The Norm of the Laplace Operator Restriction to a Homogeneous Polynomial Space

Abstract

We investigate the restriction Δ _r,μ of the Laplace operator Δ onto the space of r-variate homogeneous polynomials F of degree μ. In the uniform norm on the unit ball of ℝ^r, and with the corresponding operator norm, ‖Δ _r,μ F‖≤‖Δ _r,μ ‖⋅‖F‖ holds, where, for arbitrary F, the ‘constant’ ‖Δ _r,μ ‖ is the best possible. We describe ‖Δ _r,μ ‖ with the help of the family T _μ (σ x), $\cos\frac{\pi}{\mu}\leq\sigma\leq1$ , of scaled Chebyshev polynomials of degree μ. On the interval [−1,+1], they alternate at least (μ−1)-times, as the Zolotarev polynomials do, but they differ from them by their symmetry. We call them Zolotarev polynomials of the second kind, and calculate ‖Δ _r,μ ‖ with their help. We derive upper and lower bounds, as well as the asymptotics for μ→∞. For r≥5 and sufficiently large μ, we just get ‖Δ _r,μ ‖=(r−2)μ(μ−1). However, for 2≤r≤4 or lower values of μ, the result is more complicated. This gives the problem a particular flavor. Some Bessel functions and the φcot φ-expansion are involved.