Merging asymptotic expansions for cooperative gamblers in generalized St. Petersburg games

Merging asymptotic expansions are established for the distribution functions of suitably centered and normed linear combinations of winnings in a full sequence of generalized St. Petersburg games, where a linear combination is viewed as the share of any one of n cooperative gamblers who play with a pooling strategy. The expansions are given in terms of Fourier-Stieltjes transforms and are constructed from suitably chosen members of the classes of subsequential semistable infinitely divisible asymptotic distributions for the total winnings of the n players and from their pooling strategy, where the classes themselves are determined by the two parameters of the game. For all values of the tail parameter, the expansions yield best possible rates of uniform merge. Surprisingly, it turns out that for a subclass of strategies, not containing the averaging uniform strategy, our merging approximations reduce to asymptotic expansions of the usual type, derived from a proper limiting distribution. The Fourier-Stieltjes transforms are shown to be numerically invertible in general and it is also demonstrated that the merging expansions provide excellent approximations even for very small n.     

Fourier coefficients of continuous functions

It is well known that the Fourier coefficients of continuous functions with respect to classical orthogonal systems (trigonometric, Haar, and Walsh) can be estimated via the moduli of continuity of the functions. However, not all orthonormal systems possess this property. We obtain necessary and sufficient conditions on orthonormal systems such that the Fourier coefficients of continuous functions with respect to these orthonormal systems can be estimated via the moduli of continuity in a certain sense.     

Polynomial operators and local smoothness classes on the unit interval

We obtain a characterization of local Besov spaces of functions on [-1,1] in terms of algebraic polynomial operators. These operators are constructed using the coefficients in the orthogonal polynomial expansions of the functions involved. The example of Jacobi polynomials is studied in further detail. A by-product of our proofs is an apparently simple proof of the fact that the Cesàro means of a sufficiently high integer order of the Jacobi expansion of a continuous function are uniformly bounded.     

State Analysis and Optimal Control of Time-Varying Discrete Systems via Haar Wavelets

The state analysis and optimal control of time-varying discrete systems via Haar wavelets are the main tasks of this paper. First, we introduce the definition of discrete Haar wavelets. Then, a comparison between Haar wavelets and other orthogonal functions is given. Based upon some useful properties of the Haar wavelets, a special product matrix and a related coefficient matrix are proposed; also, a shift matrix and a summation matrix are derived. These matrices are very effective in solving our problems. The local property of the Haar wavelets is applied to shorten the calculation procedures.     

Unbiased estimation in the multivariate natural exponential family with simple quadratic variance function

We give expansions for the unbiased estimator of a parametric function of the mean vector in a multivariate natural exponential family with simple quadratic variance function. This expansion is given in terms of a system of multivariate orthogonal polynomials with respect to the density of the sample mean. We study some limit properties of the system of orthogonal polynomials. We show that these properties are useful to establish the limit distribution of unbiased estimators.     

Cross products of complete orthonormal systems of functions

The orthonormal kernel is a continuous analog for an orthonormal system of functions. The cross product of any two orthonormal systems, complete in L₂, is an example of a complete orthonormal kernel with respect to Lebesgue measure. In this note we continue our study of the properties of the cross product of a Haar system with an arbitrary orthonormal system of functions, complete in L₂, and totally bounded. We investigate certain properties of the cross product of a Haar system with another Haar system.Translated from Matematicheskie Zametki, Vol. 15, No. 2, pp. 331–340, February, 1974.The author thanks Professor N. Ya. Vilenkin for helpful discussions during the course of this work.     

Properties and calculation of Paired Haar transform

Different properties of recently introduced Paired Haar transform have been shown. Nonpolynomial Haar Pxpansion of incompletely specified Boolean functions has been presented. Based on the above properties and expansion some applications of Paired Haar spectrum have been proposed. Algorithm for the calculation of Haar Pair spectrum from disjoint cubes for systems of incompletely specified Boolean functions has also been developed.     

PROPERTIES AND CALCULATION OF PAIRED HAAR TRANSFORM

Different properties of recently introduced Paired Haar transform have been shown. Nonpolynomial Haar Pxpansion of incompletely specified Boolean functions has been presented. Based on the above properties and expansion some applications of Paired Haar spectrum have been proposed. Algorithm for the calculation of Haar Pair spectrum from disjoint cubes for systems of incompletely specified Boolean functions has also been developed.     

Приближение функций многих переменных классов H p Ω полиномами по системе Хаара

In the paper, problems of approximation of functions indicated in the title are studied in the case when a majorant of their mixed modulus of continuity is given. Two forms of approximation (linear and nonlinear) are considered: the classical linear approximation by polynomials with respect to the Haar system when the indices lie in hyperbolic crosses; and the best M-term approximation by polynomials also with respect to the Haar system. For the corresponding quantities, exact estimates are obtained as to their order of magnitude.     

О БАжИсАх В пРОстРАНс тВАх ОРлИЧА

The expansion of a given function with respect to a certain basis depends both on the properties of the function and on the metric in which the expansion is considered. Conditions are obtained in the paper that ensure the unconditional convergence of the expansions with respect to the spline systems which were introduced byZ. Ciesielski. In particular, the solution of a problem raised by P. L. Ul'janov is obtained: There exists no functionΩ(u) ↑ ∞ (u ↑ ∞) such that the condition $$\mathop \smallint \limits_0^1 \Phi (f)\omega (f)dt< \infty $$ implies the unconditionalΦ-convergence of the Haar—Fourier series of the functionf.     

Orthorecursive expansions in Hilbert spaces

Orthorecursive expansions over systems of closed subspaces of a Hilbert space are considered. Sufficient conditions for convergence of these expansions to the expanded elements are proved. The results obtained are illustrated on systems of contractions and translations of fixed functions.     

Shift Generated Haar Spaces on Compact Domains in the Complex Plane

Haar spaces are certain finite-dimensional subspaces of $\cc(K)$, where $K$ is a compact set and $\cc(K)$ is the Banach space of continuous functions defined on $K$ having values in $\C$. We characterize those Haar spaces which are generated by shifts applied to a single, analytic function for $K\subset\C$. This means that an arbitrary finite number of shifts generates Haar spaces by forming linear hulls. We have to distinguish two cases: (a) $K\not=\overline{K^\circ}$; (b) $K=\overline{K^\circ}$. It turns out that, in case (a), an analytic Haar space generator for dimensions one and two is already a universal Haar space generator for all dimensions. The geometrically simplest case that, in case (b), $K$ is convex with smooth boundary turns out to be the most difficult case. There is one numerical example in which the entire function $f:=1/\Gamma$ is interpolated in a shift generated Haar space of dimension four.     

Asymptotics of orthogonal polynomials with asymptotic Freud-like weights

We derive uniform asymptotic expansions for polynomials orthogonal with respect to a class of weight functions that are real analytic and behave asymptotically like the Freud weight at infinity. Although the limiting zero distributions are the same as in the Freud cases, the asymptotic expansions are different due to the fact that the weight functions may have a finite or infinite number of zeros on the imaginary axis. To resolve the singularities caused by these zeros, an auxiliary function is introduced in the Riemann–Hilbert analysis. Asymptotic formulas are established in several regions covering the whole complex plane. We take the continuous dual Hahn polynomials as an example to illustrate our main results. Some numerical verifications are also given.     

On constructing new expansions of functions using linear operators

Let T,U be two linear operators mapped onto the function f such that U(T(f))=f, but T(U(f))≠f. In this paper, we first obtain the expansion of functions T(U(f)) and U(T(f)) in a general case. Then, we introduce four special examples of the derived expansions. First example is a combination of the Fourier trigonometric expansion with the Taylor expansion and the second example is a mixed combination of orthogonal polynomial expansions with respect to the defined linear operators T and U. In the third example, we apply the basic expansion U(T(f))=f(x) to explicitly compute some inverse integral transforms, particularly the inverse Laplace transform. And in the last example, a mixed combination of Taylor expansions is presented. A separate section is also allocated to discuss the convergence of the basic expansions T(U(f)) and U(T(f)).     

PIλDμ controller design for integer and fractional plants using piecewise orthogonal functions

This paper deals with the design of fractional PID controller for integer and fractional plants. A new analytic method is proposed, the developments are based on the expansion of the control loop signals as well as a chosen reference model input and output over a piecewise orthogonal functions, namely, Block pulse, Walsh and Haar wavelets. The generalized operational matrices of differentiation related to these bases which are fitting the Riemann–Liouville definition accurately are used to replace the fractional differential calculus by an algebraic one easier to solve. Thereafter, the controller tuning is elaborated simply with a matrix manipulation manner. At first, a least square is drawn to find only the controller gains, then a nonlinear function defined as a matrix norm is minimized to optimize the whole parameters. A variety of examples covering both integer and fractional systems and reference models are presented to show the validity of the technique.     

Uniform Asymptotic Expansions of Symmetric Elliptic Integrals

Symmetric standard elliptic integrals are considered when two or more parameters are larger than the others. The distributional approach is used to derive seven expansions of these integrals in inverse powers of the asymptotic parameters. Some of these expansions also involve logarithmic terms in the asymptotic variables. These expansions are uniformly convergent when the asymptotic parameters are greater than the remaining ones. The coefficients of six of these expansions involve hypergeometric functions with less parameters than the original integrals. The coefficients of the seventh expansion again involve elliptic integrals, but with less parameters than the original integrals. The convergence speed of any of these expansions increases for an increasing difference between the asymptotic variables and the remaining ones. All the expansions are accompanied by an error bound at any order of the approximation. January 31, 2000. Date revised: May 18, 2000. Date accepted: August 4, 2000.     

Prolate spheroidal wave functions, an introduction to the Slepian series and its properties

For decades mathematicians, physicists, and engineers have relied on various orthogonal expansions such as Fourier, Legendre, and Chebyschev to solve a variety of problems. In this paper we exploit the orthogonal properties of prolate spheroidal wave functions (PSWF) in the form of a new orthogonal expansion which we have named the Slepian series. We empirically show that the Slepian series is potentially optimal over more conventional orthogonal expansions for discontinuous functions such as the square wave among others. With regards to interpolation, we explore the connections the Slepian series has to the Shannon sampling theorem. By utilizing Euler's equation, a relationship between the even and odd ordered PSWFs is investigated. We also establish several other key advantages the Slepian series has such as the presence of a free tunable bandwidth parameter.     

Limit-periodic arithmetical functions and the ring of finite integral adeles

In this paper, we show that the ring of finite integral adeles, together with its Borel field and its normalized Haar measure, is an appropriate probability space where limit-periodic arithmetical functions can be extended to random variables. The natural extensions of additive and multiplicative functions are studied. Besides, the convergence of Fourier expansions of limit-periodic functions is proved.     

Exponentially small expansions in the asymptotics of the Wright function

We consider exponentially small expansions present in the asymptotics of the generalised hypergeometric function, or Wright function, _pΨ_q(z) for large |z| that have not been considered in the existing theory. Our interest is principally with those functions of this class that possess either a finite algebraic expansion or no such expansion and with parameter values that produce exponentially small expansions in the neighbourhood of the negative real z axis. Numerical examples are presented to demonstrate the presence of these exponentially small expansions.     

An approximate method for analysing stochastic mechanical systems

An approximate method for analysing diffusion processes in a natural mechanical system when there are perturbing forces similar to normal white noise is proposed. It is based on orthogonal expansions of the one-dimensional probability density of the state vector in a suitable Hubert space of functions which are square-integrable with respect to a certain measure in the phase space (manifold) of the system. The method consists of solving a special system of linear ordinary differential equations for the expansion coefficients, and is suitable for computer implementation. The method is rigorously proved. The motion of a two-dimensional mathematical pendulum in a random medium is investigated as an example.