Simultaneous rational approximation to binomial functions

Diophantische Approximationen

Pade approximation and Rational interpolation.- Integral approximants for functions of higher monodromic dimension.- Asymptotics of Hermite-Pade Polynomials and related convergence results.- Rational approximation.- On the behavior of zeros and poles of best uniform polynomial and rational approximants.- Once again: the Adamjan-Arov-Krein approximation theory.- Diagonal Pade approximants, rational Chebyshev approximants and poles of functions.- On the use of the Caratheodory-Fejer method for investigating'1/9' and similar constants.- Multidimensional and Multivariate problems.- Simultaneous rational approximation to some q-hypergeometric functions.- Minimal Pade-sense matrix approximations around s = 0 and s = ?.- (Pade)y of (Pade)x approximants of F(x,y).- Different techniques for the construction of multivariate rational interpolants.- Rational approximants of hypergeometric series in ?n.- Orthogonal polynomials and the Moment problem.- Some orthogonal systems of p+1Fp-type Laurent polynomials.- The moment problem on equipotential curves.- Difference equations, continued fractions, Jacobi matrices and orthogonal polynomials.- Multipoint Pade approximation and orthogonal rational functions.- L-Polynomials orthogonal on the unit circle.- Continued fractions.- Schur's algorithm extended and Schur continued fractions.- Some recent results in the analytic theory of continued fractions.- Best a posteriori truncation error estimates for continued fractions if (an/1) with twin element regions.- Convergence acceleration for Miller's algorithm.- Convergence acceleration.- A new approach to convergence acceleration methods.- Applications.- General T-fraction solutions to Riccati differential equations.- A simple alternative principle for rational ?-method approximation.- Evaluation of Fermi-Dirac integral.- An application of operator Pade approximants to multireggeon processes.

We show that Mahler's classification of real numbers $\zeta$ with respect to the growth of the sequence $(w_{n}(\zeta))_{n\geq 1}$ is equivalently induced by certain natural assumptions on the decay of the sequence $(\lambda_{n}(\zeta))_{n\geq 1}$ concerning simultaneous rational approximation. Thereby we obtain a much clearer picture on simultaneous approximation to successive powers of a real number in general. Another variant of the Mahler classification concerning uniform approximation by algebraic numbers is shown as well. Our method has several applications to classic exponents of Diophantine approximation and metric theory. We deduce estimates on the Hausdorff dimension of well-approximable vectors on the Veronese curve and refine the best known upper bound for the exponent $\widehat{\lambda}_{n}(\zeta)$ for even $n\geq 4$.

In recursive digital filter design, the only linear technique available is probably the method of Pade approximants. Unfortunately, to obtain a Pade approximant, a formal power (Maclaurin) series must be given. If an ideal amplitude response |H(e^{j\omega})| is given, the usual method is to approximate its truncated delayed Fourier series, H_{N}(e^{j\omega}) = \Sigma\min{0}\max{2N}h_{-N+k}e^{-jk\omega} . This procedure is not desirable especially when the Pade approximant method is applied, since the first few terms in the power series (that is, h_{-N}, h_{-N+1} , ... in H N ) play the most important role in the characteristics of its Pade approximants. In this paper, we apply the idea of Hilbert transformations to obtain a complete complex frequency response H(e^{j\omega}) whose Fourier expansion gives rise to a power (Maclaurin) series. A method is given to compute this series, so that the Pade approximant technique can be applied readily.

The dominance of wireless communication (WC) is evident in all areas of life such as Information and Communication Technology (ICT), research, business, academia, etc. However, modelling analysis of WC has been a serious challenge when it involves diversity combining. Previous work has attempted to solve this problem using approximation techniques. These approximation techniques seem to be complex and may be difficult to interpret. The existing practice of using the Probability Density Function (PDF) to analyse multipath fading in a diversity-rich environment is ineffective to handle cascaded fading channels. The current work proposed the Pade Approximation (PA) technique to mitigate the problem. The PA was developed from the generated Moment Generating Function (MGF) by truncating the Taylor series to obtain a rational expression. The approximated rational expressions obtained were transformed into PDF, Cumulative Density Function (CDF) and Outage Probability (Pout). The results show that the Pout reduces as the threshold value increases. The numerical results also shows that the diversity techniques is effective in combating fading because as the number of paths increases, the Pout reduces. The Pout reduces by 17.56% at L=4 from 90.83% at L=1 when there is no diversity. PA is a useful approximation to analyse the behavior of cascaded Rayleigh-Rician channel.

In this paper, performance analysis of Optimum and Sub-optimum diversity combining receivers over generalized fading channels modeled by the three parameter Generic-Gamma model is presented. The Generic-Gamma model is versatile enough to represent short term fading such as Weibull, Nakagami-m or Rayleigh as well as shadowing. The performance measures such as amount of fading, average bit error rate, and signal outage are considered for analysis. With the aid of Moment Generating Function (MGF) approach and Pade approximation (PA) technique outage probability and Average bit error rate have been evaluated for a variety of modulation formats. PA technique has been used to derive simple-to-evaluate compact rational expressions for the MGF of output SNR. Using these novel rational expressions, the performance of multichannel receivers employing diversity combining under a range of representative channel fading conditions have been evaluated. The results have been validated through simulations which shows perfect match.

Chapter 1 The Padé Approximant Method and Some Related Generalizations

The linear, functional equation approach to the problem of the convergence of Pade approximants.- Construction of variational bounds for the N-body eigenstate problem by the method of Pade approximations.- Rational polynomial approximants in N variables.- Convergence of rows of the Pade table.- The use of Pade approximation in numerical integration.- Determination of shock waves by convergence acceleration.- Cyclic iterative method applied to transonic flow analyses.- A technique for accelerating iterative convergence in numerical integration, with application in transonic aerodynamics.- The rise of a bubble in a fluid.- Rational approximations to the solution of the blunt-body & related problems.- Wave front expansions and Pade' approximants for transient waves in linear dispersive media.- Application of methods for acceleration of convergence to the calculation of singularities of transonic flows.- The use of Pade fractions in the calculation of nozzle flows.- A bibliography on Pade approximation and some related matters.

Padé approximation of two-point functions and the calculation of resonances

For complex systems with time-varying, nonlinearity, and large time delay, the IMC-PID controller designed using the internal mode control strategy can be used to improve the control performance. Different setting formulas for IMC-PID controller parameters can be obtained by using different approximation methods for processing the time-delay process. These methods have a great influence on the control effect. The first-order Taylor, the first-order Pade, the second-order symmetric Pade and the second-order asymmetric Pade and all-pole approximation are used to obtain the parameter setting formulas, respectively. Their adaptabilities to large time-delay, large-inertia process and to large time-delay, small-inertia process are studied. The simulation results show that the all-pole approximation is more adaptive to the application of large time delay. The second-order symmetric Pade approximation is more effective in the large time-delay, small-inertia process, whereas the second-order asymmetric Pade approximation is more adaptive to large time-delay, large-inertia process.

Preface. Approximation by Functions of Nonclassical Form E.W. Cheney. Wavelets- with Emphasis on Spline-Wavelets and Applications to Signal Analysis C.K. Chui. Pade Approximation in One and More Variables A. Cuyt. Rational Hermite Interpolation in One and More Variables A. Cuyt. The Method of Alternating Orthogonal Projections F. Deutsch. Selections for Metric Projections F. Deutsch. Weighted Polynomials M. v. Golitschek. Some Aspects of Radial Basis Function Approximation W.A. Light. Using the Refinement Equation for the Construction of Pre-Wavelets VI: Shift Invariant Subspaces C.A. Micchelli. A Tutorial on Multivariate Wavelet Decomposition C.A. Micchelli. Error Estimates for Near-Minimax Approximations G.M. Phillips. Different Metrics and Location Problems E. Casini, P.L. Papini. On the Effectiveness of Some Inversion Methods for Noisy Fourier Series L. De Michele, M. Di Natale, D. Roux. A Generalization of N-Widths A.G. Aksoy. The Equivalence of the Usual and Quotient Topologies for CINFINITY(E) when EGBPsubGBPRn is Whitney p-Regular L.P. Bos, P.D. Milman. Korovkin Theorems for Vector-Valued Continuous Functions M. Campiti. On Modified Bojanic-Shisha Operators A.S. Cavaretta, S.S. Guo. A Property of Zeros and Cotes Numbers of Hermite and Laguerre Orthogonal Polynomials F. Costabile. Hermite-Fejer and Hermite Interpolation G. Criscuolo, B. Della Vecchia, G. Mastroianni. New Results on Lagrange Interpolation G. Criscuolo, G. Mastroianni. Ambiguous Loci in Best Approximation Theory F.S. De Blasi, J. Myjak. A Theorem on Best Approximations in Topological Vector Spaces E. De Pascale, G. Trombetta. On theCharacterization of Totally Positive Matrices M. Gasca, J.M. Pena. Iterative Methods for the General Order Complementarity Problem G. Isac. Wavelets, Splines, and Divergence-Free Vector Functions P-G. Lemarie-Rieusset. An Approach to Meromorphic Approximation in a Stein Manifold C.H. Lutterodt. Approximating Fixed Points for Nonexpansive Maps in Hilbert Spaces G. Marino. On Approximation and Interpolation of Convex Functions M. Neamtu. Convergence of Approximating Fixed Point Sets for Multivalued Nonexpansive Mappings P. Pietramala. A Subdivision Algorithm for Non-Uniform B-Splines R. Qu, J.A. Gregory. Some Applications of an Approximation Theorem for Fixed Points of Multi-Valued Contractions B. Ricceri. Geometrical Differentiation and High-Accuracy Curve Interpolation R. Schaback. On Best Simultaneous Approximation in Normed Linear Spaces V.M. Sehgal, S.P. Singh. Some Examples Concerning Projection Constants B. Shekhtman. Subject Index.

A new simple method for approximating certain algebraic numbers is developed. By applying this method, an effective upper bound is derived for the integral solutions of the quartic Thue equation with two parameters $tx^4 - 4sx^3 y - 6tx^2 y^2 + 4sxy^3 + ty^4 = N$ , where s > 32t 3. As an application, Ljunggren’s equation is solved in an elementary way.

We use the Pade approximation (PA) technique to obtain closed-form approximate expressions for the moment-generating function (MGF) of the Weibull random variable. Unlike previously obtained closed-form exact expressions for the MGF, which are relatively complicated as being given in terms of the Meijer G-function, PA can be used to obtain simple rational expressions for the MGF, which can be easily used in further computations. We illustrate the accuracy of the PA technique by comparing its results to either the existing exact MGF or to that obtained via Monte Carlo simulations. Using the approximate expressions, we analyze the performance of digital modulation schemes over the single channel and the multichannels employing maximal ratio combining (MRC) under the Weibull fading assumption. Our results show excellent agreement with previously published results as well as with simulations.

This paper presents a comparison system approach for the analysis of stability and ℋ︁∞performance of linear time‐invariant systems with unknown delays. The comparison system is developed by replacing the delay elements with certain parameter‐dependent Padé approximations. It is shown using the special properties of the Padé approximation to e−sthat the value sets of these approximations provide outer and inner coverings for that of each delay element and that the robust stability of the outer covering system is a sufficient condition for the stability of the original time delay system. The inner covering system, in turn, is used to provide an upper bound on the degree of conservatism of the delay margin established by the sufficient condition. This upper bound is dependent only upon the Padé approximation order and may be made arbitrarily small. In the single delay case, the delay margin can be calculated explicitly without incurring any additional conservatism. In the general case, this condition can be reduced with some (typically small) conservatism to finite‐dimensional LMIs. Finally, this approach is also extended to the analysis of ℋ︁∞performance for linear time‐delay systems with an exogenous disturbance. Copyright © 2003 John Wiley & Sons, Ltd.

This paper presents the time-domain Pade approximation and modal-Pade method for single and multivariable systems. In the first part of the paper, Pade equations are derived for SISO and MIMO systems by assuming a state-space description of the higher order system and its reduced order model. For a SISO system, it is shown that these Pade equations are the same as the Pade equations obtained by a frequency-domain procedure. However, for a multivariable system, it is shown that,. although a full Pade approximation exists in the frequency domain, the time-domain procedure leads only to a partial Pade approximation. In the next part, the time-domain modal-Pade method for SISO and MIMO systems is proposed. A relationship between the state vectors of the system and the model is derived for the modal-Pade reduction procedure and is referred to as an exact aggregation matrix.