Abstract. We construct higher order spectral shift functions, which represent the remainders of Taylor-type approximations for the value of a function at a perturbed self-adjoint operator by derivatives of the function at an initial unbounded operator. In the particular cases of the zero and the first order approximations, the corresponding spectral shift functions have been constructed by M. G...

Taylor’s polynomial is a central tool in any elementary course in mathematical analysis. Nowadays, its importance is centred on its applications, for instance, to asymptotic analysis or to obtain satisfactory numerical or integral inequalities see, e.g., 1–5 . The core of these results comes from manipulations on the explicit formula of the remainder, that is, the error estimation when consider...

We consider a family of two-point quadrature formulae and establish sharp estimates for the remainders under various regularity conditions. Improved forms of certain integral inequalities due to Hermite–Hadamard, Iyengar, Milovanović–Pec̆arić, and others are obtained as special cases. Our results may also be interpreted as analogues to a theorem of Ostrowski on the deviation of a functions from ...

We study the diffusive expansion for solutions around Maxwellian equilibrium and in a periodic box to the Vlasov-Maxwell-Boltzmann system, the most fundamental model for an ensemble of charged particles. Such an expansion yields a set of dissipative new macroscopic PDE’s, the incompressible Vlasov-Navier-Stokes-Fourier system and its higher order corrections for describing a charged fluid, wher...

We present new algorithms using structured matrix methods for manipulating polynomi-als expressed in generalized form, that is, relative to a polynomial basis fp i (x)g satisfying a three-term recurrence relation. This includes computing the Euclidean remainders as well as the greatest common divisor of two polynomials u(x) and v(x) of degrees less than n. The computational schemes involve solv...