Papers by Keyword: Resolvent

Abstract: To perform the Hilbert transform H_il of a non-integrable function φ, such as φ(x) = 1, x, in a numerical calculation-friendly way, we propose a method of rewriting H_il in terms of the resolvent for a differential operator R whose eigenfunctions satisfy the orthogonality and the completeness, so that the resolvent kernel 〈x|R^-1y〉can be given by the eigenfunction expansion. We deal with two cases for the choice of R: one is the harmonic oscillator Hamiltonian, which is commutative with the Fourier transform F; and the other is such that is commutative with H_il itself. We show how the calculation of H_ilφ is made in a numerical calculation-friendly way, to find that Π_k=0,1 H_ilf_k (f_k (x) = x^k) satisfies quite a simple relation.

Abstract: In terms of cogenerator and rsolvent of cosine functions,the contractivity and boundedness of cosine functions were characterized.

Abstract: Let E be a real q-uniformly smooth and uniformly convex Banach space and K a nonempty closed convex subset of E. Let Ti : K ! K, i = 1; 2; : : : ;N be ki-strictly asymptotically pseudocon- tractive mappings with \N i=1F (Ti) 6= ;, where F(Ti) = fx 2 K : Tix = xg. Let fxng be the sequence generated by xn+1 = (1 ¡ ®n)xn + ®nTn [n]xn; where f®ng is a sequence in [0,1] satisfying certain conditions and Tn [n] = Ti n; i = n(modN). Weak and strong convergence theorems for the iterative approximation of common ¯xed points of the family fTigN i=1 are proved.