Hardy's inequalities in finite dimensional Hilbert spaces

2020/07/20 by Dimitrov, Dimitar K., Gadjev, Ivan, Nikolov, Geno +1
#15A42 #26D15 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Primary 26D10 #Secondary 33C45

paper · doi:10.48550/arxiv.2007.10073

Abstract

We study the behaviour of the smallest possible constants d_n and c_n in Hardy's inequalities ∑_k=1ⁿ((1)/(k)∑_j=1^ka_j)²≤ d_n ∑_k=1ⁿa_k², (a₁,…,a_n) ∈ ℝⁿ and ∫₀^∞((1)/(x)∫₀^xf(t) dt)² dx ≤ c_n ∫₀^∞ f²(x) dx, f∈ H_n, for the finite dimensional spaces ℝⁿ and H_n:=\f : ∫₀^x f(t) dt =e^-x/2 p(x) : p∈ P_n, p(0)=0\, where P_n is the set of real-valued algebraic polynomials of degree not exceeding n. The constants d_n and c_n are identified as the smallest eigenvalues of certain Jacobi matrices and the two-sided estimates for d_n and c_n of the form 4-(c)/(ln n)lt; d_n, c_nlt;4-(c)/(ln² n) , cgt;0 are established.

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