex-valued function u associated with v to be where H[v] denotes the Hilbert transform 2 of v.This particular choice has indeed been used in the context of superoscillations (see, e.g., [82]).However, even if we use this method, there is a second issue that this definition does not address: it still does not capture some superoscillatory behaviours.To illustrate this, consider once more the function in Eq. ( 7), in the particular case when m = 1 and n = 2. From Eq. ( 12), we find that the complex-valued function associated with h(x, 1, 2) according to the Hilbert transform method is the one given precisely by Eq. ( 9), i.e., u 2 (x, 1, 2).Eq. ( 11) then yields a constant local wavenumber, k(x) = 2, while h(x, 1, 2) is easily seen to be bandlimited to the interval [ −3, 3].This means that the definition of superoscillations in terms of the local wavenumber would not classify it as a superoscillating function.However, in Fig. 2 we can see that h(x, 1, 2) oscillates almost synchronically with a function proportional 3 to sin(4x), whose frequency is outside of the Fourier spectrum of h(x, 1, 2), around x = π/2.This leads to the conclusion that h(x, 1, 2) is indeed a superoscillating function, and yet its superoscillatory behaviour is not captured by the local wavenumber.To reinforce this point, consider also the square of a shifted 2 The Hilbert transform of a function v is defined as [80] H and its Fourier transform satisfies [81] F cosine, which was given as an example of superoscillating function (around x = 0) in [7,46,83]: for some 0 < s < 1.We can write where the complex-valued function inside the real part was obtained using the Hilbert transform method in Eq. (12).Using this prescription to evaluate the local wavenumber, it can be shown that for all x ∈ R.This means again that with this definition of superoscillations we could end up classifying this widely-accepted superoscillating function as nonsuperoscillating.
The third problem of the definition of superoscillations based on the local wavenumber is precisely its local nature, which implies that it can generally apply to superoscillating intervals that are "too short", in the sense that the function oscillates much less than half a cycle.For instance, it was shown in [52] that one-dimensional monochromatic waves can exhibit superoscillations according to this definition, but these happen very briefly in the neighbourhood of points where the function is close to zero.Arguably, we should not expect to be able to use this kind of extremely brief superoscillation in practical applications, such as, e.g., encoding information [22,24], or superresolution [32,43].Intuitively, the reason for this is that, even though it is not impossible that some of these functions are used to, e.g., encode some messages-in the same way a string of zeros can encode one message-, it seems reasonable to expect that to be able to encode an alphabet in a wave with a certain rate, one needs at least half-wave oscillations within the distance between one sample and the next one, otherwise an encoding scheme that uses superoscillations can hardly take advantage of them to increase the information density.This is even more so for optical applications of superoscillations such as superresolution, where overcoming the diffraction limit relies on having shorter but full wavelengths for some period of time.Because of this, we argue that those functions displaying a superoscillatory behaviour that is not long-lived enough should not be considered superoscillating functions for practical purposes.
This analysis reveals that there is room to propose a new definition that can be universally applied to any given function.This is the problem that we address in the following section.

III. IDENTIFYING SUPEROSCILLATIONS
In this section, we introduce a formal definition of superoscillatory behaviour that does not exhibit the problems of previous proposals.First, we will introduce the two criteria on which the definition relies.As we will see, these two criteria allow us to discriminate whether a given real-valued one-dimensional function is superoscillating or not in certain intervals.The definition then generalizes this to arbitrary intervals.
The first criterion is designed to capture superoscillations around a specific horizontal reference line, y = c.
for each k ∈ N, be the Fourier sine coefficients over the interval [b 1 , b 2 ] around the horizontal line y = c.If we denote with k 0 ∈ N the smallest natural number that satisfies k 0 > W (b 2 − b 1 ), we define Then, we say that f is superoscillating around Notice that the Fourier sine coefficients evaluated according to Eq. ( 18) are exactly the Fourier coefficients of a periodic function with period (at most) 2(b 2 − b 1 ) that is odd with respect to the vertical line x = b 1 .In a way, we are evaluating a "local Fourier spectrum" 4 of the function with the information available in the interval [b 1 , b 2 ].Thus, the quantity Q sin is simply the ratio between two quantities: i) the (2-norm5 ) weight of the local Fourier sine coefficients associated with frequencies outside the total Fourier spectrum of the function f , and ii) the weight of all the local Fourier sine coefficients together.The criterion then establishes that if the superoscillating contributions (i.e., the ones with a frequency beyond the bandlimit) stand for more than half the weight of the contributions of all the Fourier sine coefficients, then the function should be considered to display a superoscillatory behaviour around It is worth remarking that this criterion cannot be applied to just any interval, but specifically, those for which the function has value c at its endpoints.Thus, the sine criterion alone might not be enough to characterize superoscillations when there is not a specific horizontal line of reference around which the function oscillates.That is why we introduce a second criterion, designed to capture superoscillations precisely in such scenarios.
Criterion 2 (Cosine criterion).Given a real-valued one-dimensional function f which is bandlimited to Then, we say that f is superoscillating in the interval Here, the Fourier cosine coefficients evaluated according to Eq. ( 20) are exactly the Fourier coefficients of a periodic function with period (at most) 2(b 2 − b 1 ) that is even with respect to the vertical line x = b 1 .In a way, we are again evaluating a "local Fourier spectrum" 6 of the function with the information available in the interval [b 1 , b 2 ].The quantity Q cos is, in a completely analogous fashion, a ratio between the weight of the Fourier coefficients associated with superoscillating modes and the total weight of all the coefficients together except for the zero mode (which accounts for the possible vertical translation of the function, without altering its spectrum elsewhere).Then, we consider the function to be superoscillating in the interval if the superoscillating modes have more overall weight in the local Fourier decomposition than the ones within the bandlimited Fourier spectrum of the function.This criterion, which can only be applied to intervals whose endpoints are local extrema of the function, seems intuitively less restrictive than the sine criterion.The sine criterion is indeed tailored for those situations in which the only oscillations of interest are about a particular horizontal line, while the cosine criterion allows for oscillations that do not necessarily have a prespecified horizontal line they oscillate about.Nevertheless, the cosine and the sine criteria are generally applied to different intervals, and therefore it is not possible in principle to prove an inequality that establishes a hierarchy between them.
Once we have introduced the sine and cosine criteria, we can give the following definitions of superoscillating function.

Definition (Superoscillating function in an interval).
A real-valued one-dimensional bandlimited function f is considered to display superoscillations around y = c in an interval I if there exist intervals I 1 and I 2 such that I 1 ⊆ I ⊆ I 2 , and the sine criterion for y = c is satisfied for both I 1 and I 2 .Alternatively, f is considered to display superoscillations in I if there exist intervals I 1 and I 2 such that I 1 ⊆ I ⊆ I 2 , and the cosine criterion is satisfied for both I 1 and I 2 .
As a remark, note that the inclusions I 1 ⊆ I ⊆ I 2 are not necessarily strict, and therefore if I is an interval that satisfies one of the two criteria, then the definition is automatically satisfied by choosing This definition addresses at first glance some of the issues of the traditional criteria, since 1) it is well defined for any given real-valued one-dimensional bandlimited function, and 2) they can only answer affirmatively the question of whether a function is superoscillating or not in intervals that are long enough to capture at least half a (super)oscillation.This property rules out by fiat those functions that display superoscillations that are too short-lived to be of practical use.Moreover, in the next section we will see that this definition identifies superoscillatory behaviours that the other definitions do not capture.We will also see that, since Q sin and Q cos represent overall contributions of the superoscillating modes (and in particular they are exactly zero for sines and cosines), these values can be used as measures of the degree of superoscillatory behaviour displayed by the function of interest.
Notice that this definition is intended to capture superoscillations about horizontal lines (whether specified, as in the sine criterion, or unspecified, as in the cosine one).However, it is possible to readily generalize the defini-tion to account for superoscillations about sloped lines, or even around an arbitrary polynomial p(x) by virtue of applying the criteria on the function f (x) = f (x) − p(x).

IV. EXAMPLES
In this section, we apply the definition of superoscillating function given in Sec.III to a variety of examples, both superoscillating and non-superoscillating, to check how it correctly captures the qualitative description of whether a function has superoscillations or not in a particular interval.For greater clarity, we will discuss the sine and cosine criteria separately.
For the sake of simplicity, here we only consider examples in which the oscillation happens about y = 0. Also for convenience, we will analyze functions that are odd using the sine criterion and those that are even using the cosine criterion.

A. Sine criterion
The first example that we consider here is the imaginary part of g(x, a, N ), as defined in Eq. ( 1).In particular, we will examine the cases N = 10 and N = 20, for a = 2, which are represented in Figs.3a and 4a.These functions are bandlimited to [−1, 1], but they are known to display superoscillations around x = 0, as the figures illustrate.For these functions, we evaluated Q sin in intervals of the form [0, b], for several consecutive zeros b > 0 as right endpoints.Notice that since these functions are odd, the value of Q sin in the intervals [0, b] and [−b, b] is the same.The results for Q sin are plotted in Figs.3b and 4b, and support that the functions are superoscillating around x = 0. Notice that even though the deviation of these functions from the monochromatic sin(2x) is evident relatively soon (after the first or second zero), the value of Q sin witnesses superoscillatory behaviour for much longer than this.The reason is that even if these functions stop resembling a monochromatic function relatively close to the origin, they superoscillate (i.e., locally oscillate faster than the fastest Fourier component) for longer than this.This behaviour is captured as superoscillating by the definition we propose here, but is missed by the definition based on superoscillating sequences, which estimates the superoscillating behaviour to happen only where the function is approximately monochromatic (i.e., the region which is called in [66] of 'fast superoscillations', corresponding here to |x| < √ N ).An even better example of how the definition based on superoscillating sequences could not capture general enough superoscillatory behaviours is the family of functions given by the sum g(x, 2, N ) + g(x, 3, N ), which is bandlimited to [−1, 1], and whose imaginary part for the case N = 20 is represented in Fig. 5a.Clearly, lim which is not a monochromatic function.Yet, Fig. 5a shows that Im{g(x, 2, N ) + g(x, 3, N )} oscillates synchronically with 2 sin 5x/2 around x = 0, and this superoscillatory behaviour is corroborated by the values of Q sin plotted in Fig. 5b.Notice that, near x = 0, Thus, close enough to the origin the cosine factor is approximately 1, and the function can be approximated just by the sine.This explains the synchronic oscillation of the function and 2 sin 5x/2 , but also makes it clear that, even if it is small, the cosine factor prevents the sequence g(x, 2, N ) + g(x, 3, N ) from converging to 2 sin 5x/2 in any compact interval containing x = 0, even though it resembles to it.
The next example we consider is Im{F (x, 1, 0.2)}, where, F is shown to be bandlimited in [−1, 1] in [6], and it is argued to superoscillate around x = 0, as its graph in Fig. 6a illustrates.The evaluation of Q sin yields the results plotted in Fig. 6b, which again successfully characterizes the superoscillatory behaviour.
One more example we use to test the criterion is - The particular case G(x, 1, 1) is represented in Fig. 7a.
In [84], G was shown to be bandlimited to [−1, 1], and argued to be superoscillating around x = 0, for D ≥ 1 and s ≤ 1.This superoscillatory behaviour is indeed witnessed by the Q sin values plotted in Fig. 7b.
Finally, we can revisit h(x, 1, 2), which was given in Eq. ( 7) and represented in Fig. 2, and which we showed in Sec.II that the local wavenumber definition fails to identify as superoscillating.Looking at its graph, the function clearly displays a superoscillatory behaviour around x = π/2.Since the function is odd with respect to this point, we evaluate Q sin for intervals [π/2, b], for consecutive zeros of the function b > π/2.The results are plotted in Fig. 8, showing that the sine criterion does identify the superoscillatory behaviour of this function around x = π/2.

B. Cosine criterion
The first example that we consider here is Re{g(x, a, N )}.
In particular, we study the cases N = 10 and N = 20, for a = 2, i.e., the real counterparts of the functions we examined in Sec.IV A, which are represented in Figs.9a and 10a, respectively.For these functions, we evaluated Q cos in intervals of the form  and 10b, showing that the cosine criterion correctly identifies the superoscillatory behaviour around x = 0.
The next example we consider here is Re{F (x, 1, 0.2)}, once more the real part of a function whose imaginary part we analyzed in Sec.IV A, with F (x, A, δ) being de- fined in Eq. (24).Like its imaginary counterpart, this function is bandlimited to [−1, 1] and superoscillating around x = 0, as its graph in Fig. 11a illustrates.The values of Q cos , which are plotted in Fig. 11b, indeed report this superoscillatory behaviour.
As our last example, we can revisit the square of the shifted cosine, which we denoted as h s (x, m) in Eq. ( 15).This function, represented in Fig. 12a for the particular case m 1 and s = 1/2, was used in Sec.II as another example of superoscillation that is not captured by the definition based on the local wavenumber.The results of Q cos , plotted Fig. 12b, show that the definition proposed here captures the superoscillatory behaviour of the function around x = 0.The value of Q cos drops when evaluated with the second extremum point as right endpoint.This indicates that the superoscillation is shortlived, in agreement with the graphical estimates in [7,83].It is also easy to observe in Fig. 12b that there is a no-ticeable increase of Q cos for the third and fifth extremum points (b = 5π/3 and 7π/3).This is due to the onset of a new superoscillating interval around x = 2π.While the function h 1/2 (x, 1) is not superoscillating in the whole interval [0, 5π/3], or in the whole interval [0, 7π/3], one can see how Q cos still captures this onset.This reinforces the idea that it can be used not only as a witness, but also as a measure of the degree of superoscillatory behaviour.
Finally, to further support the adequacy of the proposed definition of superoscillations, it is worth confronting our criteria against examples of nonsuperoscillating functions, to make sure they do not produce false positives.First, we can consider the case of (cos x) 2 , for which the value of Q cos can be easily calculated.Specifically, for intervals of the form [0, b], where b = απ/2 with α ∈ N, We can also check that the cosine criterion does not identify any superoscillatory behaviour around x = 0 for the functions sinc x and (sinc x) 2 , which are bandlimited to [−1, 1] and [−2, 2], respectively.Their graphical representations in Figs.13a and 14a show that these functions do not superoscillate, and their respective results for Q cos plotted in Figs.13b and 14b are in agreement with this qualitative judgement.

V. CONCLUSION AND OUTLOOK
We proposed a rigorous definition of superoscillating function in an interval that also allows us to quantify the degree of such superoscillation.We illustrated through several examples that the definition captures the superoscillatory behaviour of functions that are widely accepted in the literature as superoscillating, and classifies as non-superoscillating other functions that should not be identified as such.Moreover, it does not suffer from any of the issues that we argue are exhibited by previous proposals.
Having a quantifier of superoscillations that is connected to their potential physical relevance can open new avenues for the formal study of superoscillations, as well as for the discovery of new superoscillating functions and methods to generate them.There is also plenty of room for generalizations, including, e.g., the prospect that the phenomenon of superoscillations can manifest itself-albeit in an approximate way-outside the realm of bandlimited functions [85].

Criterion 1 (
Sine criterion for y = c).Given a realvalued one-dimensional function f which is bandlimited to [−πW, πW ] ⊂ R, and given b 1 , b 2 ∈ R such that b 1 < b 2 and f (b 1 ) = f (b 2 ) = c, let for each k ∈ N, be the Fourier cosine coefficients over the interval [b 1 , b 2 ].If we denote with k 0 ∈ N the smallest natural number that satisfies k 0 > W (b 2 − b 1 ), we define

FIG. 3 FIG. 4
FIG. 3. a) Comparison between the imaginary part of g(x, 2, 10) and sin(2x) in the interval [−2π, 2π], showing that Im{g(x, 2, 10)} is superoscillating around x = 0. b) Values of Qsin for intervals of the form [0, b], as a function of the right endpoint b > 0. The value of Qsin was plotted for consecutive zeros, until two points were reached for which Qsin < 1/2, indicating the end of the superoscillating interval.

FIG. 6
FIG. 6. a) Comparison between the imaginary part of F (x, 1, 0.2) and sin 3x/2 in the interval [−2π, 2π], showing that Im{F (x, 1, 0.2)} is superoscillating around x = 0. b) Values of Qsin for intervals of the form [0, b], as a function of the right endpoint b > 0. The value of Qsin was plotted for consecutive zeros, until two points were reached for which Qsin < 1/2, indicating the end of the superoscillating interval.
[0, b], for several consecutive extremum points b > 0 as right endpoints.Notice that, since these functions are even, the value of Q cos for the intervals [0, b] and [−b, b] are the same.The values of Q cos are plotted in Figs.9b
)Since (cos x) 2 is bandlimited to [−2, 2] (i.e., W = 2/π), the smallest natural number greater than W b = α is k 0 = α + 1, and thus Q cos = 0 for intervals of the form [0, b], with the right endpoint b being an extremum point, as the application of the cosine criterion requires.

FIG. 9 .FIG. 10 FIG. 11
FIG. 9. a) Comparison between the real part of g(x, 2, 10) and cos(2x) in the interval [−2π, 2π], showing that Re{g(x, 2, 10)} is superoscillating around x = 0. b) Values of Qcos for intervals of the form [0, b], as a function of the right endpoint b > 0. The value of Qcos was plotted for consecutive extremum points, until two points were reached for which Qcos < 1/2, indicating the end of the superoscillating interval.

FIG. 12 FIG. 13
FIG. 12. a) Comparison between h 1/2 (x, 1) and a function proportional to cos 2 (3x/2) in the interval [−π, π], showing that h 1/2 (x, 1) is superoscillating around x = 0. b) Values of Qcos for intervals of the form [0, b], as a function of the right endpoint b > 0. The value of Qcos was plotted for consecutive extremum points until past the centre of the next superoscillation (around x = 2π), in order to show how this reflects on the value of Qcos as an increase.

FIG. 14
FIG. 14. a) Comparison between (sinc x) 2 and (cos x) 2 in the interval [−2π, 2π], showing that (sinc x) 2 is not superoscillating around x = 0. b) Values of Qcos for intervals of the form [0, b], as a function of the right endpoint b > 0. The values of Qcos were plotted for four consecutive extremum points, showing that all of them (easily) satisfy Qcos < 1/2.