Orthogoal Dirichlet Polyomials with Arctaget Desity Doro S. Lubisky School of Mathematics, Georgia Istitute of Techology, Atlata, GA 3033-060 USA. Abstract Let { j } j= be a strictly icreasig sequece of positive umbers with =. We fid a simple explicit formula for the orthogoal { } Dirchlet polyomials {φ } formed from liear combiatios of, associated with the arctaget desity. Thus it j j= dt φ tφ m t π + t = δ m. We obtai formulae for their Christoffel fuctios, ad deduce their asymptotics, as well as uiversality limits, ad spacig of zeros for their reproducig kerels. We also ivestigate the relatioship betwee ordiary Dirichlet series, ad orthogoal expasios ivolvig the {φ }, ad establish Markov- Berstei iequalities. Key words: Dirichlet polyomials, orthogoal polyomials. Itroductio Throughout, let = < < 3 <.. A Dirichlet series associated with this sequece of expoets has the form a it = a e ilog t. Research supported by NSF grat DMS008 ad US-Israel BSF grat 008399 Email addresses: lubisky@math.gatech.edu Doro S. Lubisky Preprit submitted to Joural of Approximatio Theory October 4, 03
I particular, whe j = j, j, we obtai the stadard Dirichlet series, of which the Riema zeta fuctio is a special case. It was Harald Bohr [] who developed much of the theory of almostperiodic fuctios. Oe of its basic tools is that if 0 < α β <, the α it ad β it are orthoormal o, i the mea, that is lim T T T T α it β it dt = δ αβ.. Cosequetly, if {a } ad {b } are square summable, ad the f t = lim T T a it ad g t = T T f t g tdt = b it, a b. Thus oe ca idetify spaces of Dirichlet series with the sequece space l. The mai results of the theory iclude existece ad uiqueess of o-harmoic Fourier series for almost periodic fuctios, ad their approximability by oharmoic trigoometric polyomials. Notable cotributors, i additio to Bohr, iclude Bocher, Stepaov, ad Besicovitch [], []. This has led to a very rich theory, i which Dirichlet polyomials { m } L m = a it : a,a,...,a m C, m,.3 have also bee extesively studied [6], [7]. It is the purpose of this paper to ivestigate various properties of Dirichlet polyomials, usig the arctaget desity,t,. Our π+t hope is that a more direct orthoormality relatio tha., might have some advatages. Our aalysis uses the orthoormal polyomials {φ } formed by applyig the Gram-Schmidt process to { it } with respect to the arctaget desity. Thus φ L, has positive leadig coefficiet, ad dt φ t φ m t π + t = δ m, m,..4 These Dirichlet orthogoal polyomials admit a very simple explicit expressio:
Theorem.. For =, φ =, ad for, φ t = it it..5 The author has searched the extesive literature of Dirichlet polyomials, ad ot foud this, eve i the special case j = j. There of course, it it arises i oe of the stadard ways of aalytically cotiuig the Riema zeta fuctio, via summatio by parts. We believe that eve if.5 is kow, at least the applicatios below are ew. Some elemetary properties of {φ } are give i the followig propositio. I it, ad i the sequel, we use the covetio 0 = 0. Propositio.. Let. a sup φ t = t R b The zeros of φ are simple ad have the form i + +..6 kπ, k Z..7 log / c d e f sup φ t log + log =..8 t R it m = m j= φ t π + t dt = log + if c,c,...,c C it j= j j φ j t..9 c j it j log..0 dt π + t =.. 3
The machiery of orthogoal fuctios ad the simplicity of the formula above allow us to aalyze reproducig kerels, Christoffel fuctios, ad Markov-Berstei iequalities. The mth reproducig kerel is K m x,t = ad mth Christoffel fuctio is Λ m x = It admits the extremal property Λ m x = if φ xφ t, K m x,x = m φ x.. { } dt P t π+t P x : P L m..3 Christoffel fuctios are a essetial tool i aalysis of orthogoal polyomials [5]. We eed the sic ad hyperbolic sic kerels S z = siπz πz ; i describig limits of Christoffel fuctios. S z = sih πz πz.4 Theorem.3. a For s,t C, ad m, K m s,t = + 4 + i s t = [ s + t si log 4 b For real x, ad m, si [ t s 4 + i ] log ]..5 K m x,x = + = [ x + 4 si log ]..6 4
c For all real x, ad m, K m x,x + d Moreover, if as m, = + x..7 + m ad m m = + o,.8 the as m, uiformly for s,t, i compact subsets of the complex plae, K m s,t = s t + is it is t/ m log m S π log m + o Ims t / m log m S Ims t log m. π.9 e As m, uiformly for x i compact subsets of the real lie, K m x,x = + x log m + o..0 We ca deduce uiversality limits cf. [4] for the reproducig kerels, ad asymptotics for their zeros: Theorem.4. Assume.8. a We have, uiformly for α,β i compact subsets of C, ad x i compact subsets of the real lie, lim K m x + α,x + β m log m log m log m = [ + x ] α β e iα β/ S.. π b Let x R. The for each fixed iteger j = ±, ±, ±3,..., ad large eough m, K m x,t has a simple zero t m,j, which satisfies lim t m,j x log m = jπ.. m Moreover, { give r > 0, } for large eough m, the oly zeros of K m x,t i z : z x r log m are the zeros {t m,j }. 5
Next, we tur to Markov-Berstei iequalities, which estimate derivatives of Dirichlet polyomials. There is a substatial literature for such iequalities for Műtz polyomials [3], but the author has ot foud ay such results for Dirichlet polyomials. Theorem.5 Markov-Berstei Iequality. For P L m, I particular, P t / π + t dt P t / π + t dt / log m + j j..3 j + j P t / π + t dt P t / π + t dt log m + log m /..4 Propositio. e shows that this is essetially sharp with respect to the order of log m, ad moreover, just a growth factor of log m is isufficiet we eed a extra smaller term i the last right-had side. Fially, we tur to orthoormal expasios. Let { } H = f = a φ : a <..5 This is a subspace of the weighted L space cosistig of measurable fuctios f : R R with f = j= f t / π + t dt <, which we deote by G. For f H ad m, we deote the mth partial sum of its orthoormal expasio by S m [f] = a φ,.6 6
where for, a = a [f] = f tφ t π + t dt..7 The relatioship betwee formal orthoormal expasios ad formal Dirichlet series is give i: Theorem.6. expasio For, let The for m, a Let {a } C ad fdeote the formal orthoormal f = a a φ..8 b = S m [f] = m a +..9 + b it + a m m m m it m..30 b Coversely, let {b } C. Choose a C, ad for m, let m a m = m b m a..3 Defie f by the formal orthoormal expasio.8. The the partial sums S m [f] satisfy.30 for m. Uder additioal coditios, we ca give aalytic meaig to these formal idetities: Theorem.7. a Let {b } C be a sequece for which b.3 coverges. Defie a m for m by a m = m m =m b,.33 ad f by.8. The the coclusio.30 of Propositio.6a remais valid. 7
b If i additio m m b m= =m <,.34 the f defied by.8 has f H, ad this last sum equals f. c If i additio lim m a m m m m = 0,.35 we have lim m ad as fuctios i G, I particular, if S m [f] f t = P t = the Theorem.7b implies that P t l π + t dt =. Proofs of Theorems.-.4 m= m l b it L R = 0,.36 b it..37 b it, m m l b =m..38 Proof of Theorem.. Let φ # it t = it.. We use the followig simple cosequece of the residue theorem: for real µ, e iµt π + t dt = e µ.. 8
The for µ log, e iµt φ # t π + t dt = e iµ log t e iµ log t dt π + t = e µ log e µ log = e µ e µ = 0. For log < µ < log, istead φ # e iµt π + t dt = e µ log e µ log For µ log, istead φ # I summary, φ # = e µ e µ. e iµt π + t dt = e µ log e µ log e µ e µ. e iµt π + t dt = = 0, µ log e µ e µ, log µ < log. e µ, µ log.3 This immediately yields the desired orthogoality relatios for φ = φ #. Fially,.3 shows that φ # t dt π + t = φ # t ilog t e π + t dt =. 9
Proof of Propositio.. a φ t = / it + = +, with equality if t = π/log /. b This is immediate. c φ t = i it log it log,.4 so the result follows as i a. d Usig our covetio 0 = 0, j= e From.4 ad d, j j φ j t = j= it j it j = it m. φ t i = log j j φ j t log j j φ j t j= j= i = j j φ j tlog log ilog φ t. j= So by orthoormality, φ t π + t dt = j= j j log log + log. This telescopes to the right-had side of.0. 0
f Let ψ t = it = it φ t deote the th moic Dirichlet orthogoal polyomial. The orthoormality relatios show that for ay moic Dirichlet polyomial P t = ψ t + j= a jφ j t, we have P t π + t dt = + a j. j= Thus, the if over such moic polyomials P is, with equality iff P t = ψ t. Proof of Theorem.3. a Let ad τ = log. Elemetary trigoometric idetities give φ tφ s = it it +is +is [ s t +i i t ] i t = [ +i s ] +i s s t +i = 4 si t + i τ si s i τ [ s t +i = cos t s + i τ cos s + t τ ] s t log +i = 4 [si s + t 4 ] log si t s + i. 4 Now add for =,3,...,m, ad recall φ t =.
b Whe s = t = x, the above idetity simplifies to φ x = 4 [ si x log si i log = + 4 si x log Now add over =,3,...,m, ad recall φ t =. c Usig first siu u, for all real u, ad the log + u u for u 0, K m x,x + + = + = = =. ] [ [ x ] ] + 4 log / [ [ x + 4 [ [ ] ] x +. ] ] d Usig si t = t + o as t 0, we see that as m, with the o term below havig limit 0 as, K m s,t +is t + o = 4 = log [ s + t t s + i + o + o] 4 4 +is t + o = 4 = [ s + t t s + i + o + o] 4 4 is t = = [ + is t + st + o] is t = [ + is it + o], + =
uiformly for s, t i compact subsets of the plae. Agai usig.8, we cotiue this as u is t = + o [ + is it + o] du = u = m + is it u is t du + o u Imt s du = = m + is it u is t du + o = m u Imt s du. Here we are also usig that m as m, so that the o term grows at least as fast as log m. Simple calculatios show that for complex α, real oegative β, ad for T, T α u iα du = T iα/ log TS π log T ; T u β du = T β/ log T S β π log T. Hece K m s,t = + o + is it is t/ m Ims t / m log m S s t log m S Im s t π π log m log m e Settig s = t = x, we also obtai.0. Proof of Theorem.4. a We choose s = x + ad α log m ad t = x + β log m i.9. We see that + is it = + x + o, s t is t/ m S π log m = e iα β/ S The. follows from.9. α β b This follows directly from a, from Hurwitz theorem, ad the fact that the oly zeros of S z are the o-zero itegers. 3 π.
3. Proof of Theorem.5 Proof of Theorem.5. Now for j,.5 ad.4 show that φ j t + ilog j φ j t = i log j log j it j j, 3. j so φ j t + ilog j φ j t j = j j j / log j j. 3. Next if P t = we recall that φ t = ad write P t = a j φ j t, j= [ a j φ j t + ilog j φ j t ] j= =: T t + T t. a j ilog j φ j t Here, usig Cauchy-Schwarz, 3. ad the iequality log + u u,u 0, T t a j / j= j= j j P t / π + t dt j= j j j j j + j j= / j j so the triagle iequality ad orthoormality, ad our boud o T give P t / π + t dt T t / π + t dt + P t / π + t dt j=, T t / π + t dt j j j + j / / 4
+ a j log j j= / / P t π + t dt / log m + j j. j + j j= Here j= j j j + j j= j j j j= j dt j t = log m. We ote that usig our explicit expressio for φ j, it is possible to obtai a explicit orthoormal expasio for P i terms of the {φ j }. However, estimatio of that does ot seem to lead to a better estimate tha that i.3/.4. 4. Proof of Theorems.6 ad.7 Proof of Theorem.6. a From Theorem., S m [f] = = a m it [ it a it ] a + + + a m m m by a summatio by parts. Our defiitio.9 of {b } gives the result. b It is easily see from.3 that b satisfies.9 for, so the result follows from a. Proof of Theorem.7. it m, 5
a If {a m } are defied by.33 for m, the it is easily see that.9 is satisfied for, ad Theorem.6a yields the result. b Our hypothesis.34 asserts that a <, so ideed f = a φ H, ad f = a = m m m= b =m. c From.30, S m [f]t m b it L R = a m m. m m The.36 follows from.35. Moreover, the m f t b it f S m [f] + S m [f]t m / a + S m [f]t =m 0, m. b it m b it L R The.37 follows. Refereces [] A. S. Besicovitch, Almost Periodic Fuctios, Dover, New York, 954. [] H. Bohr, Almost Periodic Fuctios, Chelsea, New York, 947. [3] P. Borwei ad T. Erdelyi, Polyomials ad Polyomial Iequalities, Spriger, New York, 995. [4] P. Deift, Orthogoal Polyomials ad Radom Matrices: A Riema- Hilbert Approach, Courat Lecture Notes, Vol. 3, New York Uiversity, 999. 6
[5] P. Nevai, Geza Freud, Orthogoal Polyomials ad Christoffel Fuctios: A Case Study, J. Approx. Theory, 48 967, 3 67. [6] K. M. Seip, Estimates for Dirichlet Polyomials, CRM Notes, 0, olie at www.yumpu.com/e/documet/view/0090/estimatesfor-dirichlet-polyomials-kristia-seip-ems-. [7] M. Weber, Dirichlet polyomials: some old ad recet results, ad their iterplay i umber theory, i Depedece i probability, aalysis ad umber theory, 00, Kedrick Press, Heber City, UT, pp. 33 353. 7