Vol. 74, No. 2, 1978

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Extremal properties of real biaxially symmetric potentials in E2(α+β+2)

Peter A. McCoy

Vol. 74 (1978), No. 2, 381–389

The set consists of all real biaxially symmetric potentials U(α,β)(x,y) = n=0an(x2 + y2)nPn(α,β)(x2 y2∕x2 + y2)∕Pn(α,β)(1), α > β > 12 which are regular in the open unit sphere Σ about the origin in E2(α+β+2). Three problems appear regarding and subset 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.

Mathematical Subject Classification 2000
Primary: 31B99
Received: 25 January 1977
Published: 1 February 1978
Peter A. McCoy