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We provide a new upper bound for sampling numbers [Formula omitted] associated with the compact embedding of a separable reproducing kernel Hilbert space into the space of square integrable functions. There are universal constants [Formula omitted] (which are specified in the paper) such that gn2[less than or equal to]Clog(n)n[summation]k[greater than or equal to]âcn[right floor][sigma]k2,n[greater than or equal to]2,where [Formula omitted] is the sequence of singular numbers (approximation numbers) of the Hilbert-Schmidt embedding [Formula omitted]. The algorithm which realizes the bound is a least squares algorithm based on a specific set of sampling nodes. These are constructed out of a random draw in combination with a down-sampling procedure coming from the celebrated proof of Weaver's conjecture, which was shown to be equivalent to the Kadison-Singer problem. Our result is non-constructive since we only show the existence of a linear sampling operator realizing the above bound. The general result can for instance be applied to the well-known situation of [Formula omitted] in [Formula omitted] with [Formula omitted]. We obtain the asymptotic bound gn[less than or equal to]Cs,dn-slog(n)(d-1)s+1/2,which improves on very recent results by shortening the gap between upper and lower bound to [Formula omitted]. The result implies that for dimensions [Formula omitted] any sparse grid sampling recovery method does not perform asymptotically optimal., Author(s): Nicolas Nagel [sup.1], Martin Schäfer [sup.1], Tino Ullrich [sup.1] Author Affiliations: (1) grid.6810.f, 0000 0001 2294 5505, Faculty of Mathematics, TU Chemnitz, , 09107, Chemnitz, Germany Introduction > In [...]

The karyon-module (KM-) algorithm for expanding an algebraic number [alpha] = ([[alpha].sub.1], ..., [[alpha].sub.d]) from [R.sup.d] in a multidimensional continued fraction, i.e., in a sequence of rational numbers [P.sub.a]/[Q.sub.a] = ([P.sub.a.sub.1]/[Q.sup.a], ..., [P.sub.a.sub.d]/[Q.sup.a]) a = 1,2,3 ..., from [Q.sup.d] with numerators [P.sub.a.sub.1], ..., [P.sub.a.sub.d] [member of] Z and a common denominator [Q.sup.a] = 1, 2, 3, ... is proposed. The KM-algorithm belongs to the class of tuned algorithms and is based on constructing localized Pisot units [zeta] > 1, for which the moduli of all the conjugates [[zeta].sup.(i)] [not equal to] [zeta] are contained in the [theta]-neighborhood of the number [[zeta].sup.-1/d], where the parameter [theta] may take an arbitrary fixed value. It is proved that given a real algebraic point a of degree deg([alpha]) = d + 1, the KM-algorithm provides its approximation such that [absolute value of ([alpha] - [P.sub.a]/[Q.sub.a])] [less than or equal to] c/[Q.sup.1+1/d-[theta].sub.a] for all a [greater than or equal to] [a.sub.[alpha],[theta]], where the constants [a.sub.[alpha],[theta]] > 0 and c = [c.sub.[alpha],[theta]] > 0 are independent of a = 1, 2, 3, ..., and the convergents [P.sub.a]/[Q.sub.a] are computed from a certain recurrence relation with constant coefficients, determined by the choice of the localized unit [zeta]. Bibliography: 19 titles., UDC 511.3 INTRODUCTION 0.1. The karyon-module algorithm. The paper suggests the karyon-module (KM-) algorithm KM : [P.sub.a]/[Q.sub.a] [right arrow] [alpha] as a [right arrow] +[infinity] (0.1) for expanding an algebraic [...]

The karyon-module (KM-) algorithm for expanding an algebraic number [alpha] = ([[alpha].sub.1], ..., [[alpha].sub.d]) from [R.sup.d] in a multidimensional continued fraction, i.e., in a sequence of rational numbers [mathematical expression not reproducible] from [Q.sup.d] with numerators [P.sup.a.sub.1], ..., [P.sup.a.sub.d] [member of] Z and a common denominator [Q.sup.a] = 1, 2, 3, ... is proposed. The KM-algorithm belongs to the class of tuned algorithms and is based on constructing localized Pisot units [zeta] > 1, for which the moduli of all the conjugates [[zeta].sup.(i) [not equal to] [zeta] are contained in the [theta]-neighborhood of the number [[zeta].sup.-1/d], where the parameter [theta] may take an arbitrary fixed value. It is proved that given a real algebraic point [alpha] of degree deg([alpha]) = d + 1, the KM-algorithm provides its approximation such that [mathematical expression not reproducible] for all a [greater than or equal to] [a.sub.[alpha],[theta]], where the constants [a.sub.[alpha],[theta]] > 0 and c = [c.sub.[alpha],[theta]] > 0 are independent of a = 1, 2, 3, ..., and the convergents [[P.sub.a]/[Q.sub.a]] are computed from a certain recurrence relation with constant coefficients, determined by the choice of the localized unit [zeta]. Bibliography: 19 titles., UDC 511.3 INTRODUCTION 0.1. The karyon-module algorithm. The paper suggests the karyon-module (KM-) algorithm KM : [[P.sub.a]/[Q.sub.a] [right arrow] [alpha] as a [right arrow] + [infinity] (0.1) for expanding an [...]

A development of the simplex-module algorithm for expansion of algebraic numbers in multidimensional continued fractions is proposed. To this end, localized Pisot matrices are constructed, whose eigenvalues with moduli less than one are contained in an interval of small length. Such Pisot matrices generate continued fractions whose convergents are arbitrarily close to the best approximations. Bibliography: 16 titles., UDC 511 INTRODUCTION 0.1. The simplex-module algorithm. The simplex-module (SM-) algorithm for expansion of algebraic numbers [[alpha].sub.1], ..., [[alpha].sub.d] in multidimensional continued fractions SM: [P.sub.a]/[Q.sub.a] = ([P.sub.a,1]/[Q.sub.a], ..., [P.sub.a,d]/[Q.sub.a]) [right [...]

The paper establishes the invariance of the simplex-module algorithm for expanding real numbers [alpha] = ([[alpha].sub.1], ..., [[alpha].sub.d]) in multidimensional continued fractions under linear-fractional Transformations [alpha] = ([[alpha]'.sub.1], ..., [[alpha]'.sub.d]) = U with matrices U from the unimodular group [GL.sub.d+1] (Z). It is shown that the convergents of the transformed collections of numbers [alpha]' satisfy the same recurrence relation and have the same approximation order. Bibliography: 20 titles., UDC 511 INTRODUCTION 0.1. The simplex-module algorithm. The simplex-module algorithm (SM-algorithm) for expansion of algebraic numbers [[alpha].sub.1], ..., [[alpha].sub.d]] in multidimensional continued fractions, SM : t [P.sub.a]/[Q.sub.a] = ([P.sub.a,1]/[Q.sub.a], ..., [...]

The simplex-module algorithm for expansion of algebraic numbers a = ([[alpha].sub.1], ..., [[alpha].sub.d]) in multidimensional continued fractions is suggested. The method is based on minimal rational simplices s, where [alpha] [member of] s, and Pisot matrices [P.sub.[alpha]], for which [??] = ([[alpha].sub.1], ..., [[alpha].sub.d], 1) is an eigenvector. A multidimensional generalization of the Lagrange theorem is proved. Bibliography: 19 titles., UDC 511 Introduction 0.1. Complete points. A point [alpha] = ([[alpha].sub.1], ..., [[alpha].sub.d]) and the corresponding set {[[alpha].sub.1], ..., [[alpha].sub.d]} of real algebraic numbers are said to be complete if [...]