2013/06/30 by Chang‐Feng Dai, Sergey Tikhonov, Dai, Feng +1
Mathematics · #33C50 #33C52 #42B15 #42C10 #Advanced Harmonic Analysis Research #Approximation Theory and Sequence Spaces #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Mathematical functions and polynomials
paper · pdf · doi:10.48550/arxiv.1307.0207
openalex publication_date 2013/06/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
This paper studies the following weighted, fractional Bernstein inequality for spherical polynomials on \sph: ‖(-Δ0)r/2 f‖p,w≤ Cw nr ‖f‖p,w, ∀ f∈ Πnd, where Πnd denotes the space of all spherical polynomials of degree at most n on \sph, and (-Δ0)r/2 is the fractional Laplacian-Beltrami operator on \sph. A new class of doubling weights with conditions weaker than the Ap is introduced, and used to fully characterize those doubling weights w on \sph for which the weighted Bernstein inequality \eqref4-1-TD-ab holds for some 1≤ p≤ ∞ and all r>τ. In the unweighted case, it is shown that if 00 is not an even integer, then \eqref4-1-TD-ab with w≡ 1 holds if and only if r>(d-1)(\f 1p-1). As applications, we show that any function f∈ Lp(\sph) with 0