## 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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### References:

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