We analyze the spectrum of a certain singular integral operator on the space (L2(fi))3 where fi is contained in three dimensional Euclidean space and has a Lipschitz continuous boundary. This operator arises in the integral formulation of the magnetostatic field problem. We decompose (L2(fi))3 into invariant subspaces: in one where the operator is the zero map; in one, the identity map; and in ...

A class of multi-wavelet bases for L2 is constructed with the property that a variety of integral operators is represented in these bases as sparse matrices, to high precision. In particular, an integral operator K whose kernel is smooth except along a nite number of singular bands has a sparse representation. In addition, the inverse operator (I K) 1 appearing in the solution of a second-kind ...

We give a generalisation of the multivariate beta integral. This is used to show that the (multivariate) Bernstein–Durrmeyer operator for a Jacobi weight has a limit as the weight becomes singular. The limit is an operator previously studied by Goodman and Sharma. From the elementary proof given, it follows that this operator inherits many properties of the Bernstein–Durrmeyer operator in a nat...

Abstract In this paper, the numerical method for solving Abel’s integral equations is presented. This method is based on fractional calculus. Also, Chebyshev polynomials are utilized to apply fractional properties for solving Abel’s integral equations of the first and second kind. The fractional operator is considered in the sense of RiemannLiouville. Although Abel’s integral equations as singu...

This paper concerns a rather concrete phenomenon in abstract operator algebras. The main examples of the algebras we study are algebras of singular integral operators (pseudo-differential operators of order zero). As everyone knows, the Fredhohn index of a pseudo-differential operator depends only on its symbol and the Atiyah-Singer Index Theorem gives an explicit formula for computing the depe...

We consider a singular second order differential operator ∆ defined on ]0,∞[. We give nice estimates for the kernel which intervenes in the integral transform of the eigenfunction of ∆. Using these results, we establish Hardy type inequalities for Riemann-Liouville and Weyl transforms associated with the operator ∆.