Szeg\"o kernels and asymptotic expansions for Legendre polynomials

We present a geometric approach to the asymptotics of the Legendre polynomials $P_{k,n+1}$, based on the Szeg\"o kernel of the Fermat quadric hypersurface, and leading to complete asymptotic expansions holding on expanding subintervals of $[-1,1]$.


Introduction
The goal of this paper is to develop a geometric approach to the asymptotics of the Legendre polynomial P k,n+1 (t) for k → +∞, with t = cos(ϑ) ∈ [−1, 1] and n ≥ 1 fixed; as is well-known, P k,n+1 (t) is the restriction to S n of the Legendre harmonic, expressed in polar coordinates on the sphere. We follow here the terminology of [M], [M1] and [AH].
There is a tight relation between P k,n+1 (t) and the orthogonal projector P k,n : L 2 (S n ) → V k,n , where V k,n is the space of level-k spherical harmonics on S n ; equivalently, V k,n is the eigenspace of the (positive) Laplace-Beltrami operator on functions on S n , corresponding to its k-th eigenvalue λ k,n = k (k + n − 1). Namely, for any choice of an orthonormal basis ̺ knj N k,n j=1 of V k,n the distributional kernel P k,n (·, ·) ∈ C ∞ (S n × S n ) satisfies where q · q ′ = q t q ′ (we think of q and q ′ as columns vectors), and N k,n is the dimension of V k,n . By symmetry considerations, P k,n (q, q ′ ) only depends on q · q ′ . In fact, with the normalization P k,n+1 (1) = 1, P k,n (q, q ′ ) = N k,n vol(S n ) P k,n+1 (q · q ′ ).
Thus it equivalent to give asymptotic expansions for P k,n+1 cos(ϑ) and for P k,n (q, q ′ ) with q · q ′ = cos(ϑ). Since for any (q, q ′ ) ∈ S n × S n we have P k,n (q, q) = N k,n vol (S n ) , P k,n (q, −q ′ ) = (−1) k P k,n (q, q ′ ) , we may assume q = ±q ′ . Then there is a unique great circle parametrized by arc length going from q to q ′ in a time ϑ ∈ (0, π), and q t q ′ = cos(ϑ). Our geometric approach uses on the one hand the specific relation between spherical harmonics on S n and the Hardy space of the Fermat quadric hypersurface in P n ( [L], [G]), and the other hand the off-diagonal scaling asymptotics of the level-k Szegö kernel of polarized projective manifold ([BSZ], [SZ]).
The following asymptotic expansions involve a sequence of constants C k,n > 0 with a precise geometric meaning [G]. There is a natural conformally unitary isomorphism between the level-k Szegö kernel of the Fermat quadric F n ⊂ P n and V k,n , given by a push-forward operation, and C k,n is the corresponding conformal factor.
An asymptotic expansion for C k,n is discussed in [G], building on the theory of [L]; an alternative derivation is given in Proposition 1.1 (with an explicit computation of the leading order term).
In the following, the symbol ∼ stands for 'has the same asymptotics as'.
Proposition 1.1. For k → +∞ we have an asymptotic expansion of the form: If we insert the latter expansion in the one provided by Theorem 1.1, we obtain the following: Corollary 1.1. With the assumptions and notation of Theorem 1.1, for k → +∞ there is an asymptotic expansion where C n (ϑ, k) and D n (ϑ, k) admit asymptotic expansions similar to those of A n (ϑ, k) and B n (ϑ, k), respectively (of course, with different functions C nl and D nl , l ≥ 1).
Pairing Corollary 1.1 with (2), we obtain: Corollary 1.2. In the same situation as in Theorem 1.1, for k → +∞ there is an asymptotic expansion where again E n (ϑ, k) and F n (ϑ, k) admit asymptotic expansions similar to those of A n (ϑ, k) and B n (ϑ, k), respectively.
Let us verify that Corollary 1.2 fits with the classical asymptotics. For example, when n = 1 we obtain P k,2 cos(ϑ) ∼ cos(kϑ) + · · · , so that the leading order term is the k-th Chebychev polynomial. Since it is known that in this case the Legendre polynomial is the Chebychev polynomial ( [M], page 11), this is in fact the only term of the expansion.
Acknowledgments. I am endebted to Leonardo Colzani and Stefano Meda for very valuable comments and insights.

The geometric picture
For the following, see [G], [L]. Let S n 1 ⊂ R n+1 be the unit sphere, and let us identify the tangent and cotangent bundles of S n 1 by means of the standard Riemannian metric. The unit (co)sphere bundles of S n 1 is given by the incidence correspondence The Fermat quadric hypersurface in complex projective space is let A be the restriction to F n of the hyperplane line bundle. Given the standard Hermitian product on C n+1 , A is naturally a positive Hermitian line bundle, F n inherits a Kähler structure ω Fn (the restriction of the Fubini-Study metric), and the spaces of global holomorphic sections of higher powers of A, H 0 F n , A ⊗k , have an induced hermitian structure.
The affine cone over F n is C n = {z t z = 0} ⊂ C n+1 ; the intersection X 1 =: C n ∩ S 2n+1 1 may be viewed as the unit circle bundle in the dual line bundle A ∨ . More generally, for any r > 0 the intersection with the sphere of radius r is naturally identified with the circle bundle of radius r in A ∨ . In particular, is diffeomorphic to S * (S n ) by the map β : (q, p) → q + i p; furthermore, β is equivariant for the natural actions of O(n + 1) on S * (S n ) and X √ 2 defined by, respectively, We shall identify S * (S n ) and X √ 2 , and denote the projection by There is also a standard structure action of S 1 on X √ 2 , induced by fibrewise scalar multiplication in A ∨ , or equivalently in C n+1 . The latter action is interwined by β with the 'reverse' geodesic flow on S * (S n ) ∼ = S(S n ). The S 1 -orbits are the fibers of the circle bundle projection This holds for any r > 0; we shall denote by π r : X r → F n the projection for general r > 0.

The metric on X r
Let us dwell on the metric aspect of (5); there are two natural choices of a Riemannian metric on X r , hence of a Riemannian density, and we need to clarify the relation between the two.
There is an obvious choice of a Riemannian metric g ′ r on X r , induced by the standard Euclidean product on C n+1 . With respect to g ′ r , the S 1 orbits on X r have length 2π r. Clearly, g ′ r is homogeneous of degree 2 with respect to the dilation µ r : x ∈ X → r x ∈ X r , and therefore the corresponding volume form Υ ′ Xr on X r is homogeneous of degree dim(X) = 2n − 1. That is, An alternative and common choice of a Riemannian structure g 1 on X 1 comes from its structure of a unit circle bundle over F n . Let α ∈ Ω 1 (X 1 ) be the connection 1-form associated to the unique compatible covariant derivative on A, so that dα = 2 π * 1 (ω Fn ). Also, let denote the horizontal and vertical tangent bundles for π 1 , respectively. There is a unique Riemannian metric g 1 on X 1 such that π 1 a Riemannian submersion, and the S 1 -orbits on X 1 have unit length. The corresponding volume form on X 1 is given by where Υ Fn = ω ∧(n−1) Fn /(n − 1)! is the symplectic volume form on F n . We wish to compare the two Riemannian metrics g 1 and g ′ 1 , the corresponding volume forms, Υ ′ X 1 and Υ X 1 , and densities, dV X and d ′ V X .
thus α is the restriction of θ to X 1 . Let ω 0 be the standard symplectic structure on C n+1 . Since θ z (w) = ω 0 (z, w), we have ker(θ z ) = z ⊥ω 0 (symplectic annihilator). In other words, where z ⊥ h 0 is the Hermitian orthocomplement of z for the standard Hermitian product. Thus, if z ∈ X 1 then On the other hand, are orthogonal with respect to both g 1 (by construction) and g ′ 1 (by the previous considerations). Hence we may compare g 1 and g ′ 1 separately on H(X 1 /F n ) and V (X 1 /F n ).
On the complex vector bundle H(X 1 /F n ), g ′ 1 and g 1 are, respectively, the Euclidean scalar products associated to the restrcitions of the (1, 1)-forms Given that ω 0 and ω 1 agree on T S 2n+2 1 , g 1 = g ′ 1 on H(X 1 /F n ). On the other hand, both g 1 and g ′ 1 are S 1 -invariant, but S 1 -orbits on X 1 have length 2π for g ′ 1 and 1 for g 1 -Thus g 1 = g ′ 1 /2π on V (X 1 /F n ). The claim follows directly from this.

The Szegö kernel on X r
For every r > 0, X r is the boundary of a strictly pseudoconvex domain, and as such it carries a CR structure, a Hardy space H(X r ), and a Szegö projector Π r : L 2 (X r ) → H(X r ). We aim to relate the various Π r 's.
Let O(C n \{0}) be the ring of holomorphic functions on the conic complex For every r > 0 and k = 0, 1, 2, ; with a slight abuse of language, we shall denote by the same symbol an element of H k (X r ) and the corresponding element Setting y = r x, and using (9) together with Lemma 2.1, we get Therefore we have: restricts to an orthonormal basis of H k (X r ), with respecto to d ′ V Xr /2π.
Let now Π r,k be the level-k Szegö kernel on X r , that is, the orthogonal projector Π r,k : By Lemma 2.2, its Schwartz kernel Π r,k ∈ C ∞ (X r × X r ) is given by When pulled-back to X 1 , this is (here (14) In particular, We shall make repeated use of the following asymptotic property of Π 1,k , which follows from the microlocal description of Π as an FIO (explicit exponential estimates are discussed in [C]).
Theorem 2.1. Let dist Fn be the distance function on F n associated to the Kähler metric. Given any C, ǫ > 0, uniformly for x, x ′ ∈ X satisfying

Heisenberg local coordinates
There are two unit circle bundles in our picture: the Hopf fibration π : S 2n+1 1 → P n , and π 1 : X 1 → F n . Clearly, π 1 is the pull-back of π under the inclusion F n ֒→ P n . Both S 2n+1 1 and X 1 are boundaries of strictly pseudoconvex domains, and carry a CR structure.
On both S 2n+1 1 and X 1 , we may consider privileged systems of coordinates called Heisenberg local coordinates (HLC). In these coordinates, Szegö kernel asymptotics exhibit a 'universal' structure [SZ]; we refer to ibidem for a detailed discussion.
Given z 0 ∈ X 1 , a HLC system on X 1 centered at z 0 will be denoted in additive notation: Here θ ∈ (−π, π) is an 'angular' coordinate measuring displacement along the S 1 -orbit through z 0 (the fiber through z 0 of π 1 : We may thus think of v as a tangent vector in T [z 0 ] F n . Here this additive notation might be misleading, since X 1 ⊂ C n+1 . Therefore we shall write z 0 + X 1 (θ, v) for HLC's on X 1 centered at z 0 . We shall generally abridge notation by writing (θ, v) will denote a system of Heisenberg local coordinates on S 2n+1 1 centered at z 0 . There is in fact a natural choice of HLC on S 2n+1 1 centered at any z 0 ∈ S 2n+1 1 . Namely, let (a 1 , . . . , a n ) be an orthonormal basis of the Hermitian orthocomplement z ⊥ h 0 ⊆ C n+1 , and for w = (w j ) ∈ C n let us set Since there is a canonical unitary identification z ⊥ h 0 ∼ = T [z 0 ] P n , we shall also write this as z 0 If z 0 ∈ X 1 , HLC's on X 1 centered at z 0 can be chosen so that they agree to second order with the former HLC's on S 2n+1 1 . More precisely, we may assume that for any where R 2 is a function vanishing to second order at the origin.
Given v, w ∈ C n+1 ∼ = R 2n+2 , let us define here ω 0 is the standard symplectic structure, and · is the standard Euclidean norm. We shall make use of the following asymptotic expansion, for which we refer again to [SZ]: Theorem 2.2. Let us fix C > 0 and ǫ ∈ (0, 1/6). Then for any z ∈ X 1 , and for any choice of HLC's on X 1 centered at z, there exists polynomials P j of degree ≤ 3j and parity j on T [z] F n × T [z] F n ∼ = R 2n−2 × R 2n−2 , such that following holds. Uniformly in v 1 , v 2 ∈ T [z] F n with v j ≤ C k ǫ for j = 1, 2, and θ 1 , θ 2 ∈ (−π, π), one has for k → +∞ the following asymptotic expansion: In the given range the above is an asymptotic expansion, since 2.5 P k and Π √

2,k
As discussed in [G], the push-forward operator ν * : for every k, (19) restricts to a conformally unitary isomorphism H k (X √ 2 ) −→ V k , with a scalar conformal factor C k,n > 0. Thus we have Therefore, if (σ kj ) N k j=0 is an orthonormal basis of H k (X √ 2 ), then is an orthonormal basis of V k . It follows that P k,n in (1) is given by where ν × ν : X √ 2 × X √ 2 → S n × S n is the product projection. More explicitly, for q ∈ S n let S(q ⊥ ) ∼ = S n−1 be the unit sphere centered at the origin in the orthocomplement q ⊥ , and let dV S(q ⊥ ) be the Riemannian density on S(q ⊥ ); then P k,n (q 0 , q 1 )
Although they project down to the same locus in S n , γ + and γ − correspond to distinct fibers of the circle bundle projection π : X( √ 2) → F n . Let us express the (co)tangent lift γ ± of the geodesics γ ± in complex coordinates, and set p 1 =γ + (ϑ). . Then In view of (8), we have: On the other hand, [q 0 + i p 0 ] = [q 0 − i p 0 ] ∈ F n , since q 0 + i p 0 and q 0 − i p 0 are linearly independent in C n+1 . Thus we have: Lemma 3.1. Suppose q 0 , q 1 ∈ S n and q 1 = ±q 0 . Then the only points [z] ∈ F n such that By Theorem 2.1, for fixed p and p ′ and k → +∞ we have unless p = ±p 0 and p ′ = ±p 1 . Therefore, for a fixed ϑ ∈ (0, π) integration in (22) may be localized in a small neighborhood of (±p 0 , ±p 1 ), perhaps at the cost of disregarding a negligible contribution to the asymptotics. Since however we are allowing ϑ to approach 0 or π at a controlled rate, we need to give a more precise quantitative estimate of how small the previous neighborhood may be chosen when k → +∞.
Proposition 3.1. Let us fix C > 0, δ ∈ (0, 1/6) and ǫ > δ. Then there exist constants D, ǫ 1 > 0 such that the following holds. Suppose that In view of Theorem 2.1, Proposition 3.1 implies: Corollary 3.1. Uniformly in the range of Proposition 3.1, we have Proof of Proposition 3.1. Let us set for γ ∈ [−π, π]: Let dist Fn be the restriction to F n of the distance function on P n . Then The factor in front is needed because while the Hopf map S 2n+1 1 → P n is a Riemannian submersion, the projection S 2n+1 √ 2 → P n is so only in a conformal sense.
Given this and (34), we conclude that, under the present hypothesis, Let us now pick δ ′ with ǫ > δ ′ > δ, and assume Then This establishes (30) with ǫ 1 = ǫ − δ ′ , in the case where (36) holds. Thus we are reduced to assuming Then we also have | sin(γ)| ≤ k −δ ′ . Let us then look at the first summand on the last line of (31). We have an Hermitian orthogonal direct sum On the other hand, since sin(ϑ) vanishes exactly to first order at ϑ = 0 and ϑ = π, there exists E > 0 such that for ϑ ∈ (0, π) under the assumptions of the Lemma we have Hence, in view of (33), we have for some D 1 > 0 and k ≫ 0 since δ ′ > δ and δ < 1/6. This establishes (30) with ǫ 1 = 1/3 when (38) holds. The proof of Proposition 3.1 is complete.
Equations (26) and (27) parametrize neighborhoods of p 0 and −p 0 , respectively. Therefore, Proposition 3.1 implies that in (22) only a negligible contribution to the asymptotics is lost, if integration in p and p ′ is restricted to shrinking neighborhoods of ±p 0 and ±p 1 , of radii O k ǫ−1/2 . This may be rephrased as follows. Let ̺ ∈ C ∞ 0 (R n+1 ) be even, supported in a small neighborhood of the origin, and identically equal to one in a smaller neighborhood of the origin. Then the asymptotics of (22) are unchanged, if the integrand is multiplied by In this way the integrand splits into four summands. In fact, only two of these are non-negligible for k → +∞. Namely, consider the summand containing the factor ̺ k 1/2−ǫ (p − p 0 ) ̺ k 1/2−ǫ (p ′ + p 1 ) .
On its support, p lies in a shrinking neighborhood of p 0 , and p ′ in a shrinking neighborhood of −p 1 . Therefore, on the same support q 0 + i p lies in a shrinking neighborhood of q 0 + i p 0 , and q 1 − i p ′ lies in a shrinking neighborhood of q 1 − i p 1 . Since has unit norm, on the support of (41) [q 0 + i p] and [q 1 + i p ′ ] remain at a distance ≥ 2/3, say, in projective space. This implies that as k → +∞ uniformly in (p, p ′ ) in the support of (41). A similar argument applies to the summand containing the factor Thus we may rewrite (22) as follows: where P k,n (q 0 , q 1 ) ± =: As a further reduction, we need only deal with one of P k,n (q 0 , q 1 ) ± .
Proof of Lemma 3.3. . Let us apply the change of integration variable p → −p and p ′ → −p ′ , and apply (23). Since ̺ is even, we get

Lemma 3.3 and (43) imply
In the definition of P k (q 0 , q 1 ) + , integration is over a shrinking neighborhood of (p 0 , p 1 ) ∈ S(q ⊥ 0 )×S(q 1 ⊥ ). We can thus make use of the parametrization (26), and write in (44): It is also harmless to replace p − p j by v j in the rescaled cut-offs in (44). Let us also set z j = q j + i p j , and recall that z 1 = e −iϑ z 0 . We then obtain with Let us consider the Szegö term in the integrand. In view of (15), this is Now the sums in the previous expression are just algebraic sums in C n+1 ; in order to apply the scaling asymptotics of Theorem 2.2, we need to first express the argument of (49) in terms of local Heisenberg coordinates on X 1 centered at z 0 / √ 2.
Lemma 3.4. Suppose z = q + i p ∈ X 1 and choose a system of HLC's on X 1 centered at z. Then for δp ∼ 0 ∈ R n+1 and e iϑ ∈ S 1 such that z + i e iϑ δp ∈ X 1 we have for a suitable smooth function R 2 (θ; ·) vanishing to second order at the origin (in v).
Proof of Lemma 3.4. In view of (17), it suffices to prove the statement on S 2n+1 1 , working with the HLC's (16). Since z, z + i e iϑ δp ∈ C n , we have so that i z t δp = e iϑ δp 2 /2. Let us look for β > 0 and h ∈ z ⊥ h (Hermitian orthocomplement) such that z + i e iϑ δp = β (z + h).
If this is possible at all, then necessarily β = 1/ z + h , as z + i e iϑ δp = 1.
Assuming that (51) may be solved, then, taking the Hermitian product with z on both sides of (51) and using (50) we get With this value of β, let us set so that (51) is certainly satisfied. We need to verify that h ∈ z ⊥ h . Indeed we have Since h = i e iθ δp + R 2 (δp), the proof of the Lemma is complete.
Notice that h is given for δp ∼ 0 by an asymptotic expansion in homogeneous polynomials of increasing degree in δp of the form This holds on S 2n+1 1 , but a similar expansion obviously holds on X 1 , possibly with modified terms in higher degree.
Let us apply Lemma 3.4 with z = z 0 / √ 2 and δp j = e iθ A jk (v j )/ √ 2 / √ k (we'll set θ = 0 for j = 0 and θ = ϑ for j = 1). To this end, let us note that in view of (48) for k → +∞ there is an asymptotic expansion of the form where P j,l is a homogeneous (vector valued) polynomial function of degree l, and P j1 (v) = v. Hence Making use of (57) in (55) we obtain where Q jl (θ; ·) is a homogeneous polynomial function of degree l, and we have emphasized the dependence on k.
Thus we obtain for j = 0 (with θ = 0) that where with a k0 defined by the latter equality. Similarly, for j = 1 (with θ = ϑ) we have where again each T bl has the same parity as l and degree ≤ 3l.
Putting this all together, we obtain an asymptotic expansion for the integrand in (47): Lemma 3.5. For l ≥ 0, there exist polynomials Z l (ϑ; ·, ·) of degree ≤ 3l and parity (−1) l , with Z 0 (ϑ; ·, ·) = 1, such that Proof of Lemma 3.5. The previous arguments yield an asymptotic expansion of the given form for the first factor. We need only multiply the latter expansion by the Taylor expansion of the second factor.
Since integration in (47) takes place over a poly-ball or radius O (k ǫ ) in q ⊥ 0 ∩ q ⊥ 1 2 , the expansion may be integrated term by term. In addition, given that the exponent and the cut-offs are even functions of (v 0 , v 1 ), only terms of even parity yield a non-zero integral. Hence we may discard the half-integer powers and obtain where P l (ϑ) + =: We can slightly simplify the previous asymptotic expansion, as follows. First, as emphasized the dependence on (q 0 , q 1 ) is of course only through the angle ϑ. In particular, in (65) nothing is lost by assuming that q 0 and q 1 span the 2-plane {0} × R 2 ⊆ R n+1 , and therefore that q ⊥ 0 ∩ q ⊥ 1 = R n−1 × {0}. Furthermore, given (18), we have With the change of variables Since Z 2l (ϑ, ·, ·) is even and has degree ≤ 6l, we can write where T l (ϑ; ·, ·) is an even polynomial of degree ≤ 6l, with smooth bounded coefficients for ϑ ∈ [0, π]. Thus There is a constant C > 0 such that the support of is contained in the locus where (b 0 , b 1 ) ≥ C k ǫ sin(ϑ). Under the assumptions of the Theorem, this implies, perhaps for a different constant C > 0, that (b 0 , b 1 ) ≥ C k ǫ−δ . On the other hand, the exponent in (68) satisfies Given that ǫ > δ (statement of Proposition 3.1), we conclude that only a negligible contribution to the asymptotics is lost, if the cut-off function is omitted and integration is now extended to all of R n−1 × R n−1 . We can thus rewrite (64) as follows: where Let us set B ϑ = 1 + i cot(ϑ) I n−1 . The leading order coefficient is Given (70), (69) and (45), P k (q 0 , q 1 ) has an asymptotic expansion for k → +∞ with leading order term 2 n/2 C 2 k,n 1 sin(ϑ) (n−1)/2 cos k ϑ + ϑ 2 − π 2 (n − 1) .
For any l, we can write where T l (ϑ; ·) is an even polynomial of degree ≤ 6l. Let us introduce the Fourier transform Then (72) is the result of applying an even differential polynomial P l (D c ) of degree ≤ 6l to F (c), and then evaluating the result at c = 0.
As before, the expansion may be integrated term by term and, by parity, only the summands with l even yield a non-zero contribution. In addition, only a negligible contribution is lost if the cut off is omitted and integration is extended to all of q ⊥ ∩ p ⊥ ∼ = R n−1 . Therefore Inserting this in (76), we obtain an asymptotic expansion P k,n (q, q) ∼ vol(S n−1 ) C 2 k,n 1 √ 2 k π (n−1)/2 Comparing (75) and (80), we obtain an asymptotic expansion in descending powers of k, of the form C k,n ∼ vol(S n ) vol(S n−1 ) 2 √ 2 · (n − 1)! 1/2 (π k) −(n−1)/4 + · · ·