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The
set ℬ consists of all real biaxially symmetric potentials
U(α,β)(x,y) = ∑
n=0∞an(x2 + y2)nPn(α,β)(x2 −y2∕x2 + y2)∕Pn(α,β)(1), α > β > −1∕2
which are regular in the open unit sphere Σ about the origin in E2(α+β+2).
Three problems appear regarding ℬ and subset ℬ∗ whose members have the
first m + 1 coefficients a0,⋯,am specified. (1) For U(α,β) ∈ℬ, determine
I(U(α,β)) = inf{U(α,β)(x,y)|(x,y) ∈ Σ} as limit of a monotone sequence of constants
{λ2n(a0,⋯,an)}n=0∞ which can be computed algebraically. (2) Find U0(α,β) ∈ℬ∗
and the constant λ2m(a0,⋯,am) = sup{I(U(α.β))|U(α,β) ∈ℬ∗} = I(U0(α,β)). (3)
Determine necessary and sufficient conditions from the Fourier coefficients so that
U(α,β) ∈ℬ and U(α,β) is nonnegative in Σ. We develop solutions using operators
based on Koornwinder’s Laplace type integral for Jacobi polynomials, along with
applications of the methods of ascent and descent to the Caratheodory-Fejer and
Caratheodory-Toeplitz problems which focus on the properties of harmonic functions
in E2.
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