2. The properties of the superoscillating functions

The frequency limited functions and superoscillations occur in a number of scientific and technological applications such as foundations of quantum mechanics, information theory, optics, and signal processing which led also to work on the optimization and stability of superoscillations [8–14]. Some examples of superoscillatory functions have been proposed and investigated in the past [1–4, 8, 10, 11]. In this section, we consider the generic example of the Aharonov, Popescu, Rohlich functions f xð Þ given by [11]:

$$f(\mathbf{x}) = (\cos \mathbf{x} + i \mathbf{g} \sin \mathbf{x})^n; \mathbf{g} > 1; \mathbf{n} \gg 1 \tag{1}$$

Here, for general, gf xð Þ is a periodic function with a period 2π. It is easy to see from Eq. (1) that for g ¼ 1 we obtain from Eq. (1):

$$f(\mathbf{x}) = \exp(i\mathbf{m}\mathbf{x})\tag{2}$$

For small x ! 0, Eq. (1) yields:

superoscillating functions depend on tailoring the functions in order to reduce such an effect [1]. It has been shown that the superoscillations amplitude decreases exponentially with the length of the superoscillating stretch [2]. Nevertheless, the existence of superoscillations and the possibility of encoding arbitrary amounts of information into an arbitrary short segment of a low-bandwidth signal do not contradict the information theory [2]. Taking into account the Shannon's theorem concerning the information channel capacity, it appeared to be that the superoscillatory information can be compressed to an arbitrary extent under the condition that the signal power increases exponentially with the length of the superoscillatory part of the message [2]. Superoscillations can be designed by prescribing their amplitude and/or their derivative on a grid which is denser than the Nyquist density [3]. Four different ways to constrain the signal in order to render it superoscillatory have been described in Reference [3]: (1) amplitude constraints, without any restriction on the derivative; (2) derivative constraints, without restrictions on the amplitude; (3) the amplitude and the derivative constraints on staggered grids; and (4) the amplitude and the derivative constraints on aligned grids at one half density [3]. When a set of constraints is chosen to ensure a required superoscillation, the signal is optimized by minimizing its total energy within the subspace of all the superoscillatory functions obeying the same set of constraints [2]. Superoscillations can be constructed also by using the so-called direct approach. This approach is based on a signal that is a superposition of time shifted SINC functions which ensures its band limitation [4]. Then the coefficients of the superposition are chosen by specification of the signal values on a relatively dense set of points, which forces the required superoscillations yet leaves some degrees of freedom for optimization [4]. The propagation of the temporal optical signals with a superoscillation at an absorbing resonance of a dielectric medium has been studied theoretically [5]. The absorption acts only on the Fourier components of the band limited signals, while the superoscillation is not absorbed [5]. When the signal propagates through the medium, the superoscillation revives periodically or quasi-periodically, and a superoscillatory signal may be used in order to deliver fast oscillations to a target in a dielectric medium in the

It should be noted that it is impossible to infer the bandwidth of a finite energy signal f tð Þ from a sampled segment of length T even for a sufficiently large T because there exist signals of an arbitrary small bandwidth oscillating throughout an interval of a length T with an arbitrary small period [2]. The meaning is that we can make the superoscillatory part of a signal, T, and the corresponding frequency inside, ω, arbitrarily large. This comes at a price, that the amplitude of the signal outside the superoscillatory portion is exponentially large in the number of superoscillations present in the time interval, T, compared to the amplitude of superoscillation [1, 2]. In such a case, the standard Fourier analysis is not sufficient because we cannot locate the sharp pulses in the signal spectrum caused by the sharp changes of the signal in the time domain [6]. A small perturbation of the sinusoidal function sinð Þ ωt or cosð Þ ωt at any point of the time axis influences every point of the frequency axis and vice-versa [6]. The Fourier transform integral can be evaluated at only one frequency at a time which is not convenient for the signal processing [6]. In particular, the so-called time-frequency analysis combining both the frequency domain and the time domain analyses is necessary for superoscillation studies [2, 6]. The short-time Fourier transform (STFT) can be used for the time-frequency analysis because it permits to obtain the

frequency bands characterized by a high absorption [5].

196 Wavelet Theory and Its Applications

$$f(\mathbf{x}) \approx \left(\exp(\ln(1+i\mathbf{g}\mathbf{x}))\right)^n \approx \exp(i\mathbf{g}n\mathbf{x})\tag{3}$$

Obviously, in the limiting case, the function determined by Eq. (3) is varying faster than the function determined by Eq. (2). Consider now the Fourier series for f xð Þ given by [11]:

$$f(\mathbf{x}) = \sum\_{m=0}^{n} c\_{m} \exp(i n k\_{m} \mathbf{x}) \tag{4}$$

corresponding exponential smallness of j j f xð Þ [11]. More accurate approximation gives the following expressions for the region of fast superoscillations xfs and the number of fast

Superoscillations is a week phenomenon such that there is no slightest indication of superoscillations in the power spectrum P kð Þ of f xð Þ [11]. Indeed, using the Fourier compo-

n gð Þ <sup>2</sup> � <sup>1</sup> <sup>p</sup> ; nfs <sup>¼</sup> <sup>g</sup> ffiffiffi

<sup>≈</sup> <sup>1</sup> σ ffiffiffiffiffiffi

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð Þ <sup>k</sup> � h i<sup>k</sup> <sup>2</sup>

The asymptotic spectrum Eq. (12) is a narrow Gaussian with the center at <sup>k</sup> <sup>¼</sup> <sup>g</sup>�<sup>1</sup> representing the slow oscillations near j j¼ x π [11]. The superoscillations gradually disappear getting slower and reducing to the region j j k ≤ 1 according to the Fourier series Eqs. (4) and (5) [11].

The function f xð Þ defined by Eq. (1) is periodic. Consequently, it may represent a diffraction grating with spatial period πd that transforms an incident light plane wave into a propagating series of diffracted beams [11]. Such a grating transforms the wave Eq. (1) into the superoscillatory function Ψð Þ¼ x; 0 f xð Þ =d under the following condition for the wavenumber K:n=d < K < gn=d [11]. The grating produces a novel kind of super-resolution, that is, the subwavelength structure in the field with only propagating waves and without evanescent

In the framework of the precise classical wave model, it has been shown how superoscillations can emerge and propagate into the far-field region [14]. The band-limited superoscillatory wave (the "red light") is propagating along the x axis of a unidimensional (1D) waveguide with the a segment of the x axis (the "window") which is opened and closed as the superoscillation passes by releasing the light pulse into the two-dimensional (2D) space corresponding to x, z > 0 [14]. The wave traveling in the positive x direction with a speed c ¼ 1 can be described without loss of generality by the band limited function Eq. (1) with the replacement of the argument x by ð Þ x � t =n in expressions (1) and (4) [14]. This new function f redð Þ x; t is superoscillatory near x ¼ t [14]. It represents a rigidly moving polychromatic packet with associated frequencies ω<sup>m</sup> ¼ km [14]. The function f redð Þ x; t can be approximated by the

f red:appð Þ x; t represents superoscillations over the interval �2X < x < 2X for t ¼ 0 [14]. In order

¼

s

2X<sup>2</sup>

� �; X <sup>¼</sup> ffiffiffiffiffiffiffiffi

n g2�1

q [14]. The function

q

n p π ffiffiffiffiffiffiffiffiffiffiffiffiffi

Applications of Wavelet Transforms to the Analysis of Superoscillations

<sup>2</sup><sup>π</sup> <sup>p</sup> exp � ð Þ <sup>k</sup> � h i<sup>k</sup> <sup>2</sup>

ffiffiffiffiffiffiffiffiffiffiffiffiffi g<sup>2</sup> � 1 2ng<sup>2</sup>

2σ<sup>2</sup> ( )

<sup>g</sup><sup>2</sup> � <sup>1</sup> <sup>p</sup> (11)

http://dx.doi.org/10.5772/intechopen.76333

(12)

199

(13)

j j <sup>x</sup> <sup>&</sup>lt; xs <sup>¼</sup> <sup>1</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

<sup>m</sup>ð Þ m ¼ nð Þ 1 � k =2

2 Pn 0 c2 m

h i<sup>k</sup> <sup>¼</sup> <sup>1</sup> g ; σ �

following expression: <sup>f</sup> red:appð Þ¼ <sup>x</sup>; <sup>t</sup> exp½ � ig xð Þ � <sup>t</sup> exp ð Þ <sup>x</sup>�<sup>t</sup> <sup>2</sup>

superoscillations nfs [11]:

nents Eq. (5) we obtain [11]:

where

waves [11].

P kð Þ¼ nc<sup>2</sup>

where

$$k\_m = 1 - \frac{2m}{n}; c\_m = \frac{n!}{2^n}(-1)^m \frac{\left(g^2 - 1\right)^{n/2} [(g-1)/(g+1)]^{nk\_m/2}}{[n(1+k\_m)/2]! [n(1-k\_m)/2]!} \tag{5}$$

Equations (4) and (5) contain only wavenumbers j j km ≤ 1 [11]. Comparison of Eqs. (3–5) shows that the function f xð Þ demonstrates superoscillatory behavior with the degree of superoscillation defined by g [11]. The function f xð Þ Eq. (1) can be represented in an integral form [11]:

$$f(\mathbf{x}) = \left(\frac{\mathcal{S}}{k(\mathbf{x})}\right)^{n/2} \exp\left\{i\mathbf{n} \int\_0^\mathbf{x} d\mathbf{x}' k(\mathbf{x}')\right\} \tag{6}$$

Here, the local wavenumber k xð Þ is introduced given by [11]:

$$k(\mathbf{x}) \equiv \frac{1}{n} \text{Im} \left( \frac{\partial \ln f(\mathbf{x})}{\partial \mathbf{x}} \right) = \frac{\mathbf{g}}{\cos^2 \mathbf{x} + \mathbf{g}^2 \sin^2 \mathbf{x}} \tag{7}$$

The relationship Eq. (6) can be proved immediately taking into account that:

$$\int\_0^\chi d\mathbf{x}' k(\mathbf{x}') = \arctan(\mathbf{g} \tan \mathbf{x}) \tag{8}$$

and using the identities cos<sup>x</sup> <sup>¼</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>1</sup> <sup>þ</sup> tan<sup>2</sup><sup>x</sup> � � p �<sup>1</sup> and sin<sup>x</sup> <sup>¼</sup> tanx<sup>=</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>1</sup> <sup>þ</sup> tan<sup>2</sup><sup>x</sup> � � <sup>p</sup> . The wavenumber k xð Þ determined by Eq. (7) is varying from the superoscillatory value kð Þ¼ 0 g to the minimum value kð Þ¼ π=2 1=g. The superoscillatory region where j j k > 1 is determined by the following condition [11]:

$$|\mathbf{x}| < \mathbf{x}\_s = \operatorname{arc}\cot\left(\sqrt{\mathbf{g}}\right) \tag{9}$$

The number of oscillations nosc in the superoscillatory region is given by:

$$m\_{\text{osc}} = \frac{n}{2\pi} \int\_{-\arccos(\sqrt{a})}^{\arccos(\sqrt{a})} d\mathbf{x} k(\mathbf{x}) = \frac{n}{\pi} \arctan(\sqrt{\mathbf{g}}) \tag{10}$$

Equation (6) shows in particular that in the superoscillatory region j j k > 1, the magnitude j j f xð Þ is exponentially smaller than in the normal region j j k < 1 [11]. Consequently, n is the asymptotic parameter describing the number of oscillations in the superoscillatory region and the corresponding exponential smallness of j j f xð Þ [11]. More accurate approximation gives the following expressions for the region of fast superoscillations xfs and the number of fast superoscillations nfs [11]:

$$|\mathbf{x}| < \mathbf{x}\_s = \frac{1}{\sqrt{n(\mathbf{g}^2 - 1)}}; n\_\mathbb{\hat{\mathbb{H}}} = \frac{\mathbf{g}\sqrt{n}}{\pi\sqrt{\mathbf{g}^2 - 1}} \tag{11}$$

Superoscillations is a week phenomenon such that there is no slightest indication of superoscillations in the power spectrum P kð Þ of f xð Þ [11]. Indeed, using the Fourier components Eq. (5) we obtain [11]:

$$P(k) = \frac{nc\_m^2(m = n(1 - k)/2)}{2\sum\_{0}^{n} c\_m^2} \approx \frac{1}{\sigma\sqrt{2\pi}}\exp\left\{-\frac{(k - \langle k \rangle)^2}{2\sigma^2}\right\} \tag{12}$$

where

f xð Þ¼ <sup>X</sup><sup>n</sup>

km <sup>¼</sup> <sup>1</sup> � <sup>2</sup><sup>m</sup>

n

; cm <sup>¼</sup> <sup>n</sup>!

f xð Þ¼ <sup>g</sup>

Here, the local wavenumber k xð Þ is introduced given by [11]:

k xð Þ� <sup>1</sup> n

k xð Þ � �<sup>n</sup>=<sup>2</sup>

Im <sup>∂</sup>ln f xð Þ ∂x � �

<sup>1</sup> <sup>þ</sup> tan<sup>2</sup><sup>x</sup> � � p �<sup>1</sup>

number k xð Þ determined by Eq. (7) is varying from the superoscillatory value kð Þ¼ 0 g to the minimum value kð Þ¼ π=2 1=g. The superoscillatory region where j j k > 1 is determined by the

j j <sup>x</sup> <sup>&</sup>lt; xs <sup>¼</sup> arc cot ffiffiffi

dxk xð Þ¼ <sup>n</sup>

Equation (6) shows in particular that in the superoscillatory region j j k > 1, the magnitude j j f xð Þ is exponentially smaller than in the normal region j j k < 1 [11]. Consequently, n is the asymptotic parameter describing the number of oscillations in the superoscillatory region and the

π

arctan ffiffiffi

The relationship Eq. (6) can be proved immediately taking into account that:

ð x

0 dx<sup>0</sup>

The number of oscillations nosc in the superoscillatory region is given by:

<sup>ð</sup> arccot ffiffi a p ð Þ

�arccot ffiffi a p ð Þ

nosc <sup>¼</sup> <sup>n</sup> 2π

and using the identities cos<sup>x</sup> <sup>¼</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

following condition [11]:

where

198 Wavelet Theory and Its Applications

[11]:

m¼0

<sup>2</sup><sup>n</sup> ð Þ �<sup>1</sup> <sup>m</sup> <sup>g</sup><sup>2</sup> � <sup>1</sup> � �<sup>n</sup>=<sup>2</sup>

Equations (4) and (5) contain only wavenumbers j j km ≤ 1 [11]. Comparison of Eqs. (3–5) shows that the function f xð Þ demonstrates superoscillatory behavior with the degree of superoscillation defined by g [11]. The function f xð Þ Eq. (1) can be represented in an integral form

exp in

8 < : ð x

0 dx<sup>0</sup> k x<sup>0</sup> ð Þ

<sup>¼</sup> <sup>g</sup> cos2<sup>x</sup> <sup>þ</sup> <sup>g</sup>2sin<sup>2</sup>

cmexpð Þ inkmx (4)

½ � <sup>n</sup>ð Þ <sup>1</sup> <sup>þ</sup> km <sup>=</sup><sup>2</sup> !½ � <sup>n</sup>ð Þ <sup>1</sup> � km <sup>=</sup><sup>2</sup> ! (5)

(6)

(7)

½ � ð Þ <sup>g</sup> � <sup>1</sup> <sup>=</sup>ð Þ <sup>g</sup> <sup>þ</sup> <sup>1</sup> nkm=<sup>2</sup>

9 = ;

x

k x<sup>0</sup> ð Þ¼ arctanð Þ gtanx (8)

and sin<sup>x</sup> <sup>¼</sup> tanx<sup>=</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

<sup>1</sup> <sup>þ</sup> tan<sup>2</sup><sup>x</sup>

g � � p (9)

g � � p (10)

� � <sup>p</sup> . The wave-

$$\langle k \rangle = \frac{1}{\mathcal{S}}; \sigma \equiv \sqrt{\left(k - \langle k \rangle\right)^2} = \sqrt{\frac{\mathcal{g}^2 - 1}{2\mathfrak{n}\mathfrak{g}^2}} \tag{13}$$

The asymptotic spectrum Eq. (12) is a narrow Gaussian with the center at <sup>k</sup> <sup>¼</sup> <sup>g</sup>�<sup>1</sup> representing the slow oscillations near j j¼ x π [11]. The superoscillations gradually disappear getting slower and reducing to the region j j k ≤ 1 according to the Fourier series Eqs. (4) and (5) [11].

The function f xð Þ defined by Eq. (1) is periodic. Consequently, it may represent a diffraction grating with spatial period πd that transforms an incident light plane wave into a propagating series of diffracted beams [11]. Such a grating transforms the wave Eq. (1) into the superoscillatory function Ψð Þ¼ x; 0 f xð Þ =d under the following condition for the wavenumber K:n=d < K < gn=d [11]. The grating produces a novel kind of super-resolution, that is, the subwavelength structure in the field with only propagating waves and without evanescent waves [11].

In the framework of the precise classical wave model, it has been shown how superoscillations can emerge and propagate into the far-field region [14]. The band-limited superoscillatory wave (the "red light") is propagating along the x axis of a unidimensional (1D) waveguide with the a segment of the x axis (the "window") which is opened and closed as the superoscillation passes by releasing the light pulse into the two-dimensional (2D) space corresponding to x, z > 0 [14]. The wave traveling in the positive x direction with a speed c ¼ 1 can be described without loss of generality by the band limited function Eq. (1) with the replacement of the argument x by ð Þ x � t =n in expressions (1) and (4) [14]. This new function f redð Þ x; t is superoscillatory near x ¼ t [14]. It represents a rigidly moving polychromatic packet with associated frequencies ω<sup>m</sup> ¼ km [14]. The function f redð Þ x; t can be approximated by the following expression: <sup>f</sup> red:appð Þ¼ <sup>x</sup>; <sup>t</sup> exp½ � ig xð Þ � <sup>t</sup> exp ð Þ <sup>x</sup>�<sup>t</sup> <sup>2</sup> 2X<sup>2</sup> � �; X <sup>¼</sup> ffiffiffiffiffiffiffiffi n g2�1 q [14]. The function f red:appð Þ x; t represents superoscillations over the interval �2X < x < 2X for t ¼ 0 [14]. In order to capture the superoscillations, the region near x ¼ 0 is selected with a Gaussian window of width L which is opened and closed with a Gaussian switching function over an interval near t ¼ 0 [14]. The window must faithfully transmit the red light including the superoscillaitons [14].

For the sake of definiteness, we consider the time-dependent superoscillating signal of the type Eq. (1) assuming that:

$$\mathbf{x} = \frac{a\_0 t}{n}; \mathbf{g} = \frac{w}{w\_0} \tag{14}$$

superoscillatory mask and by increasing the dynamic range of the light detection [15]. SOL can be also used for the creation of sub-diffraction-limit optical needles [16]. An optical needle could be created by converting the central region of the SOL into an opaque area forming a shadow, and changing the diameter of the blocking region without varying the rest of SOL [16]. The possible applications of the sub-diffraction-limit optical needles are the far-field

Applications of Wavelet Transforms to the Analysis of Superoscillations

http://dx.doi.org/10.5772/intechopen.76333

201

The possible applications of superoscillations for data compressions have been discussed [8]. However, the superoscillations are unstable in a way that tiny perturbations of a band-limited superoscillating function can induce very high-frequency components [8]. For this reason, the practical use of the superoscillations in imperfect communication channels is difficult [8].

There exist different types of a wavelet transform: CWT, discrete wavelet transform (DWT) [6, 7, 17], multi-wavelets [17, 18], and complex wavelets [19]. We applied these types of wavelets to the problems related to the signal processing in optical communication systems [20–23]. We

In this section, we consider some fundamental features of CWT. Unlike the Fourier transform and STFT, the CWT is characterized by the time and frequency selectivity [6, 7]. It can localize events both in time and in frequency in the entire time-frequency plane [6, 7]. That is why CWT is unique mathematical tool for the investigation of the superoscillations where the timefrequency analysis in different regions of the spectrum is necessary as it is mentioned earlier

The CWT W að Þ ; b of any square integrable function f tð Þ with respect to a wavelet ψð Þt is

f tð Þ <sup>1</sup> ffiffiffiffiffi j j <sup>a</sup> <sup>p</sup> <sup>ψ</sup><sup>∗</sup> <sup>t</sup> � <sup>b</sup> a

Here a, b are real, the asterisk denotes complex conjugation, the energy signals f tð Þ,ψð Þ<sup>t</sup> <sup>⊂</sup> <sup>L</sup><sup>2</sup>

value continuous-time function ψð Þt is called the mother wavelet. It satisfies the following

CWT W að Þ ; b is a function of two variables: (1) the scale or dilation variable a determines the amount of time scaling or dilation and (2) the translation or time shift variable b represents the shift of ψa, <sup>0</sup>ð Þt by an amount b along the time axis and indicates the location of the wavelet window along it [6, 7]. The inverse scaling parameter 1=a is a measure of frequency [6].

∞ �∞

j j <sup>ψ</sup>ð Þ<sup>t</sup> <sup>2</sup>

� �dt (18)

dt < ∞. The real- or complex-

ð Þ R ,

have found that CWT is the most appropriate for the analysis of superoscillations.

W að Þ� ; b

ð Þ <sup>R</sup> is the set of square integrable functions such that <sup>Ð</sup>

ψð Þt dt ¼ 0 [7].

ð ∞

�∞

super-resolution microscopy and nanofabrication [16].

4. The fundamental properties of CWT and DWT

[2, 6]:

L2

condition that Ð

∞ �∞

defined as follows [7]:

Substituting relationships Eq. (14) into Eq. (1) we obtain:

$$f\_n(\omega\_0 t, \omega/\omega\_0) = \left\{ \cos\left(\frac{\omega\_0 t}{n}\right) + i \frac{\omega}{\omega\_0} \sin\left(\frac{\omega\_0 t}{n}\right) \right\}^n \tag{15}$$

Expression (15) is the signal band limited by the frequency ω<sup>0</sup> with the superoscillation manifested by a single peak of a width <sup>Δ</sup><sup>t</sup> <sup>¼</sup> <sup>ω</sup>�<sup>1</sup> while the ratio <sup>ω</sup>=ω<sup>0</sup> may be arbitrary larger than unity. It should be noted that:

$$\lim\_{n \to \infty} \text{Ref}\_n(\omega\_0 t, \omega/\omega\_0) = \cos(\omega t) \tag{16}$$

For finite n and under the first condition of Eq. (11) which now takes the form ω0t < ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n= ð Þ ω=ω<sup>0</sup> <sup>2</sup> � <sup>1</sup> <sup>r</sup> h i, expression (16) reduces to the following approximation:

$$f\_n(\omega\_0 t, \omega/\omega\_0) \approx \cos(\omega t) \tag{17}$$

Equation (17) shows that the band limited signal Eq. (15) oscillates with a frequency ω higher than the band limit ω<sup>0</sup> for the arbitrary long time depending on n.

#### 3. The possible applications of superoscillations

Optical superoscillations can be used in the subwavelength imaging [15]. This super-resolution technology is based on a superoscillatory lens (SOL) which represents a nanostructured mask [15]. SOL illuminated with a coherent light source creates a focus at a distance which is larger than the near-field of the mask [15]. Indeed, the ability to focus beyond the diffraction limit is related to the superoscillation, since the band-limited functions in such a case oscillate faster than their highest Fourier components [11]. Superoscillatory binary masks do not use evanescent waves and focus at distances tens of wavelengths away from the mask [15]. The superoscillation-based imaging has the following advantages with respect to other technologies: (1) it is non-invasive which allows to place the object at a substantial distance from SOL; (2) it can operate at the wide range of wavelengths from X-rays to microwaves; and (3) the resolution of the SOL can be improved by refining the design, increasing the size of the superoscillatory mask and by increasing the dynamic range of the light detection [15]. SOL can be also used for the creation of sub-diffraction-limit optical needles [16]. An optical needle could be created by converting the central region of the SOL into an opaque area forming a shadow, and changing the diameter of the blocking region without varying the rest of SOL [16]. The possible applications of the sub-diffraction-limit optical needles are the far-field super-resolution microscopy and nanofabrication [16].

The possible applications of superoscillations for data compressions have been discussed [8]. However, the superoscillations are unstable in a way that tiny perturbations of a band-limited superoscillating function can induce very high-frequency components [8]. For this reason, the practical use of the superoscillations in imperfect communication channels is difficult [8].
