Abstract
We present a geometric approach to the asymptotics of the Legendre polynomials , based on the Szegö kernel of the Fermat quadric hypersurface, leading to complete asymptotic expansions holding on expanding subintervals of .
1. Introduction
The search for asymptotic expansions and approximations of special functions is a very classical vein of research and is of great relevance in pure mathematics, in numerical analysis, mathematical physics, and the applied sciences (see, for instance, of course with no pretence of completion [1–4]).
The goal of this paper is to develop a geometric approach to the asymptotics of the Legendre polynomial for , with and fixed; as is well-known, is the restriction to of the Legendre harmonic, expressed in polar coordinates on the sphere. For thorough discussions and terminology, see, for instance, [1, 3, 5–7]. We obtain an asymptotic expansion holding on expanding subintervals of , rather than on fixed subintervals of the form for some given , as one typically finds in the literature.
However, the actual point of this work is neither to present essentially new results nor to give an especially economic proof of Legendre asymptotics (the use of Szegö kernel machinery is arguably not more elementary than the traditional approaches). Rather, it is motivated by the following considerations. On the one hand, there is a conceptually very appealing view on spherical harmonics, due to Lebeau and Guillemin, based on the Szegö kernel of the Fermat quadric. On the other hand, in recent years, a considerable amount of work and attention has been devoted to algebro-geometric Szegö kernel asymptotics, which have played a fundamental role in complex geometry. Therefore, it seems per se very natural and interesting to illustrate the important conceptual juncture between spherical harmonics and Szegö kernels, by revisiting classical results on Legendre asymptotics in view of these recent developments. In a broader perspective, the application of Szegö kernels to spherical harmonics seems a very promising area; a revisitation of this kind is also instrumental to the development of computational techniques that might be useful in future developments in this direction.
Let us come to a closer description of the content of this paper. There is a tight relation between and the orthogonal projectorwhere is the space of level- spherical harmonics on ; equivalently, is the eigenspace of the (positive) Laplace-Beltrami operator on functions on , corresponding to its th eigenvalue .
Namely, for any choice of an orthonormal basis of the distributional kernel satisfieswhere (we think of and as columns vectors), and is the dimension of . By symmetry considerations, only depends on . In fact, with the normalization , Thus, it is equivalent to give asymptotic expansions for and for with .
Since for any we havewe may assume . Then there is a unique great circle parametrized by arc length going from to in a time and .
Our geometric approach uses, on the one hand, the specific relation between spherical harmonics on and the Hardy space of the Fermat quadric hypersurface in [8, 9] and, on the other hand, the off-diagonal scaling asymptotics of the level- Szegö kernel of polarized projective manifold [10, 11].
The following asymptotic expansions involve a sequence of constants with a precise geometric meaning [9]. There is a natural algebraic isomorphism between the level- Szegö kernel of the Fermat quadric and , given by a push-forward operation; this isomorphism is however unitary only up to an appropriate rescaling, and is the corresponding scaling factor.
An asymptotic expansion for is discussed in [9], building on the theory of [8]; an alternative derivation is given in Proposition 2 (with an explicit computation of the leading order term).
In the following, the symbol stands for “has the same asymptotics as.”
Theorem 1. There exist smooth functions and on such that the following holds. Let us fix and . Then, uniformly in satisfying with we have for an asymptotic expansion of the form where
At the th step, we have for some constant and a similar estimate holds for the error term. Hence, the previous is an asymptotic expansion for .
As mentioned, the same techniques yield an asymptotic expansion for (see (6.18) in [9]).
Proposition 2. For , we have an asymptotic expansion of the form:
If we insert the latter expansion in the one provided by Theorem 1, we obtain the following.
Corollary 3. With the assumptions and notation of Theorem 1, for , there is an asymptotic expansionwhere and admit asymptotic expansions similar to those of and , respectively (of course, with different functions and , ).
Pairing Corollary 3 with (3), we obtain the following.
Corollary 4. In the same situation as in Theorem 1, for , there is an asymptotic expansionwhere again and admit asymptotic expansions similar to those of and , respectively.
Let us verify that Corollary 4 fits with the classical asymptotics. For example, when , we obtainso that the leading order term is the th Chebyshev polynomial. Since it is known that in this case the Legendre polynomial is the Chebyshev polynomial ([6], page 11), this is in fact the only term of the expansion.
For , we obtain the formula of Laplace (cfr [12], Section 4.6; [4], (8.01) of Ch. 4; [13], Theorem 8.21.2), but as a full asymptotic expansion holding uniformly on expanding subintervals converging to at a controlled rate, as above:
For arbitrary , is a multiple of a Gegenbauer polynomial ([2]; [6], page 16):Given the standardization for ([14], section 10.8)where is of course the Gamma function. By (35.31) in [3], for , we haveTherefore,If we use the well-known formula (see, e.g., (2) of [6])we obtain for as asymptotic expansion with leading order termin agreement with on page 198 of [14].
2. Preliminaries
2.1. The Geometric Picture
For the following, see [8, 9].
Let be the unit sphere, and let us identify the tangent and cotangent bundles of by means of the standard Riemannian metric. The unit (co)sphere bundle of is given by the incidence correspondence
The Fermat quadric hypersurface in complex projective space islet be the restriction to of the hyperplane line bundle. Given the standard Hermitian product on , is naturally a positive Hermitian line bundle, inherits a Kähler structure (the restriction of the Fubini-Study metric), and the spaces of global holomorphic sections of higher powers of , have an induced Hermitian structure.
The affine cone over is ; the intersection may be viewed as the unit circle bundle in the dual line bundle . More generally, for any , the intersectionwith the sphere of radius is naturally identified with the circle bundle of radius in . In particular, is diffeomorphic to by the map ; furthermore, is equivariant for the natural actions of on and defined by, respectively,
We shall identify and and denote the projection by
There is also a standard structure action of on , induced by fibrewise scalar multiplication in , or equivalently in . The latter action is intertwined by with the “reverse” geodesic flow on . The -orbits are the fibers of the circle bundle projection
This holds for any ; we shall denote by the projection for general .
2.2. The Metric on
Let us dwell on the metric aspect of (22); there are two natural choices of a Riemannian metric on , hence of a Riemannian density, and we need to clarify the relation between the two.
There is an obvious choice of a Riemannian metric on , induced by the standard Euclidean product on . With respect to , the orbits on have length of . Clearly, is homogeneous of degree with respect to the dilation , and therefore the corresponding volume form on is homogeneous of degree . That is,
An alternative and common choice of a Riemannian structure on comes from its structure of a unit circle bundle over . Let be the connection 1-form associated with the unique compatible covariant derivative on , so that . Also, let denote the horizontal and vertical tangent bundles for , respectively. There is a unique Riemannian metric on such that a Riemannian submersion and the -orbits on have unit length. The corresponding volume form on is given bywhere is the symplectic volume form on .
We wish to compare the two Riemannian metrics and , the corresponding volume forms, and , and densities, and .
Lemma 5. and .
Proof of Lemma 5. The connection 1-form for the Hopf map is thus, is the restriction of to . Let be the standard symplectic structure on . Since , we have (symplectic annihilator). In other words, where is the Hermitian orthocomplement of for the standard Hermitian product.
Thus, if , thenOn the other hand, . Thus, and are orthogonal with respect to both (by construction) and (by the previous considerations). Hence, we may compare and separately on and .
On the complex vector bundle , and are, respectively, the Euclidean scalar products associated with the restrictions of the -forms Given that and agree on , on .
On the other hand, both and are -invariant, but -orbits on have length for and for . Thus, on .
The claim follows directly from this.
2.3. The Szegö Kernel on
The following analysis is based on the equivariant asymptotics of the Szegö kernel of [11]. We refer the reader to [10, 11, 15] for a thorough discussion of Szegö kernels in the algebro-geometric context and to [16, 17] for the basic microlocal theory that underlies the subject (see also the neat discussion of Hardy spaces in [9]). To put things into perspective, however, let us recall that if is the boundary of a pseudoconvex domain, its Hardy space is the Hilbert space of square summable boundary values of holomorphic functions. The Szegö projector is then the orthogonal projector ; with some abuse of notation, the Szegö kernel is the corresponding distributional kernel. A description of as a Fourier integral operator was given in [16].
In the special algebro-geometric case, where is the dual unit circle bundle of a positive line bundle on a complex projective manifold, is the orthogonal direct sum of its isotypical components , under the -action; correspondingly, , where is the orthogonal projector onto ; since is finite-dimensional, is a smoothing operator; therefore, its Schwartz kernel is a function on . Many local asymptotic properties of , for , were first discovered in [10, 11, 15] building on the theory of [16]. We shall recall what is needed here shortly.
Let us now come to the specific case in point. For every , is the boundary of a strictly pseudoconvex domain, and as such it carries a CR structure, a Hardy space , and a Szegö projector . We aim to relate the various ’s.
Let be the ring of holomorphic functions on the conic complex manifold . Let be the subspace of holomorphic functions of degree of homogeneity .
For every and let be the finite-dimensional th isotypical component of with respect to the standard -action. Restriction induces an algebraic isomorphism ; with a slight abuse of language, we shall denote by the same symbol an element of and the corresponding element of .
Suppose that yields by restriction an orthonormal basis of :Setting and using (27) together with Lemma 5, we get
Therefore, we have the following.
Lemma 6. If yields by restriction an orthonormal basis of with respect to , then for every yields by restriction an orthonormal basis of , with respect to .
Let now be the level- Szegö kernel on , that is, the orthogonal projectorBy Lemma 6, its Schwartz kernel is given by
When pulled back to , this is (here )
In particular,
We shall make repeated use of the following asymptotic property of , which follows from the microlocal description of as an FIO (explicit exponential estimates are discussed in [18]).
Theorem 7. Let be the distance function on associated with the Kähler metric. Given any , uniformly for satisfyingwe havewhen .
2.4. Heisenberg Local Coordinates
There are two unit circle bundles in our picture: the Hopf fibration , and . Clearly, is the pull-back of under the inclusion . Both and are boundaries of strictly pseudoconvex domains and carry a CR structure.
On both and , we may consider privileged systems of coordinates called Heisenberg local coordinates (HLC). In these coordinates, Szegö kernel asymptotics exhibit a “universal” structure [11]; we refer to ibidem for a detailed discussion.
Given , a HLC system on centered at will be denoted in additive notation:Here is an “angular” coordinate measuring displacement along the -orbit through (the fiber through of ); instead, descends to a local coordinate on centered at , inducing a unitary isomorphism . We may thus think of as a tangent vector in .
Here this additive notation might be misleading, since . Therefore, we shall write for HLC on centered at . We shall generally abridge notation by writing for .
Similarly, will denote a system of Heisenberg local coordinates on centered at . There is in fact a natural choice of HLC on centered at any .
Namely, let be an orthonormal basis of the Hermitian orthocomplement , and for let us setSince there is a canonical unitary identification , we shall also write this as with .
If , HLC on centered at can be chosen so that they agree to second order with the former HLC on . More precisely, we may assume that for any we havewhere is a function vanishing to second order at the origin.
Given , let us definehere 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 [11].
Theorem 8. Let us fix and . Then for any , and for any choice of HLC on centered at , there exist polynomials of degree and parity on , such that the following holds. Uniformly in with for , and , one has for the following asymptotic expansion:
In the given range, the above is an asymptotic expansion, since
2.5. and
As discussed in [9], the push-forward operator yields by restriction an algebraic isomorphism for every , and (49) yields by restriction an isomorphismwhich is unitary up to a dilation by a constant factor . Thus, we have
Therefore, if is an orthonormal basis of , thenis an orthonormal basis of . It follows that in (2) is given bywhere is the product projection.
More explicitly, for let be the unit sphere centered at the origin in the orthocomplement , and let be the Riemannian density on ; then
2.6. and Conjugation
Conjugation in leaves invariant the affine cone and every . Furthermore, it yields a Riemannian isometry of into itself. For , let us setIf , then .
Hence, if yields by restriction an orthonormal basis of , then so does . Thus, for any , we have
3. Proof of Theorem 1
Proof of Theorem 1. Given with , let be the unique unit speed geodesic on such that and for some . Then The reverse geodesic satisfies , and for a unique .
Although they project down to the same locus in , and correspond to distinct fibers of the circle bundle projection . Let us express the (co)tangent lift of the geodesics in complex coordinates, and set . ThenIn view of (26), we have On the other hand, , since and are linearly independent in .
Thus, we have the following.
Lemma 9. Suppose and . Then the only points such thatare
By Theorem 7, for fixed and and , we haveunless and . Therefore, for a fixed integration in (54) may be localized in a small neighborhood of , perhaps at the cost of disregarding a negligible contribution to the asymptotics.
Since however we are allowing to approach or at a controlled rate, we need to give a more precise quantitative estimate of how small the previous neighborhood may be chosen when .
To this end, let us introduce some further notation. Given linearly independent , let us setFurthermore, for , we shall set
A straightforward computation yields the following.
Lemma 10. Assume that and with . Then any with , respectively, , may be written uniquely in the form respectively where (the Euclidean orthocomplement) has norm ≤ and
Proposition 11. Let us fix , and . Then there exist constants such that the following holds. Suppose that (1);(2) for ;(3);(4) for ;(5);(6) for , where can be either one of (Lemma 10).Thenfor every .
In view of Theorem 7, Proposition 11 implies the following.
Corollary 12. Uniformly in the range of Proposition 11, we have
Proof of Proposition 11. Let us set for Let be the restriction to of the distance function on . ThenThe factor in front is needed because while the Hopf map is a Riemannian submersion, the projection is so only after a constant rescaling of the metric.
We are reduced to proving that in the given range there exist constants such that for every and We have where Regarding the two summands on the last line of (73), we havewhere denotes the Hermitian orthocomplement. Hence,Since vanishes exactly to second order at , there exists such that for , we haveGiven this and (77), we conclude that, under the present hypothesis,Let us now pick with and assumeThenThis establishes (72) with , in the case where (80) holds. Thus, we are reduced to assuming Then we also have . Let us then look at the first summand on the last line of (73). We have an Hermitian orthogonal direct sumOn the other hand, since vanishes exactly to first order at and , there exists such that for under the assumptions of the lemma we haveHence, in view of (75), we have for some and since and . This establishes (72) with when (82) holds.
The proof of Proposition 11 is complete.
Equations (65) and (66) parametrize neighborhoods of and , respectively. Therefore, Proposition 11 implies that in (54) only a negligible contribution to the asymptotics is lost, if integration in and yields by restriction the shrinking neighborhoods of and , of radii .
This may be rephrased as follows. Let 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 (54) are unchanged, if the integrand is multiplied by
In this way, the integrand splits into four summands. In fact, only two of these are nonnegligible for . Namely, consider the summand containing the factor
On its support, lies in a shrinking neighborhood of and in a shrinking neighborhood of . Therefore, on the same support lies in a shrinking neighborhood of , and lies in a shrinking neighborhood of . Sincehas unit norm, on the support of (87) and remain at a distance ≥2/3, say, in projective space. This implies that as uniformly in in the support of (87). A similar argument applies to the summand containing the factor
Thus, we may rewrite (54) as follows:where
As a further reduction, we need only deal with one of .
Lemma 13. .
Proof of Lemma 13. Let us apply the change of integration variable and and apply (56). Since is even, we get
In the definition of , integration is over a shrinking neighborhood of . We can thus make use of the parametrization (65) and write in (92):where we have setIt is also harmless to replace by in the rescaled cut-offs in (92). Let us also set and recall that . We then obtainwhere .
Let us pass to rescaled integration variables in (97). Thenwith
Let us consider the Szegö term in the integrand. In view of (40), this is
Now the sums in the previous expression are just algebraic sums in ; in order to apply the scaling asymptotics of Theorem 8, we need to first express the argument of (100) in terms of local Heisenberg coordinates on centered at .
Lemma 14. Suppose and choose a system of HLC on centered at . Then for and such that we havefor a suitable smooth function vanishing to second order at the origin (in ).
Proof of Lemma 14. In view of (45), it suffices to prove the statement on , working with the HLC (44). Since , we haveso that .
Let us look for and (Hermitian orthocomplement) such thatIf this is possible at all, then necessarily , as . Then Assuming that (103) may be solved, then, taking the Hermitian product with on both sides of (103) and using (102), we get With this value of , let us set so that (103) is certainly satisfied. We need to verify that . Indeed we haveSince , the proof of the lemma is complete.
Notice that is given for by an asymptotic expansion in homogeneous polynomials of increasing degree in of the form This holds on , but a similar expansion obviously holds on , possibly with modified terms in higher degree.
Let us apply Lemma 14 with and (we will set for and for ). To this end, let us note that in view of (99) for there is an asymptotic expansion of the formwhere is a homogeneous (vector valued) polynomial function of degree and . Hence, Making use of (110) in (108) we obtainwhere is a homogeneous polynomial function of degree and we have emphasized the dependence on .
Thus, we obtain for (with ) thatwherewith defined by the latter equality. Similarly, for (with ), we have where
Let us return to (100). In view of Theorem 8, we get
We havewhere is a homogeneous -valued polynomial of degree . For any and , we havewhere is homogeneous of degree . Since for every , we have .
One can see from this that where is a polynomial of degree ≤3, having the same parity as .
Similarly, recalling that has the same parity as and degree ≤3, each summand in (116) gives rise to an asymptotic expansion in terms of the formwhere and are homogeneous polynomials of the given degree, , and is even. Then , and is also even. Hence, each summand () yields an asymptotic expansion of the formwhere again each has the same parity as and degree ≤3.
Putting this all together, we obtain an asymptotic expansion for the integrand in (98).
Lemma 15. For , there exist polynomials of degree ≤ and parity , with , such that
Proof of Lemma 15. 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 (98) takes place over a poly-ball or radius in , the expansion may be integrated term by term. In addition, given that the exponent and the cut-offs are even functions of , only terms of even parity yield a nonzero integral. Hence, we may discard the half-integer powers and obtainwhereWe can slightly simplify the previous asymptotic expansion as follows. First, as emphasized the dependence on is of course only through the angle . In particular, in (124) nothing is lost by assuming that and span the 2-plane , and therefore that .
Furthermore, given (46), we haveWith the change of variables we obtain Since is even and has degree ≤6, we can writewhere is an even polynomial of degree ≤6, with smooth bounded coefficients for . Thus,
There is a constant such that the support ofis contained in the locus, where . Under the assumptions of the theorem, this implies, perhaps for a different constant , that . On the other hand, the exponent in (129) satisfiesGiven that (statement of Proposition 11), 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 .
We can thus rewrite (123) as follows:where
Let us set . The leading order coefficient is
Given (134), (132), and (94), has an asymptotic expansion for with leading order term
For any , we can writewhere is an even polynomial of degree ≤6.
Let us introduce the Fourier transform
Then (136) is the result of applying an even differential polynomial of degree ≤6 to and then evaluating the result at .
Given this and (135), we conclude that wherewith and smooth functions of on .
Proof of Theorem 1 is complete.
4. Proof of Proposition 2
Proof of Proposition 2. The diagonal restriction may be computed in two different ways. On the one hand, since is constant, we have On the other hand, (54) with yieldswhere Again, integration in localizes in a shrinking neighborhood of . Hence, we may let where , and introduce the cut-off . Passing to rescaled coordinates, and setting , we getwhereBy Lemma 15 (with , , , ), we havefor certain polynomials of degree ≤3 and parity , with .
As before, the expansion may be integrated term by term and, by parity, only the summands with even yield a nonzero contribution. In addition, only a negligible contribution is lost if the cut-off is omitted and integration is extended to all of . Therefore,Inserting this in (141), we obtain an asymptotic expansionComparing (140) and (148), we obtain an asymptotic expansion in descending powers of , of the form
Conflicts of Interest
The author declares that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
The author is indebted to Leonardo Colzani and Stefano Meda for very valuable comments and insights.