“…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].…”
“…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).…”
We study pointwise bounds of orthogonal expansions on the real line for a class of exponential weights of smooth polynomial decay at infinity. As a consequence of our main results, we establish pointwise bounds for weighted Hilbert transforms which are of independent interest.
“…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].…”
“…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).…”
We study pointwise bounds of orthogonal expansions on the real line for a class of exponential weights of smooth polynomial decay at infinity. As a consequence of our main results, we establish pointwise bounds for weighted Hilbert transforms which are of independent interest.
We introduce special Hermite-Fejér and Grünwald operators at the zeros of the generalized Laguerre polynomials. We will prove that these interpolation processes are uniformly convergent in suitable weighted function spaces.
“…Subsequently, B. Muckenhoupt [15] introduced the weights u(x) = 1 + |x| b w(x) and v(x) = 1 + |x| B 1 + log + |x| η w(x), with w(x) = e −x 2 , and thus he proved inequalities of the type (1.1) S. W. Jha and D. S. Lubinsky [6] extended the results of B. Muckenhoupt to the case of Freud weights, namely they replaced w(x) = e −x 2 by w(x) = e −Q(x) (under suitable assumptions on Q).…”
In order to approximate functions defined on the real line or on the real semiaxis by polynomials, we introduce some new Fourier-type operators, connected to the Fourier sums of generalized Freud or Laguerre orthonormal systems. We prove necessary and sufficient conditions for the boundedness of these operators in suitable weighted L p -spaces, with 1 < p < ∞. Moreover, we give error estimates in weighted L p and uniform norms.
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