A lethargy result for real analytic functions

In this short note we prove that, if (C[a,b],{A_n}) is an approximation scheme and (A_n) satisfies de La Vall\'ee-Poussin Theorem, there are instances of continuous functions on [a,b], real analytic on (a,b], which are poorly approximable by the elements of the approximation scheme (A_n). This illustrates the thesis that the smoothness conditions guaranteeing that a function is well approximable must be, at least in these cases, global. The failure of smoothness at endpoints may result in an arbitrarily slow rate of approximation. A result of this kind, which is highly nonconstructive, based on different arguments, and applicable to different approximation schemes, was recently proved by Almira and Oikhberg (see arXiv:1009.5535v2).Comment: 4 pages, Submitted to a Journa

Persistence barcodes and Laplace eigenfunctions on surfaces

We obtain restrictions on the persistence barcodes of Laplace-Beltrami eigenfunctions and their linear combinations on compact surfaces with Riemannian metrics. Some applications to uniform approximation by linear combinations of Laplace eigenfunctions are also discussed.Comment: Revised version; some references adde

Negative Theorems in Approximation Theory

this paper best approximation operators are nonlinear. Linear methods of approximation are both preferable and simpler, and much effort has gone into their study. As it happens, however, linear processes often have intrinsic limitations. For example, it is quite natural to try to use interpolation to obtain good approximations. Let us assume that we are given a fixed triangular array of points with n 1 points in b] in the nth row of the array, and let p n ( f ) denote the (unique) polynomial in # n that interpolates f in C[a, b] at the points of this nth row. It was Faber who showed in a 1914 paper [7] that for every such array there always exists an f in C[a, b] for which p n ( f )