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Boundedness and uniform numerical approximation of the weighted Hilbert transform on the real line. (English) Zbl 0983.41015

Let \(I\) be the interval \((-\infty , \infty)\) and \(w\) a weight such that the power moments \(\int_I x^nw^2(x) dx\), \(n\geq 0\), are finite. The authors consider the continuous functions \(f:I\to \mathbb R\) satisfying \(\lim _ {|x |\infty }f(x)w(x)=0 \) and define the weighted Hilbert transform \[ H[ff;w^2](x):=\int _I \frac {f(t)}{t-x}w^2(t) dt =\lim _{\varepsilon \to 0+}\int _{|t-x |\geq \varepsilon} \frac {f(t)}{t-x}w^2(t) dt, \quad x\in I. \] The boundedness of this transform in suitable weighted subspaces of \(L_{\infty ,w}\) is established together with its numerical approximation by a quadrature procedure based on polynomial interpolation at the zeros of orthogonal polynomials associated with the weight \(w\). Example of weights that are studied are: (a) \(w_\alpha (x) =\exp (-|x |^\alpha)\), \(\alpha >1\), \(x\in \mathbb R\); (b)\(w_{k, \beta}(x) =\exp (- \exp(|x|^\beta))\), \(\beta >0\), \(k \geq 1\), \(x\in \mathbb R\).

MSC:

41A55 Approximate quadratures
65D30 Numerical integration
65R10 Numerical methods for integral transforms
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