Journal of Approximation Theory最新文献

Recently, several authors have considered the problem of determining optimal norm inequalities for discrete Hardy-type operators (like Cesàro or Copson). In this work, we obtain sharp bounds for the norms of the difference of the Cesàro operator with either the identity or the shift, when they are restricted to the cone of decreasing sequences in ${\ell }^p$ (which is closely related to the previously mentioned estimates). Finally, we also address the case of weighted inequalities and find an interesting contrast between the norms of these two difference operators.

This paper describes an approximation theoretic approach to the problem of completeness of a set of translates of a “Tauberian generator”, which is an integrable function whose Fourier transform does not vanish. This is achieved by the construction of finite rank operators, whose range is contained in the linear span of the translates of such a generator, and which allow uniform approximation of the identity operator over compact sets of certain Banach spaces $(B,{\Vert \cdot \Vert }_B)$. The key assumption is availability of a double module structure on $(B,{\Vert \cdot \Vert }_B)$, meaning the availability of sufficiently many smoothing operators (via convolution) and also pointwise multipliers, allowing localization of its elements. This structure is shared by a wide variety of function spaces and allows us to make explicit use of the Riesz–Kolmogorov Theorem characterizing compact subsets in such Banach spaces.

The construction of these operators is universal with respect to large families of such Banach spaces, i.e. they do not depend on any further information concerning the particular Banach space. As a corollary we conclude that the linear span of the set of the translates of such a Tauberian generator is dense in any such space $(B,{\Vert \cdot \Vert }_B)$. Our work has been inspired by a completeness result of V. Katsnelson which was formulated in the context of specific Hilbert spaces within this family and Gaussian generators.

The paper is concerned with the interconnection of the boundary behaviour of the solutions of the exterior Dirichlet, Neumann and Robin problems of harmonic analysis for the unit ball in $R^3$ with the corresponding behaviour of the associated ergodic inverse problems for the entire space. Rates of approximation play a basic role.

The solutions themselves are evaluated by means of Fourier expansions with respect to spherical harmonics. In case of the first two problems, the basis for the investigation of the approximation and ergodic behaviour is the theory of semigroups of linear operators mapping a Banach space X into itself. The connection between the semigroup property and the major premise of Huygens’ principle is emphasized.

Another tool is a Drazin-like inverse operator $A^{\mathrm{ad}}$ for the infinitesimal generator $A$ of a semigroup that arises naturally in ergodic theory. This operator is a closed, not necessarily bounded, operator. It was introduced in a paper with U. Westphal (Butzer and Westphal, 1970/71) and extended to a generalized setting with J.J. Koliha (Butzer and Koliha, 2009).

Unlike the latter two problems, the solution of Robin’s problem does not have the semigroup property and therefore the semigroup methods applied to Dirichlet’s and Neumann’s problem do not work. The authors give several hints how to overcome these difficulties.

We introduce a certain variant (or regularization) ${\tilde{\Lambda }}_n^{\mu }$ of the standard Christoffel function ${\Lambda }_n^{\mu }$ associated with a measure $\mu$ on a compact set $\Omega \subset R^d$. Its reciprocal is now a sum-of-squares polynomial in the variables $(x,\varepsilon )$, $\varepsilon >0$. It shares the same dichotomy property of the standard Christoffel function, that is, the growth with $n$ of its inverse is at most polynomial inside and exponential outside the support of the measure. Its distinguishing and crucial feature states that for fixed $\varepsilon >0$, and under weak assumptions, ${lim}_{n\to \infty }{\varepsilon }^{-d}{\tilde{\Lambda }}_n^{\mu }(\xi ,\varepsilon )=f\left({\zeta }_{\varepsilon }\right)$ where $f$ (assumed to be continuous) is the unknown density of $\mu$ w.r.t. Lebesgue measure on $\Omega$, and ${\zeta }_{\varepsilon }\in B_{\infty }(\xi ,\varepsilon )$ (and so $f\left({\zeta }_{\varepsilon }\right)\approx f\left(\xi \right)$ when $\varepsilon >0$ is small). This is in contrast with the standard Christoffel function where if ${lim}_{n\to \infty }{n}^d{\Lambda }_n^{\mu }\left(\xi \right)$ exists, it is of the form