Van Der Corput Research Articles

Van der Corput’s difference theorem for amenable groups and the left regular representation

We establish a connection between two variants of van der Corput’s Difference Theorem (vdCDT) for countably infinite amenable groups G and the ergodic hierarchy of mixing properties of a unitary representation U of G. In particular, we show that one variant of vdCDT corresponds to subrepresentations of the left regular representation, and another variant of vdCDT corresponds to the absence of finite dimensional subrepresentations. We then obtain applications for measure preserving actions of countably infinite abelian groups.

Distinguishing sets of strong recurrence from van der Corput sets

Distinguishing sets of strong recurrence from van der Corput sets

Van der Corput lemmas for Mittag-Leffler functions. I

In this paper, we study analogues of the van der Corput lemmas involving Mittag-Leffler functions. The generalisation is that we replace the exponential function with the Mittag-Leffler-type function to study oscillatory-type integrals appearing in the analysis of time-fractional partial differential equations. Several generalisations of the first and second van der Corput lemmas are proved. Optimal estimates on decay orders for particular cases of the Mittag-Leffler functions are also obtained. As an application of the above results, the generalised Riemann–Lebesgue lemma and the Cauchy problem for the time-fractional evolution equation are considered.

Explicit bounds on ζ(s) in the critical strip and a zero-free region

We derive explicit upper bounds for the Riemann zeta-function ζ(σ+it) on the lines σ=1−k/(2k−2) for integer k≥4. This is used to show that the zeta-function has no zeroes in the regionσ>1−log⁡log⁡|t|21.233log⁡|t|,|t|≥3. This is the largest known zero-free region for exp⁡(171)≤t≤exp⁡(5.3⋅105). Our results rely on an explicit version of the van der Corput AnB process for bounding exponential sums.

Robust hyperparameter tuned deep Elman neural network for the diagnosis of osteosarcoma on histology images

Osteosarcomas are a type of bone tumour that can develop anywhere in the bone but most typically do so around the metaphyseal growth plates at the ends of long bones. Death rates can be lowered by early detection. Manual osteosarcoma identification can be difficult and requires specialised knowledge. With the aid of contemporary technology, medical photographs may now be automatically analysed and categorised, enabling quicker and more effective data processing. This paper proposes a novel hyperparameter-tuned deep learning (DL) approach for predicting osteosarcoma on histology images with effective feature selection mechanism which aims to improve the prediction accuracy of the classification system for bone tumor detection. The proposed system mainly consists of ‘6’ phases: data collection, preprocessing, segmentation, feature extraction, feature selection, and classification. Firstly, the dataset of histology images is gathered from openly available sources. Then Median Filtering (MEF) is utilized as the preprocessing step that enhances the quality of the input images for accurate prediction by eliminating unwanted information from them. Afterwards, the pre-processed image was segmented using Harmonic Mean-based Otsu Thresholding (HMOTH) approach to obtain the tumor-affected regions from the pre-processed data. Then the features from the segmented tumor portions are extracted using the Self-Attention Mechanism-based MobileNet (SAMMNet) model. A Van der Corput sequence and Adaptive Inertia Weight included Reptile Search Optimization Algorithm (VARSOA) is used to select the more relevant features from the extracted features. Finally, a Hyperparameter-Tuned Deep Elman Neural Network (HTDENN) is utilized to diagnose and classify osteosarcoma, in which the hyperparameters of the neural network are obtained optimally using the VARSOA. The proposed HTDENN attains the higher accuracy of 0.9531 for the maximum of 200 epochs, whereas the existing DENN, MLP, RF, and SVM attains the accuracies of 0.9492, 0.9427, 0.9413, and 0.9387. Likewise, the proposed model attains the better results for precision (0.9511), f-measure (0.9423), sensitivity (0.9345) and specificity (0.9711) than the existing approaches for the maximum of 200 epochs. Simulation outcomes proved that the proposed model outperforms existing research frameworks for osteosarcoma prediction and classification.

ON THE N-POINT CORRELATION OF VAN DER CORPUT SEQUENCES

AbstractWe derive an explicit formula for the N-point correlation $F_N(s)$ of the van der Corput sequence in base $2$ for all $N \in \mathbb {N}$ and $s \geq 0$ . The formula can be evaluated without explicit knowledge about the elements of the van der Corput sequence. This constitutes the first example of an exact closed-form expression of $F_N(s)$ for all $N \in \mathbb {N}$ and all $s \geq 0$ which does not require explicit knowledge about the involved sequence. Moreover, it can be immediately read off that $\lim _{N \to \infty } F_N(s)$ exists only for $0 \leq s \leq 1/2$ .

A generalization of van der Corput's difference theorem with applications to recurrence and multiple ergodic averages

We prove a generalization of van der Corput's difference theorem for sequences of vectors in a Hilbert space. This generalization is obtained by establishing a connection between sequences of vectors in the first Hilbert space with a vector in a new Hilbert space whose spectral type with respect to a certain unitary operator is absolutely continuous with respect to the Lebesgue measure. We use this generalization to obtain applications regarding recurrence and multiple ergodic averages when we have measure preserving automorphisms T and S that do not necessarily commute, but T has a maximal spectral type that is mutually singular with the Lebesgue measure.

Equidistribution of Continuous Functions Along Monotone Compact Covers

Abstract We give a necessary and sufficient condition for equidistribution of continuous functions along monotone compact covers on locally compact spaces. We show the existence of equidistributed mappings along Bohr nets arising from group actions. Using almost periodic means, we give an analogue of Weyl’s equidistribution criterion for continuous functions with values in arbitrary topological groups. We prove van der Corput’s inequality on the lattice ℕm for vectors in Hilbert spaces, and use this inequality to extend Hlawka’s equidistribution theorem to functions on the lattice ℕm (m ≥ 1) with values in arbitrary topological groups.

Stationary phase method, powers of functions, and applications to functional analysis

The utility of the (weighted) van der Corput inequalities or of the stationary phase method is illustrated with various examples borrowed from: differentiability issues (Riemann’s function and related); functional analysis on Banach spaces or algebras of analytic functions (composition operators); and local Banach space geometry (Schäffer’s problem).

Asymptotics on the Fredholm integral equation with a highly oscillatory and weakly singular kernel

This paper focuses on the asymptotic behaviors of the solutions to the second-kind Fredholm integral equations with a highly oscillatory and weakly singular kernel. We employ van der Corput lemmas, resolvents of the Volterra integral equations, and the contraction mapping principle to analyse these behaviors. Numerical results are provided to show how the singularity and oscillatory parameters affect the solution.

Beyond the Weyl barrier for GL(2) exponential sums

In this paper, we use the Bessel δ-method, along with new variants of the van der Corput method in two dimensions, to prove non-trivial bounds for GL(2) exponential sums beyond the Weyl barrier. More explicitly, for sums of GL(2) Fourier coefficients twisted by e(f(n)), with length N and phase f(n)=Nβlog⁡n/2π or anβ, non-trivial bounds are established for β<1.63651..., which is beyond the Weyl barrier at β=3/2.

THE NEUTRIX PRODUCT OF THE DISTRIBUTIONS

In this paper author has obtained the neutrix product of x+ −r and δ (α) (x), where α is a positive fractional number. Key Words: Neutrix limit, fractional-differentiation, sequence, function. Ams Subject Classification- 46F 1. INTRODUCTION Neutrix N is defined by J.G. vander Corput [2] as a commutative additive group of functions ν(ξ) defined on a domain N' with values in additive group N", where further if for some ν in N, ν(ξ) = γ for all ξ in N', then γ= 0. The functions in N are called negligible functions. Now let N' be a set contained in a topological space with a limit point b which does not belong to N'. If f(ξ) is a function defined on N' with values in N" and it is possible to find a constant β such that f(ξ) –β is negligible in N, then β is called the neutrix limit or N-limit of f as ξ tends to b and we write N− lim ξ→b f(ξ) = β, where β must be unique, if it exists. Introducing the neutrix limit, Fisher [3,4] defined the neutrix product of two distributions as –

Extremals for $$\alpha $$-Strichartz Inequalities

A necessary and sufficient condition on the precompactness of extremal sequences for one-dimensional $$\alpha $$ -Strichartz inequalities, equivalently $$\alpha $$ -Fourier extension estimates, is established based on the profile decomposition arguments. One of our main tools is an operator-convergence dislocation property consequence which comes from the van der Corput Lemma. Our result is valid in asymmetric cases as well. In addition, we obtain the existence of extremals for non-endpoint $$\alpha $$ -Strichartz inequalities.

On the distribution of the van der Corput sequences

For an integer pge 2, let {x_n}_{nin {mathbb {N}}}subset {mathbb {T}} be the p-adic van der Corput sequence. For intervals [0,alpha )subset {mathbb {T}} and for positive integers N, consider the geometrically-shifted discrepancy function D_{p,N,alpha }(t)=sum _{n=0}^{N-1}mathcal {X}_{[0,alpha )}(x_n+t)-Nalpha . In this paper, we give a characterization of the asymptotic behavior of Vert D_{p,N,alpha }(cdot )Vert _{L^2({mathbb {T}})} for Nrightarrow infty that depends on the p-adic expansion of alpha .

On optimal convergence rates of Laguerre polynomial expansions for piecewise functions

By applying Hilb type formula and generalized van der Corput type Lemmas, this paper presents optimal decay rates on the expansion coefficients for given functions expanded in the form of Laguerre polynomial series, from which it leads to the convergence rates on the spectral orthogonal projections, and derives new estimates on the errors for Gauss–Laguerre quadrature. Ample numerical experiments are carried out to validate the optimality and accuracy of these theoretical results.

On the $(\mathrm{Vil}_B;\alpha)$-diaphony of the Van Der Corput sequence constructed in Cantor Systems

In the present paper the authors consider the so-called $(\mathrm{Vil}_{\mathcal{B}_s};\alpha;\gamma)$-diaphony as a suitable tool to investigate sequences constructed in arbitrary Cantor systems. The definition of this kind of the diaphony is based on using Vilenkin function system and depends on two arguments -- a vector $\alpha$ of exponential parameters and a vector $\gamma$ of coordinate weights. This diaphony is used to investigate the distribution of the points of the Van der Corput sequence $\omega_B$ constructed in the same $B$-adic Cantor system. In this way a process of synchronization between the technique of a construction of the sequence $\omega_B$ and the tool of its studying is realized. Upper and low bounds of the $(\mathrm{Vil}_B;\alpha)$-diaphony of the sequence $\omega_B$ are presented. This permit us to show the influence of the exponential parameter $\alpha$ to the exact order of the $(\mathrm{Vil}_B;\alpha)$-diaphony of this sequence. When $\alpha=2$ the exact order is $\mathcal{O}\left(\dfrac{\sqrt{\log N}}{N}\right)$ and when $\alpha&gt;2$ the exact order is $\mathcal{O}\left(\dfrac1N\right)$.

Local limit theorems for complex functions on [formula omitted

The local (central) limit theorem precisely describes the behavior of iterated convolution powers of a probability distribution on the d-dimensional integer lattice, Zd. Under certain mild assumptions on the distribution, the theorem says that the convolution powers are well-approximated by a single scaled Gaussian density which we call an attractor. When such distributions are allowed to take on complex values, their convolution powers exhibit new and striking behaviors not seen in the probabilistic setting. Following works of I. J. Schoenberg, T. N. E. Greville, P. Diaconis, and L. Saloff-Coste, the author and L. Saloff-Coste provided a complete description of local limit theorems for the class of finitely supported complex-valued functions on Z. For convolution powers of complex-valued functions on Zd, much less is known. In a previous work by the author and L. Saloff-Coste, local limit theorems were established for complex-valued functions whose Fourier transform is maximized in absolute value at so-called points of positive homogeneous type and, in that case, the resultant attractors are generalized heat kernels corresponding to a class of higher order partial differential operators. By considering the possibility that the Fourier transform can be maximized in absolute value at points of imaginary homogeneous type, a notion motivated by V. Thomée, this article extends previous work of the author and L. Saloff-Coste to broaden the class of complex-valued functions for which it is possible to obtain local limit theorems. These local limit theorems contain attractors given by certain oscillatory integrals and their convergence is established using a generalized polar-coordinate integration formula, due to H. Bui and the author, and the Van der Corput lemma. The article also extends recent results on sup-norm type estimates of H. Bui and the author.

A dynamical proof of the van der Corput inequality

We provide a dynamical proof of the van der Corput inequality for sequences in Hilbert spaces that is based on the Furstenberg correspondence principle. This is done by reducing the inequality to the mean ergodic theorem for contractions on Hilbert spaces. The key difficulty therein is that the Furstenberg correspondence principle is, a priori, limited to scalar-valued sequences. We, therefore, discuss how interpreting the Furstenberg correspondence principle via the Gelfand–Naimark–Segal construction for -algebras allows to study not just scalar but general Hilbert space-valued sequences in terms of unitary operators. This yields a proof of the van der Corput inequality in the spirit of the Furstenberg correspondence principle and the flexibility of this method is discussed via new proofs for different variants of the inequality.

Anti-concentration of polynomials: Dimension-free covariance bounds and decay of Fourier coefficients

We study random variables of the form f(X), when f is a degree d polynomial, and X is a random vector on Rn, motivated towards a deeper understanding of the covariance structure of X⊗d. For applications, the main interest is to bound Var(f(X)) from below, assuming a suitable normalization on the coefficients of f. Our first result applies when X has independent coordinates, and we establish dimension-free bounds. We also show that the assumption of independence can be relaxed and that our bounds carry over to uniform measures on isotropic Lp balls. Moreover, in the case of the Euclidean ball, we provide an orthogonal decomposition of Cov(X⊗d). Finally, we utilize the connection between anti-concentration and decay of Fourier coefficients to prove a high-dimensional analogue of the van der Corput lemma, thus partially answering a question posed by Carbery and Wright.

Some connections between discrepancy, finite gap properties, and pair correlations

A generic uniformly distributed sequence (x_n)_{n in mathbb {N}} in [0, 1) possesses Poissonian pair correlations (PPC). Vice versa, it has been proven that a sequence with PPC is uniformly distributed. Grepstad and Larcher gave an explicit upper bound for the discrepancy of a sequence given that it has PPC. As a first result, we generalize here their result to the case of alpha -pair correlations with 0< alpha < 1. Since the highest possible level of uniformity is achieved by low-discrepancy sequences it is tempting to assume that there are examples of such sequences which also have PPC. Although there are no such known examples, we prove that every low-discrepancy sequence has at least alpha -pair correlations for 0< alpha <1. According to Larcher and Stockinger, the reason why many known classes of low-discrepancy sequences fail to have PPC is their finite gap property. In this article, we furthermore show that the discrepancy of a sequence with the finite gap property plus a condition on the distribution of the different gap lengths can be estimated. As a concrete application of this estimation, we re-prove the fact that van der Corput and Kronecker sequences are low-discrepancy sequences. Consequently, it follows from the finite gap property that these sequences have alpha -pair correlations for 0< alpha < 1.