This computational essay investigates one-dimensional super-oscillations, demonstrating how band-limited functions can locally exceed their highest constituent frequencies. We trace this phenomenon from M. V. Berry’s famous example of a super-oscillatory function, then analyze super-oscillatory behavior in one-dimensional quantum wavefunctions (the particle in a box and the harmonic oscillator). Furthermore, we address the physical reality of leaky functions: because real-world spectra are almost always leaky, strict mathematical super-oscillations often do not exist, yet we demonstrate how one can still meaningfully analyze fast oscillations within localized zones. Finally, we include band-limited Slepian functions as an additional example.

Introduction

Super-oscillations are a mathematical phenomenon where a band-limited function oscillates faster than its highest constituent frequency component within a specific local interval. In simpler terms, if you have a signal made only of low-frequency waves, a super-oscillatory segment of that signal can appear to be a high-frequency wave. This seems to defy the intuitive understanding of the Fourier band-limit, which suggests that a function cannot wiggle faster than its fastest component.
​
For a function
f
, which is typically complex-valued in these contexts, the local wave-vector
k
local
(x)
represents the spatial rate of change of the function’s phase. It is defined for a 1D function as:
k
local
(x)=Im(

x
log(f))=Im

x
f
f
Or if we express the function as
f(x)=A(x)
ϕ(x)

then we see
k
local
(x)=
ϕ(x)
x
is the gradient of the phase. If the function is band-limited by a
k
max
in its Fourier transform, a region of super-oscillation is denoted for points
x
where |
k
local
(x)
|
>
k
max
Show that the definition of
k
local
is the gradient of the phase:
In[]:=
Im
DA[x]
ϕ[x]

,x
A[x]
ϕ[x]

//ComplexExpand//FullSimplify
Out[]=
′
ϕ
[x]
And the real part of the logarithmic derivative is related to amplitude change:
In[]:=
Re
DA[x]
ϕ[x]

,x
A[x]
ϕ[x]

//ComplexExpand//FullSimplify
Out[]=
′
A
[x]
A[x]

Berry’s function

A key example of super-oscillations studied by Michael Berry and based on work with Yakir Aharonov is the function
f(x)=(Cos(x/N)+a
N
Sin(x/N))
It is stated that the function
f
presents super-oscillations for
a>1
and
N>>1
. This prototype function is
Nπ
periodic
Define the Berry super-oscillatory function:
In[]:=
f[x_,a_,N_]:=
N
(Cos[x/N]+aSin[x/N])
Calculate the symbolic local wave-vector:
In[]:=
kLocal=Im
D[f[x,a,n],x]
f[x,a,n]
//ComplexExpand//FullSimplify
Out[]=
a
2
a
2
sin

x
n
+
2
cos

x
n

Symbolic local growth rate, or in other words is the local rate of amplitude change:
In[]:=
κ=Re
D[f[x,a,n],x]
f[x,a,n]
//ComplexExpand//FullSimplify
Out[]=
-
(
2
a
-1)sin
2x
n

(
2
a
-1)cos
2x
n
-
2
a
-1
The Berry’s function is band-limited, the magnitude of their wave-vector components are all less than 1
Sample an integer for the value of
N
:
In[]:=
Nsample=RandomInteger[{1,50}];
Calculate the
k
n
values present in the function:
In[]:=
Union@@Cases[FourierTransform[f[x,a,Nsample],x,k],DiracDelta[a_]:>SolveValues[a==0,k],∞]
Out[]=
-1,-
25
27
,-
23
27
,-
7
9
,-
19
27
,-
17
27
,-
5
9
,-
13
27
,-
11
27
,-
1
3
,-
7
27
,-
5
27
,-
1
9
,-
1
27
,
1
27
,
1
9
,
5
27
,
7
27
,
1
3
,
11
27
,
13
27
,
5
9
,
17
27
,
19
27
,
7
9
,
23
27
,
25
27
,1
The general result is
k
n
=-1+2n/N
, so |
k
max
|
=1
:
In[]:=
Array[-1+2#/Nsample&,Nsample+1,0]
Out[]=
-1,-
25
27
,-
23
27
,-
7
9
,-
19
27
,-
17
27
,-
5
9
,-
13
27
,-
11
27
,-
1
3
,-
7
27
,-
5
27
,-
1
9
,-
1
27
,
1
27
,
1
9
,
5
27
,
7
27
,
1
3
,
11
27
,
13
27
,
5
9
,
17
27
,
19
27
,
7
9
,
23
27
,
25
27
,1
Test the equality, we see indeed is band-limited:
In[]:=
(%==%%)
Out[]=
True

N=20,a=6

There is a super-oscillation zone centered at
x=0
, the local wave-vector exceeds the maximum wave-vector and the growth rate tends to 0 when
x->0
:
In[]:=
PlotEvaluate[{kLocal,κ,1,-1}/.{a->6,n->20}],{x,-40,40},

Out[]=
Local wave-vector
Growth rate
k=1
k=-1
Super-oscillations are notorious when approaching
x=0
, but nothing is free to get those high-frequency oscillations in the “zone,” the price to pay is to get the magnitude of the function gets exponentially smaller
Magnitude and phase plot:
AbsArgPlotf[x,6,20],{x,0,20},

Out[]=
More pronounced Arg slopes:
In[]:=
PlotArg[f[x,6,20]],{x,0,20},

Out[]=
Increased frequency in the magnitude oscillations:
In[]:=
PlotLog@Abs@Re[f[x,6,20]],{x,0,20},

Out[]=

N=10,a=0.999

For this value of
a
, we are not supposed to get super-oscillations:
In[]:=
PlotEvaluate[{kLocal,κ,1,-1}/.{a->0.999,n->10}],{x,-40,40},

Out[]=
Local wave-vector
Growth rate
k=1
k=-1
Visualize the amplitude and phase:
In[]:=
Plot[Log@Abs@Re[f[x,0.999,10]],{x,-10,10},PlotRange->All]
The chosen values generate an extension of the super-oscillation zone:
And it is interesting to observe the super-oscillations do not occur in the interval where the wave-function varies more:

Particle in a box wave-functions

While pure super-oscillations are, by definition, produced by band-limited functions (like sums of sines), they can be extended to “leaky” functions that have infinite bandwidth. A function is considered to have super-oscillations if it has a mostly low-frequency spectrum, with an extremely faint “tail” of high-frequency components that do not contribute to the overall amplitude but are responsible for rapid local variations. These can be called “quasi-super-oscillations.” , they are indistinguishable locally from true super-oscillations but lack the strict band-limited guarantee.

Two state superposition

No relative phase

Define a two state superposition:
But it is highly concentrated in the values close to 5π and 6 π :

With relative phase

Two state superposition using a relative phase:
And those zones are related to an increased phase gradient:
But it is interesting to note this is not reflected in the real and imaginary parts of the wave-function, that oscillate with the given frequencies of the sine wave-functions:

More superpositions

Balanced superposition

Define a uniform superposition:
Calculate the Fourier transform:
Also as expected for this case there are also no super-oscillations:

Binomial coefficient superposition

Consider a more complicated superposition with complex coefficients:
Build ψ(x):
Fourier transform:
Local wave-vector
Observing the phase tells us there is a change of behavior in the local wave-vector peaks, but they are not notorious enough:

Engineering super-oscillations

Sampling around the target zone:
Get the function evaluations:
Solve for the coefficients:
Engineered wavefunction:
Calculate its Fourier transform:
The local wave-vector gives us no useful information since the wave-function are real and the coefficients are also real:
But we can calculate the real part of the logarithmic derivative to get information in the change of amplitude:
This quantity has oscillations with high amplitude in the target zone:
Visualize the oscillations in the amplitude of the wave-function in the restricted zone:

Harmonic Oscillator

Quantum harmonic oscillator wave functions:

Maximum k as energy content integral

Fraction of total integral for some finite symmetric region:

Method 2 (energy integral)

Vary the precision:

Method 3

Use this estimation for the other methods:

Two state superposition

If we consider a superposition with real coefficients the local momentum will be clearly 0, so we define a superposition with relative phase:
Local momentum:
Show super-oscillatory region, which are very localized peaks:

Many wave functions superposition

Build wavefunction:
Local wave-vector:
Show super-oscillations zone:
Plot real and imaginary part:
We also see in this case fast phase changes in zones where |ψ| goes to zero, that of course correspond to the potential super-oscillations peaks:

Target super-oscillations zone

We can follow a similar approach when using the particles in a box wavefunction to try to design super-oscillations
Set up parameters:
Build the design matrix:
Obtain the coefficients using least squares and normalize them:
Construct the wavefunction:
Visualize the super-oscillation zone:
Zoom of the super-oscillation candidate zone:

Leaky Prototype function

Define the leaky function:
When β ≠0, the Fourier series terms can be increased as much as needed,showing is not band-limited:
Phase and magnitude plot:
Increased rate of change of the phase when approaching the super-oscillation zone:

Monochromatic waves

In one dimension the elementary scalar wave is

Two wave model

Define a two wave model that depends on a parameter ε :
Local wave-vector:
However if we take a small value of ϵ that is not zero we can observe interesting behavior
High phase gradient in the peaks:
Magnitude and Arg zoom plot in one of the zones:

Many waves

Definition for the sum of plane waves:
Plot the real part, imaginary part and the absolute value of the function:
Super-oscillations fraction or probability that an infinitesimal region has super-oscillations:
Find a super-oscillating peak:
Plot the phase close to the peak , we see there is a faster change in the phase:
Visualize more zones of fast phase change:

Prolate Spheroidal Functions

Plot the spheroidal function, we see the higher amplitude oscillations are concentrated in some zone:

Superposition

Plot the local wave-vector:
We see the small zones of fast phase change that reflect the many super-oscillation peaks:

Conclusions

◼
  • Super-oscillations may manifest directly in the real or imaginary components(or the magnitude) , depending on the specific superposition or function family utilized. Alternatively, they can be identified as regions of high phase gradients, typically occurring where the magnitude approaches a local minimum or zero
    • ◼
  • By applying these principles to fundamental quantum systems—such as the particle in a box and the harmonic oscillator—we demonstrated the ability to engineer super-oscillations in localized zones. Notably, these were designed to manifest as rapid amplitude fluctuations rather than a pronounced phase gradient

    • Utilities

    References

    ◼
  • Berry, M. V. (2013). Five momenta. European Journal of Physics, 34(6), 1337-1348.
    • ◼
  • Berry, M. V. (2019). Super-oscillations and leaky spectra. Journal of Physics A: Mathematical and Theoretical, 52(1), 015202.
    • ◼
  • Berry, M. V., & Dennis, M. R. (2009). Natural super-oscillations in monochromatic waves in D dimensions. Journal of Physics A: Mathematical and Theoretical, 42(2), 022003.

    • CITE THIS NOTEBOOK

    Superoscillations Part I: 1D foundations, leaky functions and quantum wavefunctions​
    by Bruno Tenorio​
    Wolfram Community, STAFF PICKS, May 9, 2026
    https://community.wolfram.com/groups/-/m/t/3713462