An operational matrix method based on generalized Bernoulli polynomials of level m is introduced and analyzed in order to obtain numerical solutions of initial value problems. The most innovative component of our method comes, essentially, from the introduction of the generalized Bernoulli polynomials of level m, which generalize the classical Bernoulli polynomials. Computational results demonstrate that such operational matrix method can lead to very ill-conditioned matrix equations.

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We prove the interior approximate controllability of the following broad class of reaction diffusion equation in the Hilbert spacesZ=L2(Ω)given byz′=−Az+1ωu(t),t∈[0,τ], whereΩis a domain inℝn,ωis an open nonempty subset ofΩ,1ωdenotes the characteristic function of the setω, the distributed controlu∈L2(0,t1;L2(Ω))andA:D(A)⊂Z→Zis an unbounded linear operator with the following spectral decomposition:Az=∑j=1∞λj∑k=1γj〈z,ϕj,k〉ϕj,k. The eigenvalues0<λ1<λ2<⋯<⋯λn→∞ofAhave finite multiplicityγjequal to the dimension of the corresponding eigenspace, and{ϕj,k}is a complete orthonormal set of eigenvectors ofA. The operator−Agenerates a strongly continuous semigroup{T(t)}given byT(t)z=∑j=1∞e−λjt∑k=1γj〈z,ϕj,k〉ϕj,k. Our result can be applied to thenD heat equation, the Ornstein-Uhlenbeck equation, the Laguerre equation, and the Jacobi equation.

In this paper we are going to study the zero location and asymptotic behavior of extremal polynomials with respect to a non-diagonal Sobolev norm in the worst case, i.e., when the quadratic form is allowed to degenerate. The orthogonal polynomials with respect to this Sobolev norm are a particular case of those extremal polynomials. The multiplication operator by the independent variable is the main tool in order to obtain our results.

We characterize the set of functions which can be approximated by continuous functions in the L N norm with respect to almost every weight. This allows to characterize the set of functions which can be approximated by polynomials or by smooth functions for a wide range of weights. r