A Nonparametric Test for the Equivalence of Populations Based on a Measure of Proximity of Samples

We propose a new measure of proximity of samples based on confidence limits for the bulk of a population constructed using order statistics. For this measure of proximity, we compute approximate confidence limits corresponding to a given significance level in the cases where the null hypothesis on the equality of hypothetical distribution functions may or may not be true. We compare this measure of proximity with the Kolmogorov–Smirnov and Wilcoxon statistics for samples from various populations. On the basis of the proposed measure of proximity, we construct a statistical test for testing the hypothesis on the equality of hypothetical distribution functions.     

Orthonormal sampling functions

We investigate functions (x) whose translates {(x−k)}, where k runs through the integer lattice , provide a system of orthonormal sampling functions. The cardinal sine, whose important role in the sampling theory of bandlimited functions is well documented, is the classic example. For the bandlimited case we provide a complete characterization of such functions and give examples with rapid decay including a construction which is symmetric. We also analyze the general case of arbitrary sampling rate, a>0, which leads to some unexpected observations.     

Approximation of bandlimited functions

Many signals encountered in science and engineering are approximated well by bandlimited functions. We provide suitable error bounds for the approximation of bandlimited functions by linear combinations of certain special functions—the prolate spheroidal wave functions of order 0. The coefficients in the approximating linear combinations are given explicitly via appropriate quadrature formulae.     

Sampling and recovery of multidimensional bandlimited functions via frames

In this paper, we investigate frames for L₂d[−π,π] consisting of exponential functions in connection to oversampling and nonuniform sampling of bandlimited functions. We derive a multidimensional nonuniform oversampling formula for bandlimited functions with a fairly general frequency domain. The stability of said formula under various perturbations in the sampled data is investigated, and a computationally manageable simplification of the main oversampling theorem is given. Also, a generalization of Kadec's 1/4 theorem to higher dimensions is considered. Finally, the developed techniques are used to approximate biorthogonal functions of particular exponential Riesz bases for L₂[−π,π], and a well-known theorem of Levinson is recovered as a corollary.     

Monotone Nonincreasing Random Fields on Partially Ordered Sets. I

For an arbitrary poset H and measure ρ on H × _R (where _R is the real axis), we construct a monotone decreasing stochastic field η_ρ and compute its finite-dimensional distributions. In the case where H is a Λ-semilattice and the measure ρ satisfies additional conditions, we compute various characteristics of the field η_ρ such as the expectation of the field value at a point, variance of the field value at a point, and correlation function of the field. The described construction of random fields gives a new method for constructing positive definite functions on posets. Bibliography: 6 titles.__________Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 301, 2003, pp. 92–143.     

Multipoint Padé-Type Approximants. Exact Rate of Convergence

We study the rate with which sequences of interpolating rational functions, whose poles are partially fixed, approximate Markov-type analytic functions. Applications to interpolating quadratures are given. January 25, 1996. Date revised: December 26, 1996.     

Boundary-value problems for x-analytical functions with weighted boundary conditions

We consider boundary-value problems for x-analytical functions of a complex variable z = x + iy in a number of domains. Limit values with the weight (ln x)^–1 are given for the real part of the x-analytical function on the sections of the boundary that follow the imaginary axis, and simple limits are given for the real part of the x-analytical functions on the part of the boundary outside the imaginary axis. The apparatus of integral representations of x-analytical functions is applied to obtain a solution of the problem in quadratures.Kiev University. Translated from Vychislitel'naya i Prikladnaya Matematika, No. 74, pp. 3–11, 1992;     

Optimally Localized Approximate Identities on the 2-Sphere

We introduce a method to construct approximate identities on the 2-sphere that have an optimal localization. This approach can be used to accelerate the calculations of approximations on the 2-sphere essentially with a comparably small increase of the error. The localization measure in the optimization problem includes a weight function that can be chosen under some constraints. For each choice of weight function, existence and uniqueness of the optimal kernel are proved as well as the generation of an approximate identity in the bandlimited case. Moreover, the optimally localizing approximate identity for a certain weight function is calculated and numerically tested.     

Dual integral equations related to the kontorovich-lebedev transform : PMM vol. 38, n 6, 1974, pp. 1090–1097

We study the dual integral equations related to the Kontorovich-Lebedev integral transforms arising in the course of solution of the problems of mathematical physics, in particular of the mixed boundary value problems for the wedge-shaped regions. We show that the solutions of these equations can be expressed in quadratures, using the auxilliary functions satisfying the integral Fredholm equation of second kind with a symmetric kernel.At present, the dual equations investigated in most detail are those connected with the Fourier and Hankel integral transforms. The results obtained and their applications are given in [1–3]. A large number of papers also deal with the theory and applications of the dual integral equations connected with the Mehler-Fock integral transform and its generalizations [4–11]., The dual integral transforms considered in the present paper belong to a more complex class than those listed above, and so far, no effective solution has been obtained for them. The only relevant results known to the authors are those in [12, 13]. In [12] a method of solving the equations (1.2) is given for a single particular value of the parameter γ = π/2, while in [13] the dual equations of the type under consideration are reduced to a solution of an infinite system of linear algebraic equations.     

Chromatic series for functions of slow growth

The theory of chromatic derivatives leads to chromatic series which replace Taylor's series for bandlimited functions. For such functions, these series have a global convergence property not shared by Taylor's series. In this work the theory is extended to bandlimited functions of slow growth. This includes many signals of practical importance such as polynomials, periodic functions and almost periodic functions. This extension also enables us to get improved local convergence results for chromatic series.     

Integration of ε-Fenchel Subdifferentials and Maximal Cyclic Monotonicity

This paper concerns the integration of ε-Fenchel subdifferentials of proper lower semicontinuous convex functions defined on arbitrary topological vector spaces. We make use of integration tools to provide a representation formula of the approximate subdifferential of convex functions, and also to identify the class of maximal cyclically monotone families of operators.     

Inequalities between remainders of quadratures

It is well-known that in the class of convex functions the (nonnegative) remainder of the Midpoint Rule of approximate integration is majorized by the remainder of the Trapezoid Rule. Hence the approximation of the integral of a convex function by the Midpoint Rule is better than the analogous approximation by the Trapezoid Rule. Following this fact we examine remainders of certain quadratures in classes of convex functions of higher orders. Our main results state that for 3-convex (5-convex, respectively) functions the remainder of the 2-point (3-point, respectively) Gauss–Legendre quadrature is non-negative and it is not greater than the remainder of Simpson’s Rule (4-point Lobatto quadrature, respectively). We also check 2-point Radau quadratures for 2-convex functions to demonstrate that similar results fail to hold for convex functions of even orders. We apply the Peano Kernel Theorem as a main tool of our considerations.     

A Modification of Hermite Sampling with a Gaussian Multiplier

The Hermite sampling series is used to approximate bandlimited functions. In this article, we introduce two modifications of Hermite sampling with a Gaussian multiplier to approximate bandlimited and non-bandlimited functions. The convergence rate of those modifications is much higher than the convergence rate of Hermite sampling. Based on complex analysis, we establish some error bounds for approximating different classes of functions by these modifications. Theoretically and numerically, we demonstrate that the approximation by these modifications is highly efficient.     

A note on the expressive power of deep rectified linear unit networks in high‐dimensional spaces

We investigate the ability of deep deep rectified linear unit (ReLU) networks to approximate multivariate functions. Specially, we establish the approximation error estimate on a class of bandlimited functions; in this case, ReLU networks can overcome the “curse of dimensionality.”     

Bounded mean oscillation and bandlimited interpolation in the presence of noise

We study some problems related to the effect of bounded, additive sample noise in the bandlimited interpolation given by the Whittaker-Shannon-Kotelnikov (WSK) sampling formula. We establish a generalized form of the WSK series that allows us to consider the bandlimited interpolation of any bounded sequence at the zeros of a sine-type function. The main result of the paper is that if the samples in this series consist of independent, uniformly distributed random variables, then the resulting bandlimited interpolation almost surely has a bounded global average. In this context, we also explore the related notion of a bandlimited function with bounded mean oscillation. We prove some properties of such functions, and in particular, we show that they are either bounded or have unbounded samples at any positive sampling rate. We also discuss a few concrete examples of functions that demonstrate these properties.     

Concentration Problems for Bandpass Filters in Communication Theory over Disjoint Frequency Intervals and Numerical Solutions

The concentration problem of maximizing signal strength of bandlimited and timelimited nature is important in communication theory. In this paper we consider two types of concentration problems for the signals which are bandlimited in disjoint frequency-intervals, which constitute a band-pass filter. For the first type the problem is to determine which members of L ²(−∞,∞) lose the smallest fraction of their energy when first timelimited and then bandlimited. For the second type the problem is to determine which bandlimited signals lose the smallest fraction of their energy when restricted to a given time interval. For both types of problems, basic theoretical properties and numerical algorithms for solution and convergence theorems are given. Orthogonality properties of analytically extended eigenfunctions over L ²(−∞,∞) are also proved. Numerical computations are carried out which corroborate the theory. Relationship between eigenvalues of these two types of problems is also established. Several properties of eigenvalues of both types of problems are proved.     

Error estimates for Gaussian quadratures of analytic functions

For analytic functions the remainder term of Gaussian quadrature formula and its Kronrod extension can be represented as a contour integral with a complex kernel. We study these kernels on elliptic contours with foci at the points ±1 and the sum of semi-axes ?>1 for the Chebyshev weight functions of the first, second and third kind, and derive representation of their difference. Using this representation and following Kronrod’s method of obtaining a practical error estimate in numerical integration, we derive new error estimates for Gaussian quadratures.     

Worst Case Complexity of Problems with Random Information Noise

We study the worst case complexity of solving problems for which information is partial and contaminated by random noise. It is well known that if information is exact then adaption does not help for solving linear problems, i.e., for approximating linear operators over convex and symmetric sets. On the other hand, randomization can sometimes help significantly. It turns out that for noisy information, adaption may lead to much better approximations than nonadaption, even for linear problems. This holds because, in the presence of noise, adaption is equivalent to randomization. We present sharp bounds on the worst case complexity of problems with random noise in terms of the randomized complexity with exact information. The results obtained are applied to thed-variate integration andL_∞-approximation of functions belonging to Hölder and Sobolev classes. Information is given by function evaluations with Gaussian noise of variance σ². For exact information, the two problems are intractable since the complexity is proportional to (1/ε)^qwhereqgrows linearly withd. For noisy information the situation is different. For integration, the ε-complexity is of order σ²/ε²as ε goes to zero. Hence the curse of dimensionality is broken due to random noise. for approximation, the complexity is of order σ²(1/ε)^q+2ln(1/ε), and the problem is intractable also with random noise.     

DEEP RELU NETWORKS OVERCOME THE CURSE OF DIMENSIONALITY FOR GENERALIZED BANDLIMITED FUNCTIONS

We prove a theorem concerning the approximation of generalized bandlimited mul-tivariate functions by deep ReLU networks for which the curse of the dimensionality is overcome.Our theorem is based on a result by Maurey and on the ability of deep ReLU networks to approximate Chebyshev polynomials and analytic functions efficiently.     

An augmented Lagrangian approach with a variable transformation in nonlinear programming

Tangent cone and (regular) normal cone of a closed set under an invertible variable transformation around a given point are investigated, which lead to the concepts of θ⁻¹-tangent cone of a set and θ⁻¹-subderivative of a function. When the notion of θ⁻¹-subderivative is applied to perturbation functions, a class of augmented Lagrangians involving an invertible mapping of perturbation variables are obtained, in which dualizing parameterization and augmenting functions are not necessarily convex in perturbation variables. A necessary and sufficient condition for the exact penalty representation under the proposed augmented Lagrangian scheme is obtained. For an augmenting function with an Euclidean norm, a sufficient condition (resp., a sufficient and necessary condition) for an arbitrary vector (resp., 0) to support an exact penalty representation is given in terms of θ⁻¹-subderivatives. An example of the variable transformation applied to constrained optimization problems is given, which yields several exact penalization results in the literature.