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Slepian concentration eigenvalues
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Numbers
$S$
$n$ 
$\Lambda_n(c)$
1
0:
0.7833687892100002235995653472178290316442546007739911300796155408843062674372290765788737569681403399
1
1:
0.2050398317739095409789810751360902522885519103571012713657843902606470294257956851947255587722016137
1
2:
0.01137398962948903340018319274082404186581303462562747170107118068809406389029606359265059215931160710
1
3:
0.0002152184969341478918491658130161768201171681661486099682895573175389645289249642546906763566776399622
1
4:
0.000002157219363795064706764014699681760110073452070460188040491556566753783660014490914162220665918770184
1
5:
0.00000001361092619341058248708051123948350034777002255728347544910982194206580121025025621464109728859725169
1
6:
0.00000000005918830805845704211810975974356552615383414438046946279654998802890888654898659408391680991861609596
1
7:
0.0000000000001882992841149929768436800154382145618578156116603375609393310193327666887251704613277987665299379095
1
8:
0.0000000000000004574337362895806905959859498301635396753335991171562654023525583731751361291593985238259975100598783
1
9:
0.0000000000000000008764633287272743474193926019574686373802292172910620514813257927422382834416735247791305182874452869
1
10:
0.000000000000000000001358573915351382351487417814904187226004620337941657507853049526904160867156529225680883755908685721
1
11:
0.000000000000000000000001738808326527382401286612991511736259042932561952967179343265816903338031637638769850350697370053126
2
0:
0.9810462777518808054449234570507141119324445809167813690791390735963205824087830102672848188048132464
2
1:
0.7496201982798724784638938214159643678521282100608305810121975540598807229272429804877166412621354120
2
2:
0.2435930155518087383372861036044517643395056828243289358062512953111366029703503343994563478313528532
2
3:
0.02464654705908988362741438141067281082061051288089573422661895752839338406757719608809150520299455199
2
4:
0.001066059753505256406683578540217330069820080280386907348227350263775349837356367459843036028342013314
2
5:
0.00002741513230855601873896862842386016906191023345575044916170494551537360747197560600145282214177937892
2
6:
0.0000004802781584393422393701192371579258788189031111950934508612604836141910878296533089056488671607092452
2
7:
0.000000006133200708281183222433886995129896449880881539159548832614553849563638054591926743635710195584703118
2
8:
0.00000000005971407396672722989219527463158171268963384447021499323656704702703647506776247910961866539948103425
2
9:
0.0000000000004582023466633272186894638517603395663387075343985365664240386041709661042709819765628482233157230668
2
10:
0.000000000000002843138438142318151120025186222922575109030114059847341497095716422639964519806606830955983860759044
2
11:
0.00000000000000001456295184857864698889696851562474307469583867092306677279934771744532466677177481719260694370918488
4
0:
0.9999427533541047123638438038441654776123675156545554954564696116027061594939375208399677555237124135
4
1:
0.9975617081851468565554251294684567922309447141857401615558469209220336440967857106682160324451936962
4
2:
0.9593903454479201925846785181311772468202447798636925993207269357918583215723023217177435961411941355
4
3:
0.7217515555417267936039461143217240108648309318116518301902425625489184288152947933541646561617608552
4
4:
0.2746660266254135967286025463629840537114615475971212344668266762701784626643843720029458111740851753
4
5:
0.04301464029939482691177987491365625238240650812094758493086418436483017809412447099931716116379851260
4
6:
0.003478238071584583894310029750314392717467888425346639329705989214567211608760391892194707695867349618
4
7:
0.0001870284936164377065989467646385319735191714881706698830687220800491698868630555723207232790691982309
4
8:
0.000007465620360493005354067154401868694246808586662223281449824875250194806169557590796250236187385766463
4
9:
0.0000002324182948167545025787312622763395366489082268575918163681935423149037196857697577024769549269705353
4
10:
0.000000005820371501359668069584140519212793531204610804166235021943527857817093043545147352513497469563279950
4
11:
0.0000000001199622845017152956449017264872715955479038256805077596455321750565397113218048943036271653779656383
8
0:
0.9999999997053919069660834112077218447080159330575939486187615491687027487743929865884853449517021969
8
1:
0.9999999723160420543367002744910109283104228487722667285351090304379178597765090972806282049267146493
8
2:
0.9999987898337612049468464647531069749937543225701762575617520798425518795536393004211649782152636228
8
3:
0.9999675542026677599314366317901604186360944752306942159113175296374798970839666379833011241117867586
8
4:
0.9994100757303827042628761834989748882350882358305397664549395429925767895223242211008941711131641302
8
5:
0.9925044995885703276417741242302040794444622674128868397206090636325323988980329057162393730740569490
8
6:
0.9366522329116018914195043091878664342987470569247473749639984756777232415028624556583529646510799694
8
7:
0.6988356135240769344377395461875925404759890731579168634464346244040948413260521176427736948614239819
8
8:
0.2993750136388687182417034426548931919162727445677681268082633872665070958858488191334082704249604367
8
9:
0.06424203744104683569750346465887732940013289312985434296457512675984385078372882496691230773551816801
8
10:
0.008194531638350736597153837210415956792743228612307632413222130270807087339309225790540283357823234002
8
11:
0.0007597356086781280361844596898755501462840924664762151045104810628378801882266221220838748556025452227
Definition
The Slepian concentration eigenvalues $\Lambda_n(c)$, ordered decreasingly from $n=0$, are the eigenvalues of $Q_cf(x)=\int_{-1}^1\frac{\sin(c(x-y))}{\pi(x-y)}f(y)\,dy$ on $L^2([-1,1])$, where $c=\pi S/2$ and the diagonal kernel is $c/\pi$ [1].
Parameters
$S$
—   Shannon number ($S>0$)
$n$
—   mode index ($n\geq0$)
Formulas
(1)
$1>\Lambda_0(c)>\Lambda_1(c)>\cdots>0$ for $c>0$ [1].
(2)
$\Lambda_n(c)=\frac{c}{2\pi}|\mu_n(c)|^2$, where $\mu_n(c)$ is defined by $\int_{-1}^1e^{icxy}\psi_n(y;c)\,dy=\mu_n(c)\psi_n(x;c)$ [1].
(3)
$\sum_{n=0}^{\infty}\Lambda_n(c)=\operatorname{tr}Q_c=\int_{-1}^1(c/\pi)\,dx=2c/\pi=S$.
Comments
(4)
This is the continuous, order-zero concentration problem. Its eigenfunctions are the prolate spheroidal wave functions associated with the differential eigenvalues $\chi_n(c)$. The real eigenfunctions $\psi_n$ may be normalized by $\int_{-1}^1\psi_n(x;c)^2\,dx=1$ and $\psi_n(1;c)>0$; the concentration eigenvalues do not depend on this normalization.
(5)
For the bandlimited extension of the $n$th eigenfunction, $\Lambda_n(c)$ is the fraction of its squared $L^2$ norm contained in $[-1,1]$ [2]. The bandwidth $c$ is an angular frequency, so the Fourier transform uses $e^{-i\omega x}$.
(6)
The Shannon number $S=2c/\pi$ is the trace of $Q_c$.
Programs
(P1)
Python
# Double-precision approximation to the largest concentration eigenvalue.
import numpy as np
from scipy.special import roots_legendre
from scipy.linalg import eigvalsh

def largest_concentration(S):
    x, w = roots_legendre(96)
    kernel = (S / 2) * np.sinc((S / 2) * (x[:, None] - x[None, :]))
    matrix = np.sqrt(w[:, None] * w[None, :]) * kernel
    return eigvalsh(matrix, subset_by_index=[95, 95])[0]

largest_concentration(2)  # approximately 0.981046277751
References
[1]
A. Osipov and V. Rokhlin, On the evaluation of prolate spheroidal wave functions and associated quadrature rules (2013), §§3.1–3.2 and 5.2. (arXiv)
Links
Similar tables
Prolate spheroidal differential eigenvalues —   The commuting differential operator has the same order-zero eigenfunctions and mode index.
Data properties
Entries are of type: real number
Table is complete: no (it holds every mode $0\leq n\leq11$ at $S\in\{1,2,4,8\}$, a finite reference selection of small integral effective dimensions that reaches both sides of the concentration transition)
How well the digits are known: heuristic (agreement-checked)
How they were obtained:

The attached generator uses the parity-separated symmetric tridiagonal differential operator in the orthonormal Legendre basis [1]. It obtains the finite-Fourier eigenvalue from the integral equation at the origin for even modes and its derivative there for odd modes. The concentration eigenvalue is then $c|\mu_n|^2/(2\pi)$. Finite Legendre truncation is described in [3].

more

Arithmetic at 160 and 220 decimal digits is compared at 80 terms per parity block. At 220 digits the truncation is independently enlarged to 120 and 180 terms. Every successive comparison must agree to at least 110 significant digits before 100 digits are retained.

The attached checks independently discretize the sinc integral operator using 120 Gauss–Legendre nodes and 170 decimal working digits. Every stored mode must agree with this second formulation to at least 110 significant digits. A constant kernel with known spectrum $2,0,0,\ldots$ and exact polynomial moments control the quadrature and eigensolver. The complete discretized spectrum is checked against the trace $S$ and against the squared-kernel double integral. Ordering and the strict bounds $0<\Lambda_n<1$ are checked throughout. The read-back check compares every database value with the independent eigenvalue within the uncertainty denoted by its last written digit.

These are numerical agreement checks. Neither infinite-operator truncation nor quadrature error has a rigorous bound in this computation.



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