Mean Convergence of Expansions in Freud-Type Orthogonal Polynomials

Let W : R → (0, 1] be continuous. Bernstein's approximation problem, posed in 1924, deals with approximation by polynomials in the weighted uniform norm f → f W L∞(R) . The qualitative form of this problem was solved by Achieser, Mergelyan, and Pollard, in the 1950's. Quantitative forms of the problem were actively investigated starting from the 1960's. We survey old and recent aspects of this topic, including the Bernstein problem, weighted Jackson and Bernstein Theorems, Markov-Bernstein and Nikolskii inequalities, orthogonal expansions and Lagrange interpolation. We present the main ideas used in many of the proofs, and different techniques of proof, though not the full proofs. The class of weights we consider is typically even, and supported on the whole real line, so we exclude Laguerre type weights on [0, ∞). Nor do we discuss Saff's weighted approximation problem, nor the asymptotics of orthogonal polynomials.MSC: 41A10, 41A17, 41A25, 42C10

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“…Then (see [90] or [141] for an accessible proof) K n (x, y) = K n,1 (x, y) + K n,2 (x, y) + K n,3 (x, y),…”

Section: Mean Convergencementioning

confidence: 99%

Weighted polynomial approximation with Freud weights

Lubinsky

Тотик

1994

Constr. Approx

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“…The first significant results dealing with mean convergence of orthonormal expansions on the line are due to Askey and Wainger for the Hermite weight w(x) = exp(−x 2 ), see [1]. Thereafter, followed related results of Muckenhoupt, see [24,25], Mhaskar and Xu, see [23] and Jha and Lubinsky, see [14].…”

Section: Background: Fourier Series/orthogonal Expansionsmentioning

confidence: 96%

“…More precisely, we will use Pollards decomposition of K as applied by Askey and Wainger, Muckenhoupt, Mhaskar and Xu, and Jha and Lubinsky in [1], [24,25], [23] and [14]. For a given t, x ∈ R, write,…”

Section: Proofs Of Theorems 1mentioning

confidence: 99%

“…Analogues of theorem 1 in L p (1 < p < ∞) are contained in[1,4,14,[23][24][25]. More precisely, in[1,24,25], Askey, Waigner and Muckenhoupt proved an L p analogue of theorem 1 for the Hermite weight ((1.6) with α = 2).…”

mentioning

confidence: 94%

“…More precisely, in[1,24,25], Askey, Waigner and Muckenhoupt proved an L p analogue of theorem 1 for the Hermite weight ((1.6) with α = 2). Subsequently in[23], Mhaskar and Xu generalized Muckenhoupts result to a larger class of weights on the line and in [14, theorem 1.2], Jha and Lubinsky obtained necessary and sufficient conditions for mean convergence. We note that in this later paper, the authors assumed both (1.14) and (1.16) for their sufficiency.…”

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confidence: 99%

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