Abstract
Let us introduce the Sobolev-type inner product , where and , , with and for all A Mehler-Heine-type formula and the inner strong asymptotics on as well as some estimates for the polynomials orthogonal with respect to the above Sobolev inner product are obtained. Necessary conditions for the norm convergence of Fourier expansions in terms of such Sobolev orthogonal polynomials are given.
Highlights
For a nontrivial probability measure σ, supported on −1, 1, we define the linear space Lp dσ of all measurable functions f on −1, 1 such that f Lp dσ < ∞, where ⎧ ⎪⎪⎪⎪⎨ 1/p f x pdσ x, if 1 ≤ p < ∞, f Lp dσ −1⎪⎪⎪⎪⎩ess sup f x, if p ∞. −1
Β > −1, we denote by pnα,β n≥0 the sequence of Jacobi polynomials which are orthonormal on −1, 1 with respect to the inner product f, g 1 f gdμα,β
From 3.8, we find that αnn−N
Summary
For a nontrivial probability measure σ, supported on −1, 1 , we define the linear space Lp dσ of all measurable functions f on −1, 1 such that f Lp dσ < ∞, where. Using the standard Gram-Schmidt method for the canonical basis xn n≥0 in the linear space of polynomials, we obtain a unique sequence up to a constant factor of polynomials Qnα,β,N n≥0 orthogonal with respect to the above inner product. In the sequel, they will called Jacobi-Sobolev orthogonal polynomials. In the Jacobi case, some analog problems have been considered in 10, 11 The aim of this contribution is to study necessary conditions for WN,p-norm convergence of the Fourier expansion in terms of Jacobi-Sobolev orthogonal polynomials. A Mehler-Heine-type formula, inner strong asymptotics, upper bounds in −1, 1 , and WN,p norms of Jacobi-Sobolev orthonormal polynomials are obtained. The notation un ∼ vn means that the sequence un/vn converges to 1 and notation un ∼ vn means c1un ≤ vn ≤ c2un for sufficiently large n
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