An error analysis is given for convolution quadratures based on strongly A-stable RungeKutta methods, for the non-sectorial case of a convolution kernel with a Laplace transform that is polynomially bounded in a half-plane. The order of approximation depends on the classical order and stage order of the Runge-Kutta method and on the growth exponent of the Laplace transform. Numerical experiment...

In this paper, we introduce a new class of p-valent harmonic functions defined by an integral operator. Coefficient inequalities, distortion bounds for the functions belonging to this class are obtained. The invariance of the class Hp(n, α) under convolution, convex linear combination and also under generalized Bernardi-Libera Livingston integral operator are discussed. 2000 Mathematics Subject...

The spherical functions of the noncompact Grassmann manifolds Gp,q(F) = G/K over the (skew-)fields F = R,C,H with rank q ≥ 1 and dimension parameter p > q can be described as Heckman-Opdam hypergeometric functions of type BC, where the double coset space G//K is identified with the Weyl chamber C q ⊂ R q of type B. The corresponding product formulas and Harish-Chandra integral representations w...

In this paper, we will present a generalized convolution quadrature for solving linear parabolic and hyperbolic evolution equations. The original convolution quadrature method by Lubich works very nicely for equidistant time steps while the generalization of the method and its analysis to non-uniform time stepping is by no means obvious. We will introduce the generalized convolution quadrature ...

We study the approximation properties of Runge-Kutta time discretizations of linear and semilinear parabolic equations, including incompressible Navier-Stokes equations. We derive asymptotically sharp error bounds and relate the temporal order of convergence, which is generally noninteger, to spatial regularity and the type of boundary conditions. The analysis relies on an interpretation of Run...

In this session we want to review the basic concepts introduced in class for convolution in continuous-time. It has been proved that in order to determine the characteristics of a linear, time-invariant (LTI) system we need to know the impulse response of the system. When we apply an input x(t) to a system that has h(t) as its impulse response, we obtain an output y(t) (see Figure 1). One of th...

Let A(p) be the class of functions f : f(z) = z + ∑∞ j=1 ajz p+j analytic in the open unit disc E. Let, for any integer n > −p, fn+p−1(z) = z p (1−z)n+p . We define f (−1) n+p−1(z) by using convolution ? as fn+p−1(z) ? f (−1) n+p−1(z) = z (1−z)n+p . A function p, analytic in E with p(0) = 1, is in the class Pk(ρ) if ∫ 2π 0 ∣∣∣Rep(z)−ρ p−ρ ∣∣∣ dθ ≤ kπ, where z = re, k ≥ 2 and 0 ≤ ρ < p. We use t...

The purpose of this paper is to obtain sufficient bound estimates for harmonic functions belonging to the classes S∗ H [A,B], KH [A,B] defined by subordination, and we give some convolution conditions. Finally, we examine the closure properties of the operator D on these classes under the generalized Bernardi integral operator.

Conditions for boundedness and compactness of product-convolution operators g —» PhCß = h ■ (/» g) on spaces L^G) are studied. It is necessary for boundedness to define a class of "mixed-norm" spaces L,p>q){G) interpolating the Lp(G) spaces in a natural way (L^^ = Z^,). It is then natural to study the operators acting between L(/1?)(G) spaces, where G has a compact invariant neighborhood. The t...