Lebesgue constants of the Fourier partial sums $L_n$

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Numbers

$n$

$L_n$

comment: $D_0(t)=1$, so $L_0=1$ exactly.

1.435991124176917432355986329959272216128106294066614638932065373915393940271872923014093390979675112

1.642188435222121136873627988922940344183823455484887430077620420770466569552502492845640833277007972

1.778322861525875862960961188959283135441615191461574508066345991237606055185700573101526397147234053

1.880080599102355399877522906972195437936885016805366551410099212508227215633808821029341733432795515

1.961360593766014947527659546541117505150139790060939077020073503041434996150826384452108667037480161

2.029036981624414960340196484147123467817324363122831601432427007913247255998768482421769483189600918

2.087015938756523923078422888383532088899899230612586680504350152918111166123188372956412206918796373

2.137730862548190651474899039688511726112033542052857475502927456836696264536402867472531219028351532

2.182800647213984296611656437925817496723730265861997869446107937704558075430011096171140568466098510

10:

2.223356924153684062305285817829282216841559968884406222452036735277608986108043285999852732791171313

11:

2.260221876745754939795283970266838497855627733230597158929239891950232661652610349932419721415918060

12:

2.294011690055894615104301331241174695939858779246712398445260016567095210295454799792717220691719563

13:

2.325200089292407089938012074049204593391218693640993659722827487541284165973374440216578051263857341

14:

2.354159126778002413132775624463603281175770381585720106423143247729942118257822870199907555040161218

15:

2.381186342918069553938974344378764498210053727712444679544408338490156264333106974627560170353906800

16:

2.406523422962313192617928017607851106293716636170455659703746125469427509322712818900553800098729336

17:

2.430369357209376260752767032525107179406812370682300156233982606459482039559419119034356030125102794

18:

2.452889940149246671393361033478507719629608449397629012208639349716055568129903364218384315855965290

19:

2.474224766519220516180093419037079815381561600471360952602843654914008211285565939084602294059814401

20:

2.494492476272470522327267282460681812376224555508294666013082144128075144655381713619739706893312392

21:

2.513794749412117000867617567772939819444408567948946843642566521847684578484595705162327634811975519

22:

2.532219392031881438936540842856836882877683046509497095689786008122809535117233788327790221119860387

23:

2.549842750889399091334764310306007523461085861353627527571643252781739338995866708873012549498481069

24:

2.566731624540150298504232818831961242965579194789542162176683200885048297649337847453486275437422895

25:

2.582944791963656755177738126544941976340770913449624619653490483020585805990850341674502714743535541

26:

2.598534247024267652375702663311459453431575622403296745085783397147629506496010347966926323339617081

27:

2.613546204185333424862611609615674498843614736333902470456321192984374150702375082814314357346475126

28:

2.628021924527690635004539947505549721903681429781006154832006422311649145526343160406807356567048724

29:

2.641998399274661002730421531472390146775311470437518765941358668804565292945855389474458688789614072

30:

2.655508919339226461314322902340896499199387598428885390438557033654122803623090224283145704956245982

31:

2.668583552965854465089702701575919742343212980039109406282806703080606780099768972222963923061168392

32:

2.681249548708174325773563009798524804583856656030439409033098850623761293629009512656026935088724087

33:

2.693531677324409928017195675529546731478569630939556546668188226287852240847299766957680690217751979

34:

2.705452523374713545548098444154964302973662465159077913498864660236694545012497814178812389929335668

35:

2.717032735146638786623897596725463888184447815759618411137556536506296929412722497528294851382747536

36:

2.728291239856877073157191931812745709165618542250121110267696240631799289268588040180324862122565318

37:

2.739245429762288106886803303885642624234453238479558904622727524704642072137840646554365201373872060

38:

2.749911323775155566485481902814550984923949744904174932656988554400588176259506298289251500555745494

39:

2.760303708352540189762507848037440437306051768705218501609868772389856831745589792210854178458979026

40:

2.770436260769626129845422945420603458012371489057544824393684842682774504003739962077113121841640229

41:

2.780321657355808653940593775502579486147679837021519893939844839538189170897144732474794093490450781

42:

2.789971668842333707900299372440826600121704666190908242531876341032861738398183983397767074103251444

43:

2.799397244620380240765005000454586788082633359310931254711915524491918132262236728340737852495845786

44:

2.808608587422200685371459909395752192556520478579509148326118649242897138223781650043049931381470411

45:

2.817615219702569989907124028194392874983926261411167213502131746866584054830513356147528978811263592

46:

2.826426042803374750427339807604892544387857217865950444625076005087144025437499692919111951071634596

47:

2.835049389822857851442066053150891851410330326718880171323011090237647994353203538939185067732033341

48:

2.843493072976611778994957665449194687250306356043025445026859629902959822776982431459604204689077581

49:

2.851764426124944173929139706784619921634696171628546847661157553079613186682776081606128727303660288

50:

2.859870343046769571397884947115856779974251593508088762288786733400206441029181520068898528869749459

51:

2.867817311960534238859490296784942392228003192127204603510686400024617342899820124028944173966830567

52:

2.875611446725290416762194094692885055133871866851999071468401135327906718395816064387466623939711916

53:

2.883258515097820247387353958659793341414823477312614228422953052174024493642380626083133796954546927

54:

2.890763964372972046906117640117797002492910374301935375584412565722691809830336669154326323385933965

55:

2.898132944692724617663005348304347404860733488579922780089031352247432859329013442425155146664830332

56:

2.905370330273799168128775473135284741713986352161061983880265752149001177906065727166149176325139646

57:

2.912480738772953286605845312885011627052672871511731017907507238488535522132333643260811730737789991

58:

2.919468548982639559202502889932920324656500540019061585781883590840955064912618593782634852655066853

59:

2.926337917026846942432370177662481021604150429831050346796939409823672873285794377925465804643905099

60:

2.933092791207127678719451060037467053658103666897431204751159669988194679677031049812045139602993410

61:

2.939736925631596413647951376163112973990944251569295269947253297746636161240666506266487002848147922

62:

2.946273892744693862137490757895060833541393213162343158973709417850935351279930527266402221500643902

63:

2.952707094862417415157101518173274143633079779791146111097357555113129542325372340718449166064907218

64:

2.959039774806267630742025027340292506536000282159505214464311540743876331568501121995731035650014612

65:

2.965275025719115979999415990761339158762462461810526058106521040068147081312542076147761879029847813

66:

2.971415800137373232306743861132420591402353923247610912274228761269815184740783843184097923486290730

67:

2.977464918386065880792292174190563685748741857658180709043406811786613909460620620717269176080208361

68:

2.983425076356570506812852776929404176463418915509366797125821494506629755523935470259521958792565240

69:

2.989298852720693662011809299745867507304386900337041338078294634276266509211924188231523953662536755

70:

2.995088715629415455625841539339220560749593443661786061015952965812208768808613733040302871126296545

71:

3.000797028939850724141254797703826911776848431981051953195283172536892136028464124230759903117820469

72:

3.006426058009746784454625746209251894359398110130881336563312459432630730874935242448569404503746638

73:

3.011977975095066048645338337828267407450162352434101353294965372539915793179925263942124172790188271

74:

3.017454864382838749757175257545420664437711586651941315456989837494381062714516666542485027362188581

75:

3.022858726688466763941212505339414122462604594054554518420040011348328480407614184072094169167879481

76:

3.028191483843971524277319954908797709797748364258937199024609231752307815209257865779712292271757056

77:

3.033454982801270333929707521041540817076937568865182434797805958453394827468271854214526741599739048

78:

3.038650999472403769948350234184323872443540999156103428373215715096578321889628890648695917238486643

79:

3.043781242326694173355297159476850910626504994943755450277898972900655840026096465674849654303060014

80:

3.048847355763066814779109087528246653522373924756004133206294413356326321106771315694639003358468116

81:

3.053850923274189617731580003986826822676475896622404929431827349348424800397115954847771630442010777

82:

3.058793470417665356820777268901298433433306606970588366602804774855038437530652177254396893090721106

83:

3.063676467608225353148288392900184833411919645139863945442307999631092228222596188483602545625381623

84:

3.068501332743711176292322000274602162109642801789893320110090695559716662513403532735320288090998818

85:

3.073269433676577772217286307980549255031261877138791849274053153684132842984427599388622695064324528

86:

3.077982090541696316831960790897167404060914749874126015100437635804109105190970959733138947589422970

87:

3.082640577950367809900208517808010247183925456624223484188222608362957641017460889304010717752764890

88:

3.087246127059669989667457073890003102613522114106449335494078310797524912473810095713649077384168535

89:

3.091799927525542590511474090060815610398939210051183920780404981814212420686789511852547965823402046

90:

3.096303129347362195896766598680969598093076993384756622587802051221901835817646889235044486498606758

91:

3.100756844611161647652582397270407184833445222023703319828795390778060465144857707644718951382744594

92:

3.105162149138104536702656222969188125629225171002305110784602550111273921616025955810641399182215151

93:

3.109520084044327701170876115762246373697772159521297199823871590832195546290545142566212888071038798

94:

3.113831657217809410541554904210893063974043252323758102883987138878011217710041954074124799955435189

95:

3.118097844717504006655036686694161680418546886495271430961893416884650425160237006589770948122062695

96:

3.122319592099601609174423072740612795614297903492811682570089928902481008369412912874445378956092919

97:

3.126497815675420850780837922072020376762415033265072862077568825292835137834327933537338520599705168

98:

3.130633403705120590761879555541709667783200075830751083095693443496520523396084358250127064787648669

99:

3.134727217531120562315788351263445748138585303746236740343734655567180721106266920082026876307926475

100:

3.138780092654848596541606121517657169026136877041098003621480972764059019024131304930531009747635422

Definition

For each nonnegative integer $n$, the Lebesgue constant of the Fourier partial sum of degree $n$ on continuous $2\pi$-periodic functions is $L_n=\frac{1}{\pi}\int_0^\pi |D_n(t)|\,dt$, where $D_n(t)=1+2\sum_{k=1}^n\cos(kt)$ is the Dirichlet kernel [3] [1].

Parameters

$n$

— Fourier degree ($n\geq0$)

Formulas

(1)

$D_n(t)=\frac{\sin((n+\tfrac12)t)}{\sin(t/2)}$ for $0<t\leq\pi$, with $D_n(0)=2n+1$.

(2)

$L_n=\frac{1}{2n+1}+\frac{2}{\pi}\sum_{k=1}^{n}\frac{\tan(\pi k/(2n+1))}{k}$ for $n\geq0$, with an empty sum for $n=0$ [2].

(3)

$L_0=1$, $L_1=\frac13+\frac{2\sqrt3}{\pi}$, and $L_2=\frac15+\frac{\sqrt{25-2\sqrt5}}{\pi}$ [4].

Comments

(4)

The Fourier partial sum is $S_n f(x)=\frac{1}{2\pi}\int_{-\pi}^{\pi}D_n(t)f(x-t)\,dt$. The constant $L_n$ is its operator norm for the uniform norm on continuous $2\pi$-periodic functions: it is the least $K$ for which $\|S_n f\|_\infty\leq K\|f\|_\infty$ for every such $f$ [2].

(5)

The index $n$ is the largest retained Fourier frequency. The sum retains the $2n+1$ complex modes with frequencies $-n,\ldots,n$; $n=0$ is the mean projection.

(6)

Lebesgue constants for interpolation nodes are norms of polynomial interpolation operators. The constants here are norms of Fourier projection operators [4].

(7)

The decimal expansions of $L_1$, $L_2$ and $L_3$ are recorded in [5], [6] and [7], respectively.

Programs

(P1)

Sage

import numberdb.sage as numberdb
from sage.rings.integer_ring import ZZ
from sage.rings.real_arb import RealBallField

def fourier_lebesgue(n, digits=100):
    n = ZZ(n)
    if n < 0:
        raise ValueError("n must be nonnegative")
    if n == 0:
        return ZZ(1)
    R = RealBallField(numberdb.bits(digits, losing=96))
    pi = R.pi()
    return R(1)/(2*n+1) + 2/pi*sum(
        ((pi*k/(2*n+1)).tan()/k for k in range(1, int(n)+1)), R(0))

fourier_lebesgue(101)  # the next degree after this table

References

[1]

Steven R. Finch, Mathematical Constants, Cambridge University Press (2003), section 4.2, pp. 250–255.

[2]

I. A. Shakirov, On two-sided estimate for norm of Fourier operator, Ufa Mathematical Journal 10(1) (2018), pp. 94–114; equations (3) and (5). (doi)

Links

[3]

DLMF: Fourier series, equation 1.8.8

[4]

MathWorld: Lebesgue constants

[5]

OEIS A226654: decimal expansion of L_1

[6]

OEIS A226655: decimal expansion of L_2

[7]

OEIS A226656: decimal expansion of L_3

Similar tables

Lebesgue_constants_of_the_classical_interpolation_node_families — operator norms for polynomial interpolation, indexed by the interpolation degree and node family

Gibbs_constant — the limiting overshoot near a jump discontinuity in Fourier partial sums

Data properties

Entries are of type: real number

digits: 100 (Unknown key)

How well the digits are known: proven

Table is complete: no (it holds every degree $0\leq n\leq100$, beginning with the mean projection and the classical small-degree cases and continuing through consecutive Fourier truncation degrees)

How they were obtained:

The finite tangent sum (2) is evaluated in Sage RealBallField at 429 bits, consisting of 333 bits for 100 significant decimal digits and 96 guard bits. Every nonexact value retains at least 420 bits of relative accuracy, and the largest computed radius is less than $6\times10^{-127}$. The value $L_0=1$ follows exactly from $D_0(t)=1$ and is stored as an integer.

All 101 entries were compared with an independent 640-bit evaluation of the defining integral. This evaluation splits at $t=2j\pi/(2n+1)$, where the sign of $D_n$ changes, and integrates its finite cosine expansion using $t+2\sum_{k=1}^n\sin(kt)/k$. The checker first reproduces the known values at $n=0,1$. Every comparison requires finite balls of radius less than $10^{-110}$. Recalculation with 160 extra working bits also agrees throughout.

The closed forms at $n=0,1,2,3$ provide further checks. The first three nontrivial values agree with all 104, 100 and 100 digits supplied in the OEIS b-files for [5], [6] and [7]. The attached checks also test strict increase and the classical logarithmic bounds on the stored range. Each written decimal denotes an interval of one unit in its last place on either side; its entire independent integral ball is required to lie inside that interval.