, 55:4 | Cite as

Cubature formulae for nearly singular and highly oscillating integrals

  • Donatella Occorsio
  • Giada Serafini


The paper deals with the approximation of integrals of the type
$$\begin{aligned} I(f;{\mathbf {t}})=\int _{{\mathrm {D}}} f({\mathbf {x}}) {\mathbf {K}}({\mathbf {x}},{\mathbf {t}}) {\mathbf {w}}({\mathbf {x}}) d{\mathbf {x}},\quad \quad {\mathbf {x}}=(x_1,x_2),\quad {\mathbf {t}}\in \mathrm {T}\subseteq \mathbb {R}^p, \ p\in \{1,2\} \end{aligned}$$
where \({\mathrm {D}}=[-\,1,1]^2\), f is a function defined on \({\mathrm {D}}\) with possible algebraic singularities on \(\partial {\mathrm {D}}\), \({\mathbf {w}}\) is the product of two Jacobi weight functions, and the kernel \({\mathbf {K}}\) can be of different kinds. We propose two cubature rules determining conditions under which the rules are stable and convergent. Along the paper we diffusely treat the numerical approximation for kernels which can be nearly singular and/or highly oscillating, by using a bivariate dilation technique. Some numerical examples which confirm the theoretical estimates are also proposed.


Cubature rules Orthogonal polynomials Approximation by polynomials 

Mathematics Subject Classification

65D32 41A05 41A10 



We want to thank the anonymous referee for the careful reading of the manuscript and for the valuable comments. We are also grateful to Professor G. Mastroianni for his helpful suggestions.


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© Springer-Verlag Italia S.r.l., part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Mathematics, Computer Sciences and EconomicsUniversity of BasilicataPotenzaItaly

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