Polynomial approximations in a generalized Nyman-Beurling criterion
aa r X i v : . [ m a t h . F A ] J un POLYNOMIAL APPROXIMATIONS IN A GENERALIZEDNYMAN-BEURLING CRITERION
F. ALOUGES – S. DARSES – E. HILLION
Abstract.
The Nyman-Beurling criterion, equivalent to the Riemann hypothesis, is an ap-proximation problem in the space of square integrable functions on (0 , ∞ ) , involving dilationsof the fractional part function by factors θ k ∈ (0 , , k ≥ . Randomizing the θ k generates newstructures and criteria. One of them is a sufficient condition that splits into (i) showing that theindicator function can be approximated by convolution with the fractional part, (ii) a control onthe coefficients of the approximation. This self-contained paper aims at identifying functions forwhich (i) holds unconditionally, by means of polynomial approximations. This yields in passinga short probabilistic proof of a known consequence of Wiener’s Tauberian theorem. In order totackle (ii) in the future, we give some expressions of the scalar products. New and remarkablestructures arise for the Gram matrix, in particular moment matrices for a suitable weight thatmay be the squared Ξ -function for instance. Introduction
The Riemann hypothesis (RH) is equivalent to the Nyman-Beurling (NB) criterion, which isan approximation problem of the indicator function χ of (0 , in the space of square integrablefunctions on (0 , ∞ ) , involving dilations of the fractional part function {·} by factors θ k ∈ (0 , : Theorem 1 ([BDBLS00]) . RH holds if and only if, given ε > , there exist n ≥ , coefficients c , . . . , c n ∈ R , and θ , . . . , θ n ∈ (0 , such that (1.1) Z ∞ χ ( t ) − n X k =1 c k (cid:26) θ k t (cid:27)! dt < ε. Baez-Duarte [BD03] showed that it is possible to specify θ k = k in this criterion. The coefficients ( c k ) k n leading to the best approximation, then solve the linear system Gx = b, where x = ( c , · · · , c n ) T , b = ( b , · · · , b n ) T , with b k = R (cid:8) kt (cid:9) dt and G is the Gram matrix G k,l = Z + ∞ (cid:26) kt (cid:27) (cid:26) lt (cid:27) dt, for ≤ k, l ≤ n . The computation of G is possible through a formula due to Vasyunin [Vas95]. The evaluation ofthe distance in (1.1) may be then approximately evaluated by numerical means [BDBLS00].To what extent is it possible to relax these intricate structures?Recently, randomizing the θ k has produced new characterizations and structures [DH19]. Amongothers, the following generalization of the if part is obtained. From now on, we write E Z for theexpectation of a random variable (r.v.) Z . Theorem 2.
Let ( Z k ) k ≥ be positive r.v. satisfying, for any α ≥ , E Z αk ≪ α k α , k ≥ . (1.2) Keywords:
Polynomial approximation; Zeta function; Nyman-Beurling criterion; Probability.
If there exist real coefficients ( c k,n ) ≤ k ≤ n,n ≥ such that D n = Z ∞ χ ( t ) − n X k =1 c k,n E (cid:26) Z k t (cid:27)! dt −−−−→ n →∞ (1.3) n X k =1 kc k,n P ( Z k ≥ M n ) −−−−→ n →∞ , for any M n → ∞ , (1.4) then RH holds. The relevancy of the former theorem stems from the fact that it is possible to show the converseholds for some specific structures, as dilated or concentrated r.v., see [DH19] for more details.Notice that, for r.v. Z k with densities φ k , the involved functions(1.5) E (cid:26) Z k t (cid:27) = Z ∞ n xt o φ k ( x ) dx, are almost multiplicative convolutions. Indeed, if, for g : R + → R , we denote by g × ( t ) itsmultiplicative convolution with the fractional part defined by(1.6) g × ( t ) = Z ∞ n xt o g ( x ) dxx = (cid:18)(cid:26) · (cid:27) ∗ g (cid:19) ( t ) , then we have E (cid:26) Z k t (cid:27) = g × k ( t ) with g k ( x ) = xφ k ( x ) . Eventually, we observe that this latter expression allows for a generalizationwith functions g k that possibly change sign. Definition 1.1.
We say that a sequence of real function ( g k ) k , resp. a sequence of r.v. ( Z k ) k ,verifies g NB if there exist coefficients ( c k,n ) ≤ k ≤ n,n ≥ such that D n = Z ∞ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) χ ( t ) − n X k =1 c k,n g × k ( t ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dt −−−−→ n →∞ , (1.7) resp. if (1.3) holds. The main purpose of this paper is to identify classes of functions g k or corresponding r.v. Z k that verify gNB, unconditionally (i.e. without assuming RH). In each case, we also provide someexpressions of the scalar products G k,j = h g × k , g × j i , which may be necessary to tackle Condition(1.4) in the future. 2. Preliminaries
We say that g = R + → R verifies Assumption (M) if Z ∞ | g ( x ) | dx < ∞ and Z ∞ | g ( x ) | dxx < ∞ .We write a complex number s = σ + it , σ, t ∈ R . The previous assumption allows to define theMellin transform b g of g in the critical strip < σ < : b g ( s ) = Z ∞ g ( x ) x s − dx, since Z ∞ | g ( x ) | x σ − dx ≤ (cid:18)Z ∞ | g ( x ) | dx (cid:19) σ (cid:18)Z ∞ | g ( x ) | dxx (cid:19) − σ , due to Hölder inequality. As-sumption (M) is sufficient but not necessary for the Mellin transform to be defined, and we recallthat Mellin-Plancherel theory allows to define b g ( s ) whenever g ∈ L (0 , ∞ ) . In this case, we havethe isometry: Z ∞ | g ( x ) | dx = 12 π Z ∞−∞ (cid:12)(cid:12)(cid:12)(cid:12)b g (cid:18)
12 + it (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) dt. OLYNOMIAL APPROXIMATIONS IN A GENERALIZED NYMAN-BEURLING CRITERION 3
We finally recall the fundamental identity on which relies the NB criterion, see [Tit86, (2.1.5)]: Z ∞ (cid:26) x (cid:27) x s − dx = − ζ ( s ) s , < σ < . In the case where g ( x ) = xφ ( x ) and φ is a density of a r.v. Z ≥ , Assumption (M) translateseasily as E Z < ∞ . The following lemma is standard, but we give a proof in our framework for thesake of completeness. Lemma 2.1.
Let g = R + → R satisfying (M). Then g × ∈ L ( R + ) and its Mellin transform c g × ( s ) is well defined for σ ∈ (0 , . Moreover, c g × ( s ) = − ζ ( s ) s b g ( s ) , < σ < . If Z ≥ is an integrable r.v. then t h × ( t ) := E (cid:8) Zt (cid:9) belongs to L ( R + ) and c h × ( s ) = − ζ ( s ) s E Z s , < σ < . Proof.
Writing | g ( x ) | = | g ( x ) | / | g ( x ) | / and using Cauchy-Schwarz, one obtains: Z ∞ | g × ( t ) | dt ≤ Z ∞ (cid:18)Z ∞ n xt o | g ( x ) | dxx (cid:19) dt ≤ Z ∞ Z ∞ | g ( x ) | dxx Z ∞ n xt o | g ( x ) | dxx dt = Z ∞ | g ( x ) | dxx Z ∞ | g ( x ) | dx Z ∞ (cid:26) t (cid:27) dt < ∞ , due to assumption (M). We already noticed that for all σ ∈ (0 , , Z ∞ | g ( x ) | x σ − dx < + ∞ , so Z ∞ Z ∞ (cid:12)(cid:12)(cid:12)(cid:12)n xt o g ( x ) x t s − (cid:12)(cid:12)(cid:12)(cid:12) dxdt ≤ Z ∞ (cid:26) t (cid:27) t σ − dt Z ∞ | g ( x ) | x σ − dx < + ∞ . Hence c g × is well defined and we can apply Fubini’s theorem, which justifies: c g × ( s ) = (cid:18)(cid:26) · (cid:27) ∗ g (cid:19) V ( s ) = d(cid:26) · (cid:27) ( s ) b g ( s ) = − ζ ( s ) s b g ( s ) . Now, let Z ≥ be a integrable r.v. and set g × ( t ) = E (cid:8) Zt (cid:9) . The situation reads even simpler: Z ∞ | g × ( t ) | dt ≤ E Z ∞ (cid:26) Zt (cid:27) dt = E Z Z ∞ (cid:26) t (cid:27) dt < ∞ . Again, c g × is well defined by Fubini, and the result simply follows from a change of variable: c g × ( s ) = Z ∞ E (cid:26) Zt (cid:27) t s − dt = E Z ∞ (cid:26) Zt (cid:27) t s − dt = E (cid:20) Z s Z ∞ (cid:26) u (cid:27) u s − dt (cid:21) = − ζ ( s ) s E Z s , without even assuming that Z has a density. (cid:3) Remark 2.2.
Notice that weaker assumptions, such as E Z σ < ∞ , < σ < , are sufficient toshow g × ∈ L , following the first lines of the proof with a different use of Cauchy-Schwarz. Since the methodology developed hereafter mainly uses density of polynomials in a suitableweighted L space, we recall the classical following result: Lemma 2.3 ([Nik12], (c) p.76) . Let ν be a positive measure on R with Z R e at dν ( t ) < ∞ for some a > . Then the polynomials are dense into L ( R , ν ) . F. ALOUGES – S. DARSES – E. HILLION
Finally, we generalize Theorem 2 replacing E { Z k /t } by functions g × k , which allows us to obtainsome sharper estimates. Theorem 3.
Let ( g k ) k ≥ be functions verifying Assumption (M). Let φ be an integrable functionon R + such that b φ exists and does not vanish in / < σ < , and M Z + ∞ M φ ( t ) dt −−−−→ M →∞ . If there exist coefficients ( c k,n ) ≤ k ≤ n such that for any σ ∈ (1 / , , there exist M n → ∞ with M σ − n Z ∞ φ ( t ) − n X k =1 c k,n g × k ( t ) ! dt −−−−→ n →∞ (2.8) M σ − n Z ∞ M n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n X k =1 c k,n g k ( t ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dt −−−−→ n →∞ , (2.9) then RH holds.Proof. We write for simplicity g = n X k =1 c k,n g k and n X k =1 c k,n g × k = (cid:26) x (cid:27) ∗ g = g × . The first lines of the proof follows the same ones as those of the Nyman-Beurling criterion. As-suming that ζ ( s ) = 0 , / < σ < , we show that b φ ( s ) = R + ∞ (cid:0) φ − (cid:8) x (cid:9) ∗ g (cid:1) t s − dt tends to as n → ∞ , contradicting b φ ( s ) = 0 . Following the proof in [DH19], we write: (cid:12)(cid:12)(cid:12)(cid:12)Z + ∞ (cid:0) φ − g × (cid:1) t s − dt (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z M n (cid:0) φ − g × (cid:1) t s − dt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)Z + ∞ M n (cid:0) φ − g × (cid:1) t s − dt (cid:12)(cid:12)(cid:12)(cid:12) = I + II .
We start to estimate I , by Cauchy-Schwarz: I ≪ M σ − / n Z M n (cid:0) φ − g × (cid:1) dt ! / . We now turn to II . Using the triangular inequality, n xt o = xt if x ≤ t , and ≤ n xt o tx ≤ , II ≤ Z + ∞ M n | φ | t σ − dt + Z + ∞ M n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z M n n xt o g ( x ) x dx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) t σ − dt + Z + ∞ M n (cid:12)(cid:12)(cid:12)(cid:12)Z + ∞ M n n xt o g ( x ) x dx (cid:12)(cid:12)(cid:12)(cid:12) t σ − dt ≤ M σ − n Z + ∞ M n | φ ( t ) | dt + M σ − n − σ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z M n g ( x ) dx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + M σ − n − σ Z + ∞ M n | g ( x ) | dx. We can estimate R M n g ( x ) dx as follows. On one hand, still using n xt o = xt if x ≤ t ,(2.10) Z + ∞ M n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z M n n xt o g ( x ) x dx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dt = Z + ∞ M n dtt Z M n g ( x ) dx ! = M − n Z M n g ( x ) dx ! , OLYNOMIAL APPROXIMATIONS IN A GENERALIZED NYMAN-BEURLING CRITERION 5 while, on the other hand, with ( a + b + c ) ≤ a + b + c ) and again using ≤ n xt o tx ≤ , Z + ∞ M n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z M n n xt o g ( x ) x dx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dt ≤ Z + ∞ M n (cid:18)Z ∞ n xt o g ( x ) x dx − φ ( t ) (cid:19) dt + 3 Z + ∞ M n φ ( t ) dt +3 Z + ∞ M n (cid:12)(cid:12)(cid:12)(cid:12)Z + ∞ M n n xt o g ( x ) x dx (cid:12)(cid:12)(cid:12)(cid:12) dt ≤ Z + ∞ M n (cid:0) φ ( t ) − g × ( t ) (cid:1) dt + 3 Z + ∞ M n φ ( t ) dt +3 M − n (cid:18)Z + ∞ M n | g ( x ) | dx (cid:19) . Due to (2.10), this leads to Z M n g ( x ) dx ! ≤ M n Z + ∞ M n (cid:0) φ ( t ) − g × ( t ) (cid:1) dt + 3 M n Z + ∞ M n φ ( t ) dt +3 (cid:18)Z + ∞ M n | g ( x ) | dx (cid:19) , which, put back in the bound of II with √ a + b + c ≤ √ a + √ b + √ c , gives II ≪ M σ − n Z + ∞ M n | g ( x ) | dx + M σ − n (cid:18)Z + ∞ M n (cid:0) φ ( t ) − g × ( t ) (cid:1) dt (cid:19) / + M σ − n (cid:18)Z + ∞ M n φ ( t ) dt (cid:19) / + M σ − n Z + ∞ M n | φ ( t ) | dt . We thus end up with (cid:12)(cid:12)(cid:12)(cid:12)Z + ∞ (cid:0) φ − g × (cid:1) t s − dt (cid:12)(cid:12)(cid:12)(cid:12) ≪ M σ − n Z + ∞ M n | g ( x ) | dx + M σ − n (cid:13)(cid:13) φ − g × (cid:13)(cid:13) L + M σ − n (cid:18)Z + ∞ M n φ ( t ) dt (cid:19) + M σ − n Z + ∞ M n | φ ( t ) | dt . We eventually notice that the four last terms tend to as n → ∞ by the hypotheses. (cid:3) Class of Inverse Gamma distributions
Inverse Gamma distributions verify gNB.Theorem 4.
Let Y ≥ be a r.v. satisfying (M). For k ≥ , let X k be a Γ( k, -distributed r.v.independent from Y . Set Z k = Y /X k . Then ( Z k ) k ≥ verifies gNB and Condition (1.2).Proof. We have for all k ≥ , E Z k < ∞ , so we can apply Lemma 2.1. For k = 1 , we onlyhave E (1 /X ) σ < ∞ for all σ ∈ (0 , . By the remark 2.2, this is also sufficient to ensure that g × ∈ L ( R + ) . Then for all k ≥ , c g × k ( s ) = − ζ ( s ) s b g k ( s ) , < σ < , where, since X k and Y are independent, b g k ( s ) = E Z sk = E Y s E X − sk , < σ < . Recalling that the density of the Γ( k, distribution is f k ( x ) = k ) x k − e − x , we may compute E X − sk = 1Γ( k ) Z ∞ x k − s − e − x dx = Γ( k − s )Γ( k ) . F. ALOUGES – S. DARSES – E. HILLION
Notice that, for k ≥ and < σ < , Γ( k − s ) = ( k − − s )Γ( k − − s )= ( k − − s ) · · · (1 − s )Γ(1 − s )= P k − ( s )Γ(1 − s ) , where P = 1 and P k ( s ) = ( k − s )( k − − s ) · · · (1 − s ) is a polynomial of degree k , known as aPochhammer symbol.We then write for some coefficients c k,n , setting s = 1 / it , D n = Z ∞ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) χ ( t ) − n X k =1 c k,n g × k ( t ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dt = 12 π Z ∞−∞ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)b χ ( s ) − n X k =1 c k,n c g × k ( s ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dt = 12 π Z ∞−∞ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) s + n X k =1 c k,n P k − ( s ) ζ ( s ) s E Y s Γ(1 − s ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dt = 12 π Z ∞−∞ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) sϕ ( s ) + n X k =1 c k,n P k − ( s ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) | ϕ ( s ) | dt , where ϕ ( s ) = ζ ( s ) s E Y s Γ(1 − s ) .We now notice that ζ ( s ) = O ( t ) (see e.g. [Ten95, Corollary 3.7 p.234]), | E Y s | ≤ E √ Y , and bythe complement formula | Γ(1 − s ) | = Γ (cid:18) − it (cid:19) Γ (cid:18)
12 + it (cid:19) = π sin (cid:0) π + iπt (cid:1) ≪ e − πt . Hence, we can apply Lemma 2.3 with the measure ν ( dt ) = | ϕ ( s ) | dt and the function t / ( sϕ ( s )) that belongs to L ( R , ν ) . Therefore, there exist coefficients c k,n such that D n → , which meansthat the family ( Y /X k ) k ≥ verifies gNB. (cid:3) As a consequence of Wiener’s Tauberian theorem (see e.g. [Bal00, Theorem 2 p.25]), thefollowing result is known:
Theorem 5.
Given ε > , there exist m ≥ , c ′ , . . . , c ′ m ∈ R , and θ > , . . . , θ m > such that (3.11) Z ∞ χ ( t ) − m X l =1 c ′ l (cid:26) θ l t (cid:27)! dt < ε. Interestingly, the existence of r.v. verifying gNB in Theorem 4 provides a short probabilisticproof of this fact. The basic idea is to approximate E { Z k /t } in D n by N P Nj =1 { Z k,j /t } where ( Z k,j ) j are independent copies of Z k defined on a probability space (Ω , F , P ) . Then we use Erdosprobabilistic method through the following instance: if E V ≤ ε for a r.v. V ≥ , then there exists ω ∈ Ω , s.t. V ( ω ) ≤ ε . Notice that Z k = X/Y k are fully supported on R + , and this is why the θ ’sdo not lie in (0 , . Proof.
Fix ε > . We know by Theorem 4 that there exist n and coefficients c , . . . , c n ∈ R (thatare fixed now) such that D n = Z ∞ χ ( t ) − n X k =1 c k E (cid:26) Z k t (cid:27)! dt < ε . OLYNOMIAL APPROXIMATIONS IN A GENERALIZED NYMAN-BEURLING CRITERION 7
Then set d n,N = Z ∞ χ ( t ) − n X k =1 c k /N N X j =1 { Z k,j /t } dt , and write, using ( a + b ) ≤ a + 2 b , E d n,N ≤ D n + 2 E Z ∞ n X k =1 c k E { Z k /t } − N N X j =1 { Z k,j /t } dt = 2 D n + 2 R n,N . Using now Cauchy-Schwarz, R n,N ≤ n X k =1 c k n X k =1 Z ∞ E E { Z k /t } − N N X j =1 { Z k,j /t } dt ≤ N n X k =1 c k n X k =1 Z ∞ Var( { Z k /t } ) dt. We can then choose N sufficiently large so that R n,N ≤ ε . Hence E d n,N ≤ ε , so there exist ω ∈ Ω such that d n,N ( ω ) ≤ ε , which concludes the proof, the desired θ l ’s being the Z k,j ( ω ) . (cid:3) Remark on a specific distribution tail.
The preceding arguments generalize a remarkdue to Vincent Alouin who noticed that the distribution tail x (1 + x ) − k has a Mellin transformwhich satisfies, by the change of variables u = 1 / (1 + x ) , i.e. x = 1 /u − , the identity(3.12) Z ∞ x s − (1 + x ) k dx = − Z (cid:18) u − (cid:19) s − u k du − u = Z (1 − u ) s − u k − s − du = Γ( s )Γ( k − s )Γ( k ) , which is a product of a polynomial in s by a fixed (independent of k ) function. It turns our thatthose r.v. are of the form Z k ∼ Y /X k where Y ∼ E (1) .3.3. Computation of the Gram matrix.
We now want to compute the corresponding scalarproducts. We introduce the functions ρ ( t ) = E (cid:26) Yt (cid:27) , t > ,A ( u ) = Z ∞ ρ ( ut ) ρ ((1 − u ) t ) dt, < u < . Two interesting particular cases are:(i) when Y ∼ δ , we have ρ ( t ) = (cid:8) t (cid:9) ,(ii) when Y ∼ E ( λ ) , we have ρ ( t ) = 1 e λt − − λt .Recall that g × k ( t ) = E n YX k t o where Y is a r.v. satisfying (M) independent from X k ∼ Γ( k, . Proposition 3.1.
For m, n ≥ , we have (cid:10) g × n +1 , g × m +1 (cid:11) = Z B m + nn ( u ) A ( u ) du, where B m + nn ( u ) = (cid:18) n + mn (cid:19) u n (1 − u ) m is an elementary Bernstein polynomial.Proof. We first notice that, as X n and Y are independent, g × n ( t ) = E (cid:26) YX n t (cid:27) = E [ ρ ( X n t )]= 1( n − Z ∞ ρ ( xt ) x n − e − x dx = 1( n − t n Z ∞ ρ ( x ) x n − e − xt dx. F. ALOUGES – S. DARSES – E. HILLION
We then have, by Fubini, (cid:10) g × n +1 , g × m +1 (cid:11) = 1 m ! n ! Z ∞ Z ∞ ρ ( x ) ρ ( y ) x n y m Z ∞ t m + n +2 e − x + yt dt dx dy = 1 m ! n ! Z ∞ Z ∞ ρ ( x ) ρ ( y ) x n y m ( m + n )!( x + y ) n + m − dx dy , where we used the elementary formula R ∞ t α e − βt dt = Γ( α − β α − . We now consider the changeof variables u = xx + y , z = x + y , which gives x = uz , y = (1 − u ) z . We thus have (cid:10) g × n +1 , g × m +1 (cid:11) = ( m + n )! m ! n ! Z ∞ Z ρ ( uz ) ρ ((1 − u ) z )( uz ) n ((1 − u ) z ) m z m + n +1 z dz du = ( m + n )! m ! n ! Z u n (1 − u ) m Z ∞ ρ ( uz ) ρ ((1 − u ) z ) dz du , as desired. (cid:3) Generalized function and remarkable Gram matrix
A recursive scheme verifying gNB.Theorem 6.
Let g ∈ C ∞ ( R + ) such that for all k ≥ and for some α > , lim x → x k − α g ( k )0 ( x ) = lim x →∞ x k +1+ α g ( k )0 ( x ) = 0 . Let ( r k ) be a sequence of real numbers. Define by induction g k +1 ( x ) = − xg ′ k ( x ) − r k g k ( x ) . (4.13) Then c g × k is well defined with c g × k ( s ) = − k Y j =0 ( s − r j ) ζ ( s ) s b g ( s ) , < σ < . (4.14) In particular, if b g (cid:18)
12 + it (cid:19) ≪ e − δ | t | , δ > , then ( g k ) k ≥ verifies gNB.Proof. One can show by induction on k that there exists a family of numbers ( a l,k ) k ≥ , ≤ l ≤ k , with a k,k = ( − k such that ∀ k ≥ , g k ( x ) = k X l =0 a l,k x l g ( l )0 ( x ) . (4.15)Due to the assumption on g , we thus have lim x → , ∞ x σ g k ( x ) = 0 and g k verifies Assumption (M).Moreover, d g k +1 is well defined and, by integration by parts − Z ∞ x s − xg ′ k ( x ) dx = − Z ∞ x s g ′ k ( x ) dx = − [ x s g k ( x )] ∞ + Z ∞ sx s − g k ( x ) dx = s b g k ( s ) . Hence d g k +1 ( s ) = ( s − r k ) b g k ( s ) , and then d g k +1 ( s ) = k Y j =0 ( s − r j ) b g ( s ) . We found again the polynomial structure of c g × k , as in the proof of Theorem 4. We can then followthese lines to conclude. (cid:3) OLYNOMIAL APPROXIMATIONS IN A GENERALIZED NYMAN-BEURLING CRITERION 9
Gram matrix.Proposition 4.1.
For j, k ≥ , we have (cid:10) g × k , g × j (cid:11) + (cid:10) g × k +1 , g × j − (cid:11) = (1 − r k ) (cid:10) g × k , g × j − (cid:11) . Proof.
We compute the scalar products as: Z ∞ Z ∞ n xt o g k ( x ) dxx Z ∞ n yt o g j ( y ) dyy dt = Z ∞ Z ∞ { x } { y } Z ∞ g k ( tx ) g j ( ty ) dt dxx dyy . Moreover, using g k +1 ( t ) = − tg ′ k ( t ) − r k g k ( t ) , we compute I k,j = I k,j ( x, y ) = Z ∞ g k ( tx ) g j ( ty ) dt = Z ∞ g k ( tx )( − tyg ′ j − ( ty ) − r k g j − ( ty )) dt. By integrating by parts, one has Z ∞ g k ( tx ) tyg ′ j − ( ty ) dt = [ tg k ( tx ) g j − ( ty )] ∞ − Z ∞ ( txg ′ k ( tx ) + g k ( tx )) g j − ( ty ) dt = Z ∞ ( g k +1 ( tx ) + ( r k − g k ( tx )) g j − ( ty ) dt = I k +1 ,j − + ( r k − I k,j − . Hence, I k,j = − I k +1 ,j − − ( r k − I k,j − − r k I k,j − = (1 − r k ) I k,j − − I k +1 ,j − , and the result follows due to the linearity of the double integral. (cid:3) We now set G k,j = (cid:10) g × k , g × j (cid:11) . Choosing r k = 1 / in the previous porposition yields to aremarkable structure, i.e. G k,j + G k +1 ,j − = 0 . In particular, setting j = k + 1 , we obtain G k,k +1 = 0 . This is not a surprise due to the following computation by means of Mellin transformand (4.14). We set s = 1 / it and use Mellin-Plancherel isometry: G k,k +1 = Z ∞−∞ c g × k ( s ) d g × k +1 ( s ) dt = Z ∞−∞ ( s − / k ( s − / k +1 (cid:12)(cid:12)(cid:12)(cid:12) ζ ( s ) s b g ( s ) (cid:12)(cid:12)(cid:12)(cid:12) dt = ( − k i Z ∞−∞ t k +1 (cid:12)(cid:12)(cid:12)(cid:12) ζ ( s ) s b g ( s ) (cid:12)(cid:12)(cid:12)(cid:12) dt = 0 . The Gram matrix G , sort of "alternate" Hankel matrix, is then only determined through itsdiagonal entries: G k,k = (cid:10) g × k , g × k (cid:11) = Z ∞−∞ t k (cid:12)(cid:12)(cid:12)(cid:12) ζ ( s ) s b g ( s ) (cid:12)(cid:12)(cid:12)(cid:12) dt = Z ∞−∞ t k dν ( t ) . The Gram matrix then reads: G = G − G · · · G − G · · ·− G G · · · − G G · · · ... ... ... ... . . . . (4.16) Renumbering the rows and the columns, making appear first the odd indices and then the evenones, leads to the equivalent matrix e G = G − G · · · · · ·− G G · · · · · · ... ... . . . · · · G − G · · · − G G · · · ... ... ... ... ... . . . , (4.17)which has the remarkable structure of a block Hankel matrix.Notice that if one wants to evaluate the squared distance (1.3) by computing the determinantof G , we can obtain the determinant of a moment matrix H k,j = R ∞−∞ t k + j dν ( t ) by multiplyingthe rows of e G by ( − k and the columns by ( − j . The study of such determinant falls into thetheory of Fisher-Hartwig’s singularities, see e.g. [Kra07], though the infinity of zeros of ζ (1 / it ) seems to be a challenging issue.It is also possible to provide simpler expressions for the scalar products b k = h χ, g × k i . Indeed: b k = Z ∞ χ ( t ) Z ∞ n xt o g k ( x ) dxx dt = Z ∞ { x } Z ∞ χ ( t ) g k ( tx ) dt dxx . But Z g k +1 ( tx ) dt = − Z txg ′ k ( tx ) dt − Z g k ( tx ) dt = − [ tg k ( tx )] + Z g k ( tx ) dt − Z g k ( tx ) dt = − g k ( x ) + 12 Z g k ( tx ) dt. Therefore b k +1 = − Z ∞ { x } g k ( x ) dxx + 12 b k . Examples for g and comments. The Ξ -function. As an answer to a suggestion by P. Biane and C. Delaunay, it is possibleto use the induction (4.13) to find a g that produces within ν the Ξ -function: Ξ( t ) = ξ ( s ) = 12 s ( s − π − s/ Γ (cid:16) s (cid:17) ζ ( s ) , s = 1 / it. Indeed, to construct g such that b g ( s ) = ( s − s Γ (cid:0) s (cid:1) , we define first h ( t ) = e − t , so that c h ( s ) = Z ∞ t s − e − t dt = Z ∞ u s/ − / e − u √ u du = 12 Γ (cid:16) s (cid:17) . Then, in order to have b g ( s ) = ( s − s ) c h ( s ) , we compute g ( t ) = (cid:18) ( − t ddt ) − ( − t ddt ) (cid:19) h ( t )= (8 t − t + 12 t ) e − t . Hence, again in the case r k = 1 / , we can obtain, taking π − / g , G k,k = Z ∞−∞ t k Ξ( t ) dt. As an historical nod, note that quantities as R ∞−∞ t k Ξ( t ) dt or related to, have been used by Pólyaand Hardy to study the zeros of ζ on the critical line, see e.g. [Tit86, 10.2-10.4, p.256-260]. OLYNOMIAL APPROXIMATIONS IN A GENERALIZED NYMAN-BEURLING CRITERION 11
Seed with compact support.
If the seed g has a compact support, say (0 , M ) , then the g k ’sare also supported on (0 , M ) . This removes the control condition (2.9) on the coefficient c k,n assoon as M n ≥ M . If one wants to prove RH, one then only needs a density result.Amazingly, in this non compact support case, we then lose the density of the polynomials in thewhole space L ( ν ) . Indeed, Ingham [Ing34] remarked that if g has compact support, we cannothave b g ( s ) ≪ e − δ | t | , δ > , where again s = 1 / it . More precisely, for any decreasing function ε ( t ) = o (1) , there exists a compactly supported function g such that b g ( s ) ≪ e − ε ( | t | ) | t | if andonly if Z ∞ ε ( t ) t dt < ∞ (See [Swa19, Annexe] for a nice account on these results). But this isprecisely incompatible with the condition that ensures the density of the polynomials in weighted L ( R ) -spaces, namely Z ∞−∞ log w ( t )1 + t dt = ∞ . The link between the weight w and the function ε is log w ( t ) = ε ( t ) t here. See [Nik12, 4.8.3 p.77] for many aspects regarding such theorems.Let us stress that Mellin isometry involves an integration on the whole real line. On the halfline, a density result is obtained by Mergelyan [Mer58] with the condition Z ∞ log w ( t ) t / dt = ∞ .In our framework, Borichev [Bor20] proved that for all ε > there exist Q ∈ C [ X ] such that Z ∞ (cid:12)(cid:12)(cid:12)(cid:12) sϕ ( s ) − Q ( s ) (cid:12)(cid:12)(cid:12)(cid:12) | ϕ ( s ) | dt < ε, where ϕ ( s ) = ζ ( s ) s b g ( s ) , s = 1 / it , g has compact support and verifies b g ( s ) ≪ e −| t | / log | t | .Although we do not have the density on the whole line ( −∞ , + ∞ ) , there exist results regardingthe closure of the polynomials, see e.g. [Bor01]. So, we ask for the following question:What is the closure of the space generated by the polynomials in L (cid:0) R , | ϕ ( s ) | dt (cid:1) when g hascompact support? Acknowledgement
The authors are very grateful to Vincent Alouin for his remark (3.12). This was an importantstart for our study. The second author thanks Philippe Biane, Alexander Borichev, ChristopheDelaunay, Sophie Grivaux, Igor Krasovsky, Pierre Lazag and Olivier Ramaré for helpful referencesand interesting discussions.
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