# Aicke HinrichsJohannes Kepler University Linz | JKU · Institute of Analysis

Aicke Hinrichs

Univ.-Prof. Dr.

## About

127

Publications

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Introduction

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October 2014 - present

April 2013 - September 2014

## Publications

Publications (127)

We prove the curse of dimensionality in the worst case setting for numerical integration for a number of classes of smooth d-variate functions. Roughly speaking, we consider different bounds for the directional or partial derivatives of f is an element of C-k(D-d) and ask whether the curse of dimensionality holds for the respective classes of funct...

We consider optimal importance sampling for approximating integrals I(f)=∫Df(x)ϱ(x)dx of functions f in a reproducing kernel Hilbert space H⊂L1(ϱ) where ϱ is a given probability density on D⊆Rd. We show that there exists another density ω such that the worst case error of importance sampling with density function ω is of order n−1/2.As a result, fo...

Abstract We show new lower bounds for the star-discrepancy and its inverse for subsets of the unit cube. They are polynomial in the quotient d=n of the number,n of sample points and the dimension d. They provide the best known,lower bounds for n not too large compared,with d.

We study integration and $L^2$-approximation of functions of infinitely many variables in the following setting: The underlying function space is the countably infinite tensor product of univariate Hermite spaces and the probability measure is the corresponding product of the standard normal distribution. The maximal domain of the functions from th...

In recent years finite tensor products of reproducing kernel Hilbert spaces (RKHSs) of Gaussian kernels on the one hand and of Hermite spaces on the other hand have been considered in tractability analysis of multivariate problems. In the present paper we study countably infinite tensor products for both types of spaces. We show that the incomplete...

Function values are, in some sense, “almost as good” as general linear information for L2-approximation (optimal recovery, data assimilation) of functions from a reproducing kernel Hilbert space. This was recently proved by new upper bounds on the sampling numbers under the assumption that the singular values of the embedding of this Hilbert space...

We study the circumradius of the intersection of an $m$-dimensional ellipsoid $\mathcal E$ with half axes $\sigma_1\geq\dots\geq \sigma_m$ with random subspaces of codimension $n$. We find that, under certain assumptions on $\sigma$, this random radius $\mathcal{R}_n=\mathcal{R}_n(\sigma)$ is of the same order as the minimal such radius $\sigma_{n+...

In recent years finite tensor products of reproducing kernel Hilbert spaces (RKHSs) of Gaussian kernels on the one hand and of Hermite spaces on the other hand have been considered in tractability analysis of multivariate problems. In the present paper we study countably infinite tensor products for both types of spaces. We show that the incomplete...

We study the circumradius of a random section of an $\ell_p$-ellipsoid, $0<p\le \infty$, and compare it with the minimal circumradius over all sections with subspaces of the same codimension. Our main result is an upper bound for random sections, which we prove using techniques from asymptotic geometric analysis if $1\leq p \leq \infty$ and compres...

Function values are, in some sense, "almost as good" as general linear information for $L_2$-approximation (optimal recovery, data assimilation) of functions from a reproducing kernel Hilbert space. This was recently proved by new upper bounds on the sampling numbers under the assumption that the singular values of the embedding of this Hilbert spa...

Let $0<p,q\leq \infty$ and denote by $\mathcal{S}_p^N$ and $\mathcal{S}_q^N$ the corresponding Schatten classes of real $N\times N$ matrices. We study the Gelfand numbers of natural identities $\mathcal{S}_p^N\hookrightarrow \mathcal{S}_q^N$ between Schatten classes and prove asymptotically sharp bounds up to constants only depending on $p$ and $q$...

High-dimensional systems are frequent in mathematics and applied sciences, and the understanding of high-dimensional phenomena has become increasingly important. The mathematical subdisciplines most strongly related to such phenomena are functional analysis, convex geometry, and probability theory. In fact, a new area emerged, called asymptotic geo...

We prove lower bounds for the worst case error of quadrature formulas that use given sample points Xn={x1,…,xn}. We are mainly interested in optimal point sets Xn, but also prove lower bounds that hold with high probability for sets of independently and uniformly distributed points. As a tool, we use a recent result (and extensions thereof) of Vybí...

Let $0<p,q\leq \infty$ and denote by $\mathcal{S}_p^N$ and $\mathcal{S}_q^N$ the corresponding Schatten classes of real $N\times N$ matrices. We study the Gelfand numbers of natural identities $\mathcal{S}_p^N\hookrightarrow \mathcal{S}_q^N$ between Schatten classes and prove asymptotically sharp bounds up to constants only depending on $p$ and $q$...

In this paper we study the extreme and the periodic $L_2$ discrepancy of plane point sets. The extreme discrepancy is based on arbitrary rectangles as test sets whereas the periodic discrepancy uses "periodic intervals", which can be seen as intervals on the torus. The periodic $L_2$ discrepancy is, up to a multiplicative factor, also known as diap...

We study approximation and integration problems and compare the
quality of optimal information with the quality of random information.
For some problems random information is almost optimal and for some
other problems random information is much worse than optimal information. We prove new results and give a short survey of
known results.

We prove lower bounds for the worst case error of quadrature formulas that use given sample points ${\mathcal X}_n = \{ x_1, \dots , x_n \}$. We are mainly interested in optimal point sets ${\mathcal X}_n$, but also prove lower bounds that hold for most randomly selected sets. As a tool, we use a recent result (and extensions thereof) of Vyb\'iral...

The dispersion of a point set in [0,1]d is the volume of the largest axis parallel box inside the unit cube that does not intersect with the point set. We study the expected dispersion with respect to a random set of n points determined by an i.i.d. sequence of uniformly distributed random variables. Depending on the number of points n and the dime...

We study the periodic L2-discrepancy of point sets in the d-dimensional torus. This discrepancy is intimately connected with the root-mean-square L2-discrepancy of shifted point sets, with the notion of diaphony, and with the worst-case error of cubature formulas for the integration of periodic functions in Sobolev spaces of mixed smoothness. In di...

We show that the minimal discrepancy of a point set in the d-dimensional unit cube with respect to Orlicz norms can exhibit both polynomial and weak tractability. In particular, we show that the ψ α-norms of exponential Orlicz spaces are polynomially tractable.

We study the size of the largest rectangle containing no point of a given point set in the two-dimensional torus, the dispersion of the point set. A known lower bound for the dispersion of any point set of cardinality n ≥ 2 in this setting is 2/n. We show that if n is a Fibonacci number then the Fibonacci lattice has dispersion exactly 2/n meeting...

We study the periodic $L_2$-discrepancy of point sets in the $d$-dimensional torus. This discrepancy is intimately connected with the root-mean-square $L_2$-discrepancy of shifted point sets, with the notion of diaphony, and with the worst case error of cubature formulas for the integration of periodic functions in Sobolev spaces of mixed smoothnes...

The dispersion of a point set in $[0,1]^d$ is the volume of the largest axis parallel box inside the unit cube that does not intersect with the point set. We study the expected dispersion with respect to a random set of $n$ points determined by an i.i.d.\ sequence of uniformly distributed random variables. Depending on the number of points $n$ and...

We show that the minimal discrepancy of a point set in the $d$-dimensional unit cube with respect to Orlicz norms can exhibit both polynomial and weak tractability. In particular, we show that the $\psi_\alpha$-norms of exponential Orlicz spaces are polynomially tractable.

The aim of the present article is to introduce a concept which allows to generalise the notion of Poissonian pair correlation, a second-order equidistribution property, to higher dimensions. Roughly speaking, in the one-dimensional setting, the pair correlation statistics measures the distribution of spacings between sequence elements in the unit i...

We study the size of the largest rectangle containing no point of a given point set in the two-dimensional torus, the dispersion of the point set. A known lower bound for the dispersion of any point set of cardinality $n\ge 2$ in this setting is $2/n$. We show that if $n$ is a Fibonacci number then the Fibonacci lattice has dispersion exactly $2/n$...

We study integration and L 2 -approximation on countable tensor products of function spaces of increasing smoothness. We obtain upper and lower bounds for the minimal errors, which are sharp in many cases including, e.g., Korobov, Walsh, Haar, and Sobolev spaces. For the proofs we derive embedding theorems between spaces of increasing smoothness an...

In this manuscript we introduce and study an extended version of the minimal dispersion of point sets, which has recently attracted considerable attention. Given a set $\mathscr P_n=\{x_1,\dots,x_n\}\subset [0,1]^d$ and $k\in\{0,1,\dots,n\}$, we define the $k$-dispersion to be the volume of the largest box amidst a point set containing at most $k$...

We study approximation and integration problems and compare the quality of optimal information with the quality of random information. For some problems random information is almost optimal and for some other problems random information is much worse than optimal information. We prove new results and give a short survey of known results.

The paper considers linear problems on weighted spaces of high-dimensional functions. The main questions addressed are: When is it possible to approximate the original function of very many variables by the same function; however with all but the first $k$ variables set to zero, so that the corresponding error is small? What is the truncation dimen...

We study the circumradius of the intersection of an $m$-dimensional ellipsoid $\mathcal E$ with half axes $\sigma_1\geq\dots\geq \sigma_m$ with random subspaces of codimension $n$. We find that, under certain assumptions on $\sigma$, this random radius $\mathcal{R}_n=\mathcal{R}_n(\sigma)$ is of the same order as the minimal such radius $\sigma_{n+...

We study integration and $L_2$-approximation on countable tensor products of function spaces of increasing smoothness. We obtain upper and lower bounds for the minimal errors, which are sharp in many cases including, e.g., Korobov, Walsh, Haar, and Sobolev spaces. For the proofs we derive embedding theorems between spaces of increasing smoothness a...

The aim of the present article is to introduce a concept which allows to generalise the notion of Poissonian pair correlation, a second-order equidistribution property, to higher dimensions. Roughly speaking, in the one-dimensional setting, the pair correlation statistics measures the distribution of spacings between sequence elements in the unit i...

We prove the curse of dimensionality in the worst case setting for multivariate numerical integration for various classes of smooth functions. We prove the results when the domains are isotropic convex bodies with small diameter satisfying a universal $\psi_2$-estimate. In particular, we obtain the result for the important class of volume-normalize...

In this manuscript we introduce and study an extended version of the minimal dispersion of point sets, which has recently attracted considerable attention. Given a set $\mathscr P_n=\{x_1,\dots,x_n\}\subset [0,1]^d$ and $k\in\{0,1,\dots,n\}$, we define the $k$-dispersion to be the volume of the largest box amidst a point set containing at most $k$...

In this manuscript we introduce and study an extended version of the minimal dispersion of point sets, which has recently attracted considerable attention. Given a set $\mathscr P_n=\{x_1,\dots,x_n\}\subset [0,1]^d$ and $k\in\{0,1,\dots,n\}$, we define the $k$-dispersion to be the volume of the largest box amidst a point set containing at most $k$...

In 2004 the second author of the present paper proved that a point set in [0, 1]d which has star-discrepancy at most ε must necessarily consist of at least cabs
dε
−1 points. Equivalently, every set of n points in [0, 1]d must have star-discrepancy at least cabs
dn
−1. The original proof of this result uses methods from Vapnik–Chervonenkis theory a...

We study the weighted star discrepancy of the Halton sequence. In particular, we show that the Halton sequence achieves strong polynomial tractability for the weighted star discrepancy for product weights $(\gamma_j)_{j \ge 1}$ under the mildest condition on the weight sequence known so far for explicitly constructive sequences. The condition requi...

Let $0<p,q \leq \infty$ and denote by $\mathcal S_p^N$ and $\mathcal S_q^N$ the corresponding finite-dimensional Schatten classes. We prove optimal bounds, up to constants only depending on $p$ and $q$, for the entropy numbers of natural embeddings between $\mathcal S_p^N$ and $\mathcal S_q^N$. This complements the known results in the classical se...

We prove estimates for the expected value of operator norms of Gaussian random matrices with independent (but not necessarily identically distributed) and centered entries, acting as operators from \(\ell_{p^{{\ast}}}^{n}\) to ℓ
q
m
, 1 ≤ p
∗ ≤ 2 ≤ q < ∞.

We mainly study numerical integration of real valued functions defined on the $d$-dimensional unit cube with all partial derivatives up to some finite order $r\ge1$ bounded by one. It is well known that optimal algorithms that use $n$ function values achieve the error rate $n^{-r/d}$, where the hidden constant depends on $r$ and $d$. Here we prove...

We consider -weighted anchored and ANOVA spaces of functions with mixed first order partial derivatives bounded in a weighted norm with . The domain of the functions is , where is a bounded or unbounded interval. We provide conditions on the weights that guarantee that anchored and ANOVA spaces are equal (as sets of functions) and have equivalent n...

In 2004 the second author of the present paper proved that a point set in $[0,1]^d$ which has star-discrepancy at most $\varepsilon$ must necessarily consist of at least $c_{abs} d \varepsilon^{-1}$ points. Equivalently, every set of $n$ points in $[0,1]^d$ must have star-discrepancy at least $c_{abs} d n^{-1}$. The original proof of this result us...

We investigate quasi-Monte Carlo rules for the numerical integration of
multivariate periodic functions from Besov spaces $S^r_{p,q}B(\mathbb{T}^d)$
with dominating mixed smoothness $1/p<r<2$. We show that order 2 digital nets
achieve the optimal rate of convergence $N^{-r} (\log N)^{(d-1)(1-1/q)}$. The
logarithmic term does not depend on $r$ and h...

We study embeddings and norm estimates for tensor products of weighted reproducing kernel Hilbert spaces. These results lead to a transfer principle that is directly applicable to tractability studies of multivariate problems as integration and approximation, and to their infinite-dimensional counterparts. In an application we consider weighted ten...

The discrepancy function of a point set in the $d$-dimensional unit cube is a normalized measure for the deviation of the proportion of the number of points of the point set in an axes-parallel box anchored at the origin and the volume of this box. Taking a norm of the discrepancy function gives a quantitative measure for the irregularity of distri...

We mainly study numerical integration of real valued functions defined on the d-dimensional unit cube with all partial derivatives up to some finite order r ≥ 1 bounded by one. It is well known that optimal algorithms that use n function values achieve the error rate n −r/d , where the hidden constant depends on r and d. Here we prove explicit erro...

We prove estimates for the expected value of operator norms of Gaussian random matrices with independent and mean-zero entries, acting as operators from $\ell^m_{p^*}$ to $\ell_q^n$, $1\leq p^* \leq 2 \leq q \leq \infty$.

The $L_p$-discrepancy is a quantitative measure for the irregularity of distribution modulo one of infinite sequences. In 1986 Proinov proved for all $p>1$ a lower bound for the $L_p$-discrepancy of general infinite sequences in the $d$-dimensional unit cube, but it remained an open question whether this lower bound is best possible in the order of...

We investigate
quasi-Monte
Carlo (QMC) integration of bivariate periodic functions with dominating mixed smoothness of order one. While there exist several QMC constructions which asymptotically yield the optimal rate of convergence of \(\mathscr {O}(N^{-1}\log (N)^{\frac{1}{2}})\), it is yet unknown which point set is optimal in the sense that it...

We prove that for any two quasi-Banach spaces $X$ and $Y$ and any $\alpha>0$
there exists a constant $c_\alpha>0$ such that $$ \sup_{1\le k\le
n}k^{\alpha}e_k(T)\le c_\alpha \sup_{1\le k\le n} k^\alpha c_k(T) $$ holds for
all linear and bounded operators $T:X\to Y$. Here $e_k(T)$ is the $k$-th
entropy number of $T$ and $c_k(T)$ is the $k$-th Gelfan...

We give an improved lower bound for the $L_2$-discrepancy of finite point
sets in the unit square.

The problem of finding the largest empty axis-parallel box amidst a point
configuration is a classical problem in computational complexity theory. It is
known that the volume of the largest empty box is of asymptotic order $1/n$ for
$n \to \infty$ and fixed dimension $d$. However, it is natural to assume that
the volume of the largest empty box inc...

We consider weighted anchored and ANOVA spaces of functions with first order
mixed derivatives bounded in $L_p$. Recently, Hefter, Ritter and Wasilkowski
established conditions on the weights in the cases $p=1$ and $p=\infty$ which
ensure equivalence of the corresponding norms uniformly in the dimension or
only polynomially dependent on the dimensi...

We prove sharp upper bounds on the entropy numbers $e_k(S^{d-1}_p,\ell_q^d)$
of the $p$-sphere in $\ell_q^d$ in the case $k \geq d$ and $0< p \leq q \leq
\infty$. In particular, we close a gap left open in recent work of the second
author, T. Ullrich and J. Vybiral. We also investigate generalizations to
spheres of general finite-dimensional quasi-...

We study properties of the so-called inner and outer successive radii of special families of convex bodies. First we consider the balls of the \$p\$-norms, for which we show that the precise value of the outer (inner) radii when \$p\geq 2\$ (\$1\leq p\leq 2\$), as well as bounds in the contrary case \$1\leq p\leq 2\$ (\$p\geq 2\$), can be obtained...

It is well known that the two-dimensional Hammersley point set consisting of
$N=2^n$ elements (also known as Roth net) does not have optimal order of
$L_p$-discrepancy for $p \in (1,\infty)$ in the sense of the lower bounds
according to Roth (for $p \in [2,\infty)$) and Schmidt (for $p \in (1,2)$). On
the other hand, it is also known that slight mo...

We investigate quasi-Monte Carlo (QMC) integration of bivariate periodic
functions with dominating mixed smoothness of order one. While there exist
several QMC constructions which asymptotically yield the optimal rate of
convergence of $\mathcal{O}(N^{-1}\log(N)^{\frac{1}{2}})$, it is yet unknown
which point set is optimal in the sense that it is a...

We consider the problem of integration of d-variate analytic functions
defined on the unit cube with directional derivatives of all orders bounded by
1. We prove that the Clenshaw Curtis Smolyak algorithm leads to weak
tractability of the problem. This seems to be the first positive tractability
result for the Smolyak algorithm for a normalized and...

In this survey paper we discuss some tools and methods which are of use in
quasi-Monte Carlo (QMC) theory. We group them in chapters on Numerical
Analysis, Harmonic Analysis, Algebra and Number Theory, and Probability Theory.
We do not provide a comprehensive survey of all tools, but focus on a few of
them, including reproducing and covariance kern...

We prove the curse of dimensionality in the worst case setting for numerical inte-gration for a number of classes of smooth d-variate functions. Roughly speaking, we consider different bounds for the directional or partial derivatives of f ∈ C k (D d) and ask whether the curse of dimensionality holds for the respective classes of functions. We alwa...

We study properties of the so called in-and outer successive radii of special families of convex bodies. First we consider the balls of the p-norms, for which we show that the precise value of the outer (inner) radii when p ≥ 2 (1 ≤ p ≤ 2), as well as bounds otherwise, can be obtained as consequences of known results on Gelfand and Kolmogorov numbe...

We study the complexity of Banach space valued integration in the randomized
setting. We are concerned with $r$-times continuously differentiable functions
on the $d$-dimensional unit cube $Q$, with values in a Banach space $X$, and
investigate the relation of the optimal convergence rate to the geometry of
$X$. It turns out that the $n$-th minimal...

The discrepancy function of a point distribution measures the deviation from the uniform distribution. Different versions of the discrepancy function capture this deviation with respect to different geometric objects. Via Koksma-Hlawka inequalities the norm of the discrepancy function in a function space is intimately connected to the worst case in...

We believe that discontinuous linear information is never more powerful than
continuous linear information for approximating continuous operators. We prove
such a result in the worst case setting. In the randomized setting we consider
compact linear operators defined between Hilbert spaces. In this case, the use
of discontinuous linear information...

We establish optimal estimates of Gelfand numbers or Gelfand widths of absolutely convex hulls cov(K) of precompact subsets \({K\subset H}\) of a Hilbert space H by the metric entropy of the set K where the covering numbers \({N(K, \varepsilon)}\) of K by \({\varepsilon}\) -balls of H satisfy the Lorentz condition$$ \int\limits_{0}^{\infty} \left(\...

In recent time much attention has been devoted to the study of entropy of convex hulls in Hilbert and Banach spaces and their applications in different branches of mathematics. In this paper we show how the rate of decay of the dyadic entropy numbers of a precompact set A of a Banach space X of type p, 1<p≤2, reflects the rate of decay of the dyadi...

We investigate the limit behavior of the average L-p-B-discrepancy for 0 < p < infinity if the number of sample points tends to infinity. We adopt a recent result of Steinerberger and give asymptotic expressions for several types of discrepancy functions studied in the literature. This also leads to a new proof for the average L-p-star discrepancy....

We prove a variant of a Johnson-Lindenstrauss lemma for matrices with circulant structure. This approach allows to minimise the randomness used, is easy to implement and provides good running times. The price to be paid is the higher dimension of the target space $k=O(\epsilon^{-2}\log^3n)$ instead of the classical bound $k=O(\epsilon^{-2}\log n)$.

We study the integration and approximation problems for monotone or convex bounded functions that depend on d variables, where d can be arbitrarily large. We consider the worst case error for algorithms that use finitely many function values. We prove that these problems suffer from the curse of dimensionality. That is, one needs exponentially many...

Denote by B
n
the unit ball in the Euclidean space \({\mathbb{R}^n}\) and define
$$ M(B^n) = \sup \int_{B^n} \int_{B^n}\| x - y \| \, d\mu(x)d\mu(y),$$where the supremum is taken over all finite signed Borel measures μ on B
n
of total mass 1. In this paper, the value of M(B
n
) is computed explicitly for all n, and it is shown that for n > 1 no mea...

We recall an open problem on the error of quadrature formulas for the integration of functions from some finite dimensional spaces of trigonometric functions posed by Novak (1999) in [8] ten years ago and summarised recently in Novak and Woźniakowski (2008) [9]. It is relatively easy to prove an error formula for the best quadrature rules with posi...

The L2-discrepancy measures the irregularity of the distribution of a finite point set. In this note, we prove lower bounds for the L2-discrepancy of arbitrary NN-point sets. Our main focus is on the two-dimensional case. Asymptotic upper and lower estimates of the L2-discrepancy in dimension 2 are well known, and are of the sharp order logN. Never...

Given a primitive positive integer vector a, the Frobenius number F(a) is the largest integer that cannot be represented as a non-negative integral combination of the coordinates of a. We show that for large instances the order of magnitude of the expected Frobenius number is (up to a constant depending only on the dimension) given by its lower bou...

A celebrated result of G. Pisier states that the notions of B-convexity and K-convexity coincide for Banach spaces. We complement this in the setting of linear and bounded operators between Banach spaces. Our approach is local and even yields inequalities between gradations of K-convexity norms and Walsh type norms of operators. Our method combines...

We study the discrepancy function of two-dimensional Hammersley type point sets in the unit square. It is well-known that the symmetrized Hammersley point set achieves the asymptotically best possible rate for the L2-norm of the discrepancy function. In this paper we consider the norm of the discrepancy function of Ham¬mersley type point sets in Be...

The by now classical theory of p -nuclear operators with 0 < p ≤ 1 was founded in Grothendieck's thesis (1953). Since that time only little progress has been achieved. This article describes the present state of the art. We improve Grothendieck's multiplication theorem, characterize the p -integral operators, and give a long list of challenging ope...

Let a normed space X possess a tiling T consisting of unit balls. We show that any packing P of X obtained by a small perturbation of T is completely translatively saturated; that is, one cannot replace finitely many elements of P by a larger number of unit balls such that the resulting arrangement is still a packing.In contrast with that, given a...

Let s = (sn) be an injective s-number sequence in the sense of Pietsch. We show the following Weyl inequality between geometric means of eigenvalues and
s-numbers for a Riesz-operator T: X → X acting on a (complex) Banach space of weak type 2: for any 0 < δ ≤ 1 and all n ∈ ℕ, we have , where wC2(X) is the weak cotype 2 constant of X, nδ ≔ [n/(1+δ)]...

We study algorithms for the approximation of functions, the error is measured in an L 2 norm. We consider the worst-case setting for a general reproducing kernel Hilbert space of functions. We analyze algorithms that use standard information consisting in n function values and we are interested in the optimal order of convergence. This is the maxim...

Tractability properties of various notions of discrepancy have been intensively studied in the last decade. In this paper we consider the so-called weighted star discrepancy which was introduced by Sloan and Woźniakowski. We show that under a very mild condition on the weights one can obtain tractability with s-exponent zero (s is the dimension of...

The paper investigates the asymptotic behaviour of entropy and approximation numbers of compact embeddings between weighted modulation spaces.

Let (sn) be an s-number sequence. We show for each k = 1, 2, . . . and n ≥ k + 1 the inequality between the eigenvalues and s-numbers of a compact operator T in a Banach space. Furthermore, the constant (k + 1)1/2 is optimal for n = k + 1 and k = 1, 2, . . .. This inequality seems to be an appropriate tool for estimating the first single eigenvalue...

It is shown that a Banach space X has Fourier type p with respect to a locally compact abelian group G if and only if the dual space X′ has Fourier type p with respect to G if and only if X has Fourier type p with respect to the dual group of G. This extends previously known results for the classical groups and the Cantor group to the setting of ge...

Complementing and generalizing classical as well as recent results, we prove asymptotically optimal formulas for the Gelfand
and approximation numbers of identities En ↪ Fn, where En and Fn denote the n-th sections of symmetric quasi-Banach
sequence spaces E and F satisfying certain interpolation assumptions. We illustrate our
results by considerin...

Dedicated to Professor Albrecht Pietsch on the occasion of his 70th birthday. Abstract. A well-known multiplicative Weyl inequality states that the se- quence of eigenvalues (¸k(T)) and the sequence of approximation numbers (ak(T)) of any compact operator T in a Banach space satisfy n Y k=1 |¸k(T)| · n n/2 n Y k=1 ak(T) for all n. We prove here tha...