Abstract
The purpose of the paper is to study the coefficient multipliers of the Hardy spaces H^{p} associated with Jacobi expansions of exponential type. The main results are about the boundedness from H^{p} to ell ^{q} of the multiplier operators in terms of Jacobi expansions of exponential type for (i) p=1, 2le q<infty ; (ii) gamma ( alpha ,beta )^{-1}< p<1le q<infty , under appropriate conditions, where gamma (alpha ,beta )in (1,infty ] is a number depending on the parameters of the Jacobi system.
Highlights
Introduction and main results1.1 Jacobi expansions of exponential type Assume that α, β > –1
In analogy to the relation of cos nt and sin nt, the system {R(nα–+11,β+1)(cos t) sin t}∞ n=1 is introduced in [8], which is a conjugate one of {Rn(α,β)(cos t)}∞ n=0 based on a pair of genera{Elin(zαe,βd)(Ct)a}u∞ n=c–h∞y–aRs iienm[9a]n,nbyeqEu0(αa,βti)o=ns1./√Th2i,saanldlofwosr us to define n ≥ 1, an exponential type system
The purpose of the paper is to study the coefficient multipliers of the real Hardy spaces Hp(–π, π) associated with Jacobi expansions of exponential type
Summary
1.1 Jacobi expansions of exponential type Assume that α, β > –1. Let Rn(α,β)(x) be the Jacobi polynomial on [–1, 1] of degree n normalized so that Rn(α,β)(1) = 1. It follows that the system {Rn(α,β)(cos t)}∞ n=0 is orthogonal over [0, π ] with respect to the weight sin2α+1(t/2) cos2β+1(t/2). In analogy to the relation of cos nt and sin nt, the system {R(nα–+11,β+1)(cos t) sin t}∞ n=1 is introduced in [8], which is a conjugate one of {Rn(α,β)(cos t)}∞ n=0 based on a pair of genera{Elin(zαe,βd)(Ct)a}u∞ n=c–h∞y–aRs iienm[9a]n,nbyeqEu0(αa,βti)o=ns1./√Th2i,saanldlofwosr us to define n ≥ 1, an exponential type system. For f ∈ L(–π, π), its Jacobi expansion of exponential type is defined by. –π where cn(f ) are called the Fourier–Jacobi coefficients of f
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