Uber einen Satz von van der Corput

Ohne Zusammenfassung     

Eine mehrdimensionale Version der Ungleichung von le Veque

In this short note we give a proof of le Veque's inequality $$D_N \leqq (6/\pi ^2 )\sum\limits_{h = 1}^\infty {h - 2\left| {(1/N)\sum\limits_{k = 1}^N {e^{2\pi ihx_k } |2)1/3} } \right|} $$ for dimensions≧2.     

Eine Ungleichung der Integralrechnung

Ohne Zusammenfassung     

Eine Verallgemeinerung der Ungleichung vom arithmetischen und geometrischen Mittel

Ohne ZusammenfassungDiese Arbeit wurde durch ein Stipendium der Max-Kade-Stiftung gefördert.     

Eine Ungleichung der Potentialtheorie und ihre Anwendung in der Theorie der meromorphen Funktionen

Ohne Zusammenfassung     

Eine Bemerkung zu Verallgemeinerungen der Ungleichung vom arithmetischen und geometrischen Mittel

In the literature there are known homogeneous polynomialsP(x ₁,...,x _n) with real coefficients, for which \(P(x_1 ,...,x_n ) \leqslant P(\bar x,...,\bar x)\) for allx _i≥0, and \(\bar x = (x_1 + ... + x_n )/n\) . This paper gives two theorems, which lead to new polynomials of this kind.     

Ergodic characterization of van der Corput sets

We prove an analogue to the well-known equivalence of intersective sets and Poincaré recurrent sets, in a stronger setting. We show that a set D is van der Corput, if and only if for each Hilbert space H, unitary operator U, and ${x \in H}$ such that the projection of x to the kernel of (U ? I) is nonvanishing, there exists ${d \in D\,}$ such that (U ^d x, x)≠ 0. We also characterize the smallest such d.     

Multidimensional van der Corput and sublevel set estimates

Van der Corput's lemma gives an upper bound for one-dimensional oscillatory integrals that depends only on a lower bound for some derivative of the phase, not on any upper bound of any sort. We establish generalizations to higher dimensions, in which the only hypothesis is that a partial derivative of the phase is assumed bounded below by a positive constant. Analogous upper bounds for measures of sublevel sets are also obtained. The analysis, particularly for the sublevel set estimates, has a more combinatorial flavour than in the one-dimensional case.

     

Sharp van der Corput estimates and minimal divided differences

We find the sharp constant in a sublevel set estimate which arises in connection with van der Corput's lemma. In order to do this, we find the nodes that minimise divided differences. We go on to find the sharp constant in the first instance of the van der Corput lemma. With these bounds we improve the constant in the general van der Corput lemma, so that it is asymptotically sharp.

     

Pointwise van der Corput lemma for functions of several variables

A multidimensional version of the well-known van der Corput lemma is presented. A class of phase functions is described for which the corresponding oscillatory integrals satisfy a multidimensional decay estimate. The obtained estimates are uniform with respect to parameters on which the phases and amplitudes may depend.     

Distribution functions for subsequences of the van der Corput sequence

For an integer b>1$b>1$ let _(?b(n))n≥0${\left({?}_b\left(n\right)\right)}_{n\ge 0}$ denote the base b$b$ van der Corput sequence in [0,1)$[0,1)$. Answering a question of O. Strauch, C. Aisleitner and M. Hofer showed that the distribution function of _{(?b(n),?b(n+1),…,?b(n+s−1))n≥0}${({?}_b\left(n\right),{?}_b(n+1),\dots ,{?}_b(n+s-1))}_{n\ge 0}$ on [0,1)^s${[0,1)}^s$ exists and is a copula. In this note we show that this phenomenon extends to a broad class of subsequences of the van der Corput sequences.     

Regularities of the Distribution of β-adic van der Corput Sequences

For Pisot numbers β with irreducible β-polynomial, we prove that the discrepancy function D(N, [0,y)) of the β-adic van der Corput sequence is bounded if and only if the β-expansion of y is finite or its tail is the same as that of the expansion of 1. If β is a Parry number, then we can show that the discrepancy function is unbounded for all intervals of length y ? \Bbb Q(b) y \notin {\Bbb Q}(\beta) . We give explicit formulae for the discrepancy function in terms of lengths of iterates of a reverse β-substitution.     

Zur isodiametrischen und isoperimentrischen Ungleichung in der Relativgeometrie

Commentarii Mathematici Helvetici -     

Eine Funktionalgleichung von Abel und die Grundgleichung der Information

Ohne Zusammenfassung     

On the distribution of the van der Corput sequence in arbitrary base

A central limit theorem with explicit error bound, and a large deviation result are proved for a sequence of weakly dependent random variables of a special form. As a corollary, under certain conditions on the function \(f:[0,1] \rightarrow \mathbb {R}\) a central limit theorem and a large deviation result are obtained for the sum \(\sum _{n=0}^{N-1} f(x_n)\), where \(x_n\) is the base b van der Corput sequence for an arbitrary integer \(b \ge 2\). Similar results are also proved for the \(L^p\) discrepancy of the same sequence for \(1 \le p < \infty \). The main methods used in the proofs are the Berry–Esseen theorem and Fourier analysis.     

Precise distribution properties of the van der Corput sequence and related sequences

The discrepancy is a quantitative measure for the irregularity of distribution of sequences in the unit interval. This article is devoted to the precise study of L_p–discrepancies of a special class of digital (0,1)–sequences containing especially the van der Corput sequence. We show that within this special class of digital (0,1)–sequences over ℤ₂ the van der Corput sequence is the worst distributed sequence with respect to L₂–discrepancy. Further we prove that the L_p–discrepancies of the van der Corput sequence satisfy a central limit theorem and we study the discrepancy function of (0,1)–sequences pointwise.     

Eine Erweiterung der Streifentheorie und Verallgemeinerung von Cesàrostreifen

Ohne Zusammenfassung