Power Series Type Research Articles

On projections in power series spaces and the existence of bases

Mityagin posed the problem, whether complemented subspaces of nuclear infinite type power series spaces have a basis. A related more general question was asked by Pełczyński. It is well known for a complemented subspace E E of a nuclear infinite type power series space, that its diametral dimension can be represented by Δ E = Δ Λ ∞ ( α ) \Delta E = \Delta {\Lambda _\infty }(\alpha ) for a suitable sequence α \alpha with α j ≥ ln ⁡ ( j + 1 ) {\alpha _j} \geq \ln (j + 1) . In this article we prove the existence of a basis for E E in case that α j ≥ j {\alpha _j} \geq j and sup α 2 j α j > ∞ \sup \tfrac {{{\alpha _{2j}}}}{{{\alpha _j}}} > \infty .

Parameter identification technique for a one-degree-of-freedom nonlinear system.

A parameter identification technique is presented for a one-degree-of-freedom system with a power series type nonlinear spring. The proposed technique consists of the following two steps. In the first step, a small impact is given to the system and the linear parameters are identified by using the usual modal analysis technique. In the second step, an impact large enough to produce the nonlinearity of the system is given. From both the free response due to it and a harmonic balance approximate solution, the nonlinear parameters are identified by using the least-squares method. Examples of numerical simulation and experiments are given to demonstrate the accuracy of the technique, and good results are shown.

Complete Class Theorems for Estimation of Multivariate Poisson Means and Related Problems

Basic decision theory for discrete random variables of the multivariate geometric (power series) type is developed. Some properties of Bayes estimators that carry over in the limit to admissible estimators are obtained. A stepwise generalized Bayes representation of admissible estimators is developed with estimation of the mean of a multivariate Poisson random variable in mind. The development carries over to estimation of the mean of a multivariate negative Binomial random variable. Due to the natural boundary of the parameter space there is an interesting pathology illustrated to some extent by the examples given. Examples include one to show that admissible estimators with somewhere infinite risk do exist in two or more dimensions.

Tameness of pairs of nuclear power series spaces and related topics

The equivalence of the following six assertions is proved: (i) The set of the finite limit points of the ratios α m / β n , n , m ∈ N {\alpha _m}/{\beta _n},n,m \in {\mathbf {N}} , is bounded, (ii) Every operator from Λ ∞ ( β ) {\Lambda _\infty }(\beta ) to Λ 1 ( α ) {\Lambda _1}(\alpha ) is compact, (iii) The pair ( Λ ∞ ( β ) , Λ 1 ( α ) ) ({\Lambda _\infty }(\beta ),\,{\Lambda _1}(\alpha )) is tame, i.e., for every operator T T from Λ ∞ ( β ) {\Lambda _\infty }(\beta ) to Λ 1 ( α ) {\Lambda _1}(\alpha ) there is a positive integer a a such that for every k ∈ N k \in {\mathbf {N}} there is a constant C k {C_k} such that | | T x | | k ⩽ C k | x | a k ||Tx|{|_k} \leqslant {C_k}|x{|_{ak}} for every x ∈ Λ ∞ ( β ) x \in {\Lambda _\infty }(\beta ) . (iv) Every short exact sequence 0 → Λ τ ( β ) → X → Λ 1 ( α ) → 0 0 \to {\Lambda _\tau }(\beta ) \to X \to {\Lambda _1}(\alpha ) \to 0 , where τ = 1 \tau = 1 or ∞ \infty , splits. (v) Λ 1 ( α ) × Λ ∞ ( β ) {\Lambda _1}(\alpha ) \times {\Lambda _\infty }(\beta ) has a regular basis, (vi) Λ 1 ( α ) ⊗ Λ ∞ ( β ) {\Lambda _1}(\alpha ) \otimes {\Lambda _\infty }(\beta ) has a regular basis. Also the finite type power series spaces that contain subspaces isomorphic to an infinite type power series space are characterized as well as the infinite type power series spaces that have finite type quotient spaces.

Basic sequences in stable infinite type power series spaces

Basic sequences in stable infinite type power series spaces

Quotientenr�ume von stabilen Potenzreihenr�umen endlichen Typs

In this article we characterize the quotient Kothe-spaces of a stable finite type power series space. The characterization is in terms of a nuclearity condition and a new topological property\(\bar \Omega \).

Basis sequences in a stable finite type power series space

Basis sequences in a stable finite type power series space

Borel summability and indeterminacy of the Stieltjes moment problem: Application to the anharmonic oscillators

An indeterminacy criterion is proven for the moment problem associated with the coefficients of a Borel summable power series of Stieltjes type which diverge faster than (2n) !. As an application we show that the Stieltjes type continued fraction corresponding to the Rayleigh–Schrödinger perturbation expansions for the energy eigenvalues of the anharmonic oscillators (x2(m+1) and in any finite number of dimensions) does not converge to the eigenvalues if m≳2. In particular, this implies the nonconvergence of the Padé approximants to the eigenvalues of p2+x2+λx2(m+1) if m≳2.

Higher dimensional polynomials of binomial type and formal power series inversion

Higher dimensional polynomials of binomial type and formal power series inversion

Compact maps and embeddings from an infinite type power series space to a finite type power series space.

Compact maps and embeddings from an infinite type power series space to a finite type power series space.

An Extended Padé Procedure for Constructing Global Approximations from Asymptotic Expansions: An Explication with Examples

The problem of generating approximations to functions, specified by asymptotic expansions about one or more points, is considered. The development is limited to expansions of the power series type ...

Infinite type power series subspaces of infinite type power series spaces

Infinite type power series subspaces of infinite type power series spaces

Convergence and divergence of series conjugate to a convergent multiple Fourier series

In this note we consider to what extent the classical theorems of Plessner and Kuttner comparing the set of convergence of a trigonometric series with that of the conjugate trigonometric series can be generalized to higher dimensions. We show that if a function belongs to L p , p > 1 {L^p},p > 1 , of the 2 2 -torus, then the convergence (= unrestricted rectangular convergence) of the Fourier series on a set implies its three conjugate functions converge almost everywhere on that set. That this theorem approaches the best possible may be seen from two examples which show that the dimension may not be increased to 3, nor the required power of integrability be decreased to 1. We also construct a continuous function having a boundedly divergent Fourier series of power series type and an a.e. circularly convergent double Fourier series whose y y -conjugate diverges circularly a.e. Our L p {L^p} result depends on a theorem of L. Gogöladze (our proof is included for the reader’s convenience), work of J. M. Ash and G. Welland on ( C , 1 , 0 ) (C,1,0) summability, and on a result deducing the boundedness of certain partial linear means from convergence of those partial means. The construction of the counterexamples utilizes examples given by C. Fefferman, J. Marcinkiewicz, A. Zygmund, D. Menšov, and the present authors’ earlier work.

Padé approximation and gaussian quadrature

For certain types of formal power series, including the series of Stieltjes, we prove that the [n, n+j], j ≥ −1, Padé approximants coincide with certain gaussian quadrature formulae and hence, convergence of these approximants follows immediately.

Infinite type power series subspaces of finite type power series spaces

We consider the problem of finding all power series subspaces of a given infinite type power series space. A necessary condition is obtained which is similar to a property of complemented subspaces of nuclear Frechet spaces. In some cases, a complete solution is obtained and this leads to new information on a conjecture of C. Bessaga.

A divergent multiple Fourier series of power series type

A divergent multiple Fourier series of power series type

Interaction Of Newly Defined Stress IntensityFactors For Angular Corners In Two Diamond-Shaped Inclusions

This paper deals with a row of equally spaced equal diamond-shaped inclusions with angular corners under various loading conditions. The problem is formulated as a system of singular integral equations with Cauchy-type singularities, where the unknown functions are the densities of body forces distributed in infinite plates having the same elastic constants of the matrix and inclusions. In order to analyze the problems accurately, the unknown functions of the body force densities are expressed as a linear combination of two types of fundamental density functions and power series, where the fundamental density functions are chosen to represent the symmetric stress singularity of \(1/r^{1 - \lambda _1 } \) and the skew-symmetric stress singularity of \(1/r^{1 - \lambda _2 } \). Then, newly defined stress intensity factors for angular corners are systematically calculated for various shapes, spacings, elastic constants and numbers of the diamond-shaped inclusions in a plate subjected to uniaxial tension, biaxial tension and in-plane shear. For all types of diamond-shaped inclusions, the stress intensity factor is shown to be linearly related to the reciprocal of the number of diamond-shaped inclusions.

Magnetohydrodynamic Flows in Closed Regions with Complete Symmetry about an Axis

A formal perturbational technique was developed to obtain solutions for magnetohydrodynamic flow problems in closed regions with symmetric properties of flow and boundary conditions about an axis. The system of highly nonlinear partial differential equations, governing the problems of this class, was found to reduce to systems of simultaneous linear partial differential equations, thus creating a possibility of obtaining solutions by classical mathematical methods. The technique used was based on a well-known power series type of perturbational expansion in terms of a small parameter, which in the limit was assumed to become the solution of the original system. The general technique was applied to the magnetohydrodynamic flow in a toroid under the influence of a uniform applied magnetic field along the direction of the main axis of the toroid. It was found that the secondary flow caused by the centrifugal body force on the fluid element due to the curvature created induced electromagnetic components and a pressure variation along the radial direction. The solutions were carried out to include completely the second perturbational equations and the electromagnetic and flow fields were discussed for different values of characteristic parameters appearing in the solutions.

The Use of Volterra Series to Find Region of Stability of a Non-linear Differential Equation†

ABSTRACT This report discusses a special problem which illustrates a new approach for discussing stability of non-linear systems under arbitrary input disturbances. The system is assumed to be described by a non-linear differential equation with forcing term. This equation is solved by 8 type of power series (Volterra series) which gives output explicitly in terms of input. Proof of convergence of this series gives criteria for stability of the system.

LIV. The summation of a particular type of power series

LIV. The summation of a particular type of power series