Nikol’skii constants for compact homogeneous spaces

In this paper, we study the sharp 𝐿𝑝-Nikol’skii constants for the case of Riemannian symmetric manifolds M𝑑 of rank 1. These spaces are fully classified and include the unit Euclidean sphere S𝑑, as well as the projective spaces P𝑑(R), P𝑑(C), P𝑑(H), P16(Ca). There is a common harmonic analysis on these manifolds, in particular, the subspaces of polynomials Π𝑛(M𝑑) of order at most 𝑛 are defined. In the general case, the sharp 𝐿𝑝-Nikol’skii constant for the subspace 𝑌 ⊂ 𝐿∞ is defined by the equality
𝒞(𝑌,𝐿𝑝) = sup
𝑓∈(𝑌 ∩𝐿𝑝)∖{0}
‖𝑓‖∞
‖𝑓‖𝑝
.
V.A. Ivanov (1983) gave the asymptotics
𝒞(Π𝑛(M𝑑),𝐿𝑝(M𝑑)) ≍ 𝑛𝑑/𝑝, 𝑛 → ∞, 𝑝 ∈ [1,∞).
For the case of a sphere, this result was significantly improved by the author together with F. Dai and S. Tikhonov (2020):
𝒞(Π𝑛(S𝑑),𝐿𝑝(S𝑑)) = 𝒞(ℰ𝑑
1 ,𝐿𝑝(R𝑑))𝑛𝑑/𝑝(1 + 𝑜(1)), 𝑛 → ∞, 𝑝 ∈ (0,∞),
where ℰ𝑑
1 is the set of entire functions of exponential spherical type at most 1 bounded on R𝑑.
M.I. Ganzburg (2020) transferred this equality to the case of the multidimensional torus T𝑑 and trigonometric polynomials. For 𝑑 = 1, these results follow from the fundamental work of
E. Levin and D. Lubinsky (2015).
In a joint work of the author and I.A. Martyanov (2020), the following explicit boundaries of the spherical Nikol’skii constant were proved, which refine the above results for 𝑝 ⩾ 1:
𝑛𝑑/𝑝 ⩽ 𝒞(Π𝑛(S𝑑),𝐿𝑝(S𝑑))
𝒞(ℰ𝑑
1 ,𝐿𝑝(R𝑑))
⩽
(︀
𝑛 + 2⌈𝑑+1
2𝑝 ⌉
)︀𝑑/𝑝
, 𝑛 ∈ Z+, 𝑝 ∈ [1,∞).
This result was proved using a one-dimensional version of the problem for the case of a periodic Gegenbauer weight.
The development of this method allows us to prove the following general result: for 𝑝 ⩾ 1
𝑛𝑑/𝑝 ⩽ 𝒞(Π𝑛(M𝑑),𝐿𝑝(M𝑑))
𝒞(ℰ𝑑
1 ,𝐿𝑝(R𝑑))
⩽
(︀
𝑛 + ⌈𝛼𝑑+3/2
𝑝 ⌉ + ⌈𝛽𝑑+1/2
𝑝 ⌉
)︀𝑑/𝑝
,
where 𝛼𝑑 = 𝑑/2 − 1, 𝛽𝑑 = 𝑑/2 − 1, −1/2, 0, 1, 3 respectively for S𝑑, P𝑑(R), P𝑑(C), P𝑑(H), P16(Ca). The proof of this result is based on the connection of harmonic analysis on M𝑑 with Jacobi analysis on [0, 𝜋] and T with periodic weight
⃒⃒
2 sin 𝑡
2
⃒⃒
2𝛼+1⃒⃒
cos 𝑡
2
⃒⃒
2𝛽+1. Also we give related results for the trigonometric Nikol’skii constants in 𝐿𝑝 on T with Jacobi weight and Nikol’skii
constants for entire functions of exponential type in 𝐿𝑝 on R with power weight.

1. Arestov, V.V. & Deikalova, M.V. 2014. “Nikol’skii inequality for algebraic polynomials on a multidimensional Euclidean sphere”, Proc. Steklov Inst. Math. (Suppl.), vol. 284, no. suppl. 1,

2. pp. 9–23.

3. Arestov, V. & Deikalova, M. 2016. “Nikol’skii inequality between the uniform norm and 𝐿𝑞-norm with Jacobi weight of algebraic polynomials on an interval”, Anal. Math., vol. 42, no. 2,

4. pp. 91–120.

5. Vo Thi Cuc. 2012. “Generalized translation operators in 𝐿𝑝-spaces on torus with Jacobi weight and their application”, Izv. Tul. Gos. Univ. Estestv. Nauki, no. 1, pp. 5–27. (in Russ.)

6. Dai, F., Gorbachev, D. & Tikhonov, S. 2020. “Nikolskii constants for polynomials on the unit sphere”, J. d’Anal. Math., vol. 140, no. 1, pp. 161–185.

7. Gangolli, R. 1967. “Positive definite kernels on homogeneous spaces and certain stochastic processes related to L´evy’s Brownian motion of several parameters”, Ann. Inst. H. Poincar´e,

8. vol. 3, no. 2, pp. 121–226.

9. Ganzburg, M.I. 2020. “Sharp constants of approximation theory. I. Multivariate Bernstein–Nikolskii type inequalities”, J. Fourier Anal. Appl., vol. 26, no. 11.

10. Gorbachev, D.V. 2005. “An integral problem of Konyagin and the (𝐶,𝐿)-constants of Nikol’skii”, Proc. Steklov Inst. Math., vol. Suppl. 2, pp. S117–S138.

11. Gorbachev, D.V., & Mart’yanov, I.A. 2020. “Bounds of the Nikol’skii polynomial constants in 𝐿𝑝 with Gegenbauer weight”, Trudy Inst. Mat. i Mekh. UrO RAN, vol. 26, no. 4, pp. 126–137.

12. (In Russ.)

13. Ivanov, V.A. 1983. “On the Bernstein–Nikol’skii and Favard inequalities on compact homogeneous spaces of rank 1”, Russian Math. Surveys, vol. 38, no. 3, pp. 145–146.

14. Ivanov, V.A. 1993. “Precise results in the problem of the Bernstein–Nikol’skij inequality on compact symmetric Riemannian spaces of rank 1”, Proc. Steklov Inst. Math., vol. 194, pp. 115–124.

15. Jaming, P. & Speckbacher, M. 2021. “Concentration estimates for finite expansions of spherical harmonics on two-point homogeneous spaces via the large sieve principle”, Sampl. Theory Signal Process. Data Anal., vol. 19, no. 9.

16. Levin, E. & Lubinsky, D. 2015. “Asymptotic behavior of Nikolskii constants for polynomials on the unit circle”, Comput. Methods Funct. Theory, vol. 15, no. 3, pp. 459–468.

17. Martyanov, I.A. 2020. “Nikolskii constant for trigonometric polynomials with periodic Gegenbauer weight”, Chebyshevskii Sbornik, vol. 21, no. 1, pp. 247–258. (In Russ.)

18. Helgason, S. 1962. “Differential geometry and symmetric spaces”, Academic Press, New York.

19. Chertova, D.V. 2009. “Jackson theorems in 𝐿𝑝-spaces, 1 ⩽ 𝑝 ⩽ 2, with periodic Jacobi weight”, Izv. Tul. Gos. Univ. Estestv. Nauki, no. 1, pp. 5–27. (in Russ.)