A van der Corput lemma for the -adic numbers

We prove a version of van der Corput's lemma for polynomials over the -adic numbers.

     

Kloosterman double sums

In this paper estimates of incomplete Kloosterman double sums with weights are obtained. Translated fromMatematicheskie Zametki, Vol. 66, No. 5, pp. 682–687, November, 1999.     

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.

     

On the gauss trigonometric sum

In this paper it is proved that the arguments of the Gauss sum associated with sixth powers do not constitute an everywhere dense set. A lower bound to the corresponding trigonometric sums is also obtained. Translated fromMatematicheskie Zametki, Vol. 68, No. 2, pp. 173–178, August, 2000.     

Estimates from above of certain double trigonometric sums

In this paper we consider double trigonometric sums. Expressions of this type appear in some problems of quantum chaos and number theory. We are interested in rotation numbers of bounded type. We prove a uniform linear bound on double trigonometric sums along the subsequence of denominators of the continued fraction. The proof uses elementary techniques and the analysis of cancellations in sums of certain oscillatory functions over rotations. We also include a proof of a result on discrepancy for rotations of bounded type and in the Appendix we give an elementary proof of a result by Hardy and Littlewood.     

关于Van der Corput不等式

本文给出Van der Corput一个不等式的改进:设 ,则我们有: 其中γ为Euler常数.若我们以(n+1)代替(A)中右边 ,则(A)为Van der Corput不等式,(A)的证明仍然用Van der Corput的方法。     

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.     

Lp- and Sp,qrB-discrepancy of the symmetrized van der Corput sequence and modified Hammersley point sets in arbitrary bases

We study the local discrepancy of a symmetrized version of the well-known van der Corput sequence and of modified two-dimensional Hammersley point sets in arbitrary base $b$. We give upper bounds on the norm of the local discrepancy in Besov spaces of dominating mixed smoothness $S_{p,q}^{r}B\left({[0,1)}^s\right)$, which will also give us bounds on the $L_p$-discrepancy. Our sequence and point sets will achieve the known optimal order for the $L_p$- and $S_{p,q}^{r}B$-discrepancy. The results in this paper generalize several previous results on $L_p$- and $S_{p,q}^{r}B$-discrepancy estimates and provide a sharp upper bound on the $S_{p,q}^{r}B$-discrepancy of one-dimensional sequences for $r>0$. We will use the $b$-adic Haar function system in the proofs.     

一类离散变量函数展开的方法

给出一类离散变量函数展开的方法,给出对Van der Corput不等式的一个改进;将这个方法扩展后可以应用于更多的离散变量函数的展开与研究,例如对Stirling公式的改进.     

Deterministic sampling of sparse trigonometric polynomials

One can recover sparse multivariate trigonometric polynomials from a few randomly taken samples with high probability (as shown by Kunis and Rauhut). We give a deterministic sampling of multivariate trigonometric polynomials inspired by Weil’s exponential sum. Our sampling can produce a deterministic matrix satisfying the statistical restricted isometry property, and also nearly optimal Grassmannian frames. We show that one can exactly reconstruct every M-sparse multivariate trigonometric polynomial with fixed degree and of length D from the determinant sampling X, using the orthogonal matching pursuit, and with |X| a prime number greater than (MlogD)². This result is optimal within the (logD)² factor. The simulations show that the deterministic sampling can offer reconstruction performance similar to the random sampling.     

Estimates of complete rational trigonometric sums with a prime denominator

Upper bounds are proved for special complete rational exponential sums with a prime denominator.     

Positivity and boundedness of trigonometric sums

We give a systematic account of results which assure positivity and boundedness of partial sums of cosine or sine series. New proofs of recent results are sketched.     

Error estimates for some quadrature rules with maximal trigonometric degree of exactness

In this paper, we give error estimates for quadrature rules with maximal trigonometric degree of exactness with respect to an even weight function on ( ? π,π) for integrand analytic in a certain domain of complex plane. Copyright © 2013 John Wiley & Sons, Ltd.     

Fibonacci, Van der Corput and Riesz-Nágy

What do the three names in the title have in common? The purpose of this paper is to relate them in a new and, hopefully, interesting way. Starting with the Fibonacci numeration system — also known as Zeckendorff's system — we will pose ourselves the problem of extending it in a natural way to represent all real numbers in (0,1). We will see that this natural extension leads to what is known as the ?-system restricted to the unit interval. The resulting complete system of numeration replicates the uniqueness of the binary system which, in our opinion, is responsible for the possibility of defining the Van der Corput sequence in (0,1), a very special sequence which besides being uniformly distributed has one of the lowest discrepancy, a measure of the goodness of the uniformity.Lastly, combining the Fibonacci system and the binary in a very special way we will obtain a singular function, more specifically, the inverse of one of the family of Riesz-Nágy.     

Estimates for sums of moduli of blocks in trigonometric Fourier series

We consider the following two problems. Problem 1: what conditions on a sequence of finite subsets A _k ? ? and a sequence of functions λ _k : A _k → ? provide the existence of a number C such that any function f ∈ L ₁ satisfies the inequality ‖U _A,Λ(f)‖ _p ≤ C‖f‖₁ and what is the exact constant in this inequality? Here, \(U_{\mathcal{A},\Lambda } \left( f \right)\left( x \right) = \sum\nolimits_{k = 1}^\infty {\left| {\sum\nolimits_{m \in A_k } {\lambda _k \left( m \right)c_m \left( f \right)e^{imx} } } \right|}\) and c _m (f) are Fourier coefficients of the function f ∈ L ₁. Problem 2: what conditions on a sequence of finite subsets A _k ? ? guarantee that the function \(\sum\nolimits_{k = 1}^\infty {\left| {\sum\nolimits_{m \in A_k } {c_m \left( h \right)e^{imx} } } \right|}\) belongs to L _p for every function h of bounded variation?     

Analogs of the jackson-nikol'skii inequalities for trigonometric polynomials in spaces with nonsymmetric norm

The paper is concerned with the evaluation of where ▮·▮p ₁, p₂ is a nonsymmetric norm. The order of this number is obtained. Lower bounds involve new polynomials whose properties are studied in detail. In the casep ₁=p ₂,q ₁=q ₂, the estimate obtained is reduced to the well-known Jackson-Nikol'skii inequality. Translated fromMatematicheskie Zametki, Vol. 61, No. 5, pp. 687–699, May, 1997. Translated by N. K. Kulman     

Uniqueness for multiple trigonometric and Walsh series with convergent rearranged square partial sums

If at each point of a set of positive Lebesgue measure every rearrangement of a multiple trigonometric series square converges to a finite value, then that series is the Fourier series of a function to which it converges uniformly. If there is at least one point at which every rearrangement of a multiple Walsh series square converges to a finite value, then that series is the Walsh-Fourier series of a function to which it converges uniformly.

     

Exponential Sums and the Riemann Zeta Function V

A Van der Corput exponential sum is S = exp (2 i f(m)) wherem has size M, the function f(x) has size T and = (log M) / log T < 1. There are different bounds for S in differentranges for . In the middle range where is near 1/over 2, . This bounds the exponent of growthof the Riemann zeta function on its critical line Re s = 1/over2. Van der Corput used an iteration which changed at each step.The Bombieri–Iwaniec method, whilst still based on meansquares, introduces number-theoretic ideas and problems. TheSecond Spacing Problem is to count the number of resonancesbetween short intervals of the sum, when two arcs of the graphof y = f'(x) coincide approximately after an automorphism ofthe integer lattice. In the previous paper in this series [Proc.London Math. Soc. (3) 66 (1993) 1–40] and the monographArea, lattice points, and exponential sums we saw that coincidenceimplies that there is an integer point close to some ‘resonancecurve’, one of a family of curves in some dual space,now calculated accurately in the paper ‘Resonance curvesin the Bombieri–Iwaniec method’, which is to appearin Funct. Approx. Comment. Math. We turn the whole Bombieri–Iwaniec method into an axiomatisedstep: an upper bound for the number of integer points closeto a plane curve gives a bound in the Second Spacing Problem,and a small improvement in the bound for S. Ends and cusps ofresonance curves are treated separately. Bounds for sums oftype S lead to bounds for integer points close to curves, andanother branching iteration. Luckily Swinnerton-Dyer's methodis stronger. We improve from 0.156140... in the previous paperand monograph to 0.156098.... In fact (32/205 + , 269/410 +) is an exponent pair for every > 0. 2000 Mathematics SubjectClassification 11L07 (primary), 11M06, 11P21, 11J54 (secondary).