Asymptotic behavior of singular values and singular functions of certain convolution operators

Singular values of convex functions of matrices

‎Let $A_{i},B_{i},X_{i},i=1,dots,m,$ be $n$-by-$n$ matrices such that $‎sum_{i=1}^{m}leftvert A_{i}rightvert ^{2}$ and $‎sum_{i=1}^{m}leftvert B_{i}rightvert ^{2}$  are nonzero matrices and each $X_{i}$ is‎ ‎positive semidefinite‎. ‎It is shown that if $f$ is a nonnegative increasing ‎convex function on $left[ 0,infty right) $ satisfying $fleft( 0right)‎ ‎=0 $‎, ‎then  $$‎2s_{j}left( fleft( fra...

On the asymptotic expansion of certain plane singular integral operators

*Correspondence: [email protected] Chair of Differential Equations, Belgorod National Research University, Studencheskaya 14/1, Belgorod, 308007, Russia Abstract We discuss the problem of the asymptotic expansion for some operators in a general theory of pseudo-differential equations on manifolds with borders. Using the distribution theory one obtains certain explicit representation...

Iteration of certain meromorphic functions with unbounded singular values

Let M= { f (z)= (zm/sinhm z) for z ∈ C | either m or m/2 is an odd natural number}. For each f ∈M, the set of singularities of the inverse function of f is an unbounded subset of the real line R. In this paper, the iteration of functions in oneparameter family S = { fλ(z)= λ f (z) | λ ∈ R \ {0}} is investigated for each f ∈M. It is shown that, for each f ∈M, there is a critical parameter λ > 0 ...

Inequalities for Strongly Singular Convolution Operators

Singular Convolution Operators on the Heisenberg Group

1. Statement of results and outline of method. The purpose of this note is to announce results dealing with convolution operators on the Heisenberg group. As opposed to the well-known situation where the kernels are homogeneous and C°° away from the origin, the kernels we study are homogeneous but have singularities on a hyperplane. Convolution operators with such kernels arise in the study of ...