Inversion of Vandermonde-Like Matrices

Let {x }_=0,...,n be a set of distinct nodes, and let {p _n } _n0 be a sequence of polynomials, degp _n =n, whose elements satisfy a three-term recurrence relation. We derive explicit representations for the inverse of a Vandermonde-like matrix V _n =(p (x ))_,=0,...,n and present some numerical results.     

On the free subset property at singular cardinals

We give a proof ofTheorem 1. Let be the smallest cardinal such that the free subset property Fr (, ₁)holds. Assume is singular. Then there is an inner model with ₁ measurable cardinals.     

Nondiagonal Hermite–Sobolev Orthogonal Polynomials

In this work, we study algebraic and analytic properties for the polynomials { Q _n } _{n 0}, which are orthogonal with respect to the inner product where , R such that – ² > 0.     

On a Modification of the Jacobi Linear Functional: Asymptotic Properties and Zeros of the Corresponding Orthogonal Polynomials

The paper deals with orthogonal polynomials in the case where the orthogonality condition is related to semiclassical functionals. The polynomials that we discuss are a generalization of Jacobi polynomials and Jacobi-type polynomials. More precisely, we study some algebraic properties as well as the asymptotic behaviour of polynomials orthogonal with respect to the linear functional U U=J _,+A ₁(x–1)+B ₁(x+1)–A ₂(x–1)–B ₂(x+1), where J _, is the Jacobi linear functional, i.e. J _,,p›=_–1 ¹ p(x)(1–x)(1+x)dx,,>–1, pP, and P is the linear space of polynomials with complex coefficients. The asymptotic properties are analyzed in (–1,1) (inner asymptotics) and C[–1,1] (outer asymptotics) with respect to the behaviour of Jacobi polynomials. In a second step, we use the above results in order to obtain the location of zeros of such orthogonal polynomials. Notice that the linear functional U is a generalization of one studied by T. H. Koornwinder when A ₂=B ₂=0. From the point of view of rational approximation, the corresponding Markov function is a perturbation of the Jacobi–Markov function by a rational function with two double poles at ±1. The denominators of the [n–1/n] Padé approximants are our orthogonal polynomials.     

Аппроксимация субга рмонических функций

The following statement is proved. Letu be a subharmonic function in the region and _u the associated measure. Then there exists a functionf holomorphic in and such that if _f is the associated measure of the function in ¦f¦, then ¦u(z)–ln¦f(z)¦ A¦ln s¦+B¦ln diam¦+ s(¦lns¦+1)+C. hold at every point z for which the setsD(z, t)={w: ¦w–z¦},t(0,s) lie in and satisfy(D(z, t))t both for= _u and for= _f . In the case where is an unbounded region, In diam should be replaced by ln ¦z¦. The constants, , do not depend on andu.

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Asymptotic Density and the Asymptotics of Partition Functions

Let A be a set of positive integers with gcd (A) = 1, and let p _A (n) be the partition function of A. Let c ₀ = 2/3. If A has lower asymptotic density and upper asymptotic density , then lim inf log p _A (n)/c ₀ n and lim sup log p _A (n)/c ₀ n . In particular, if A has asymptotic density > 0, then log p _A (n) c₀n. Conversely, if > 0 and log p _A (n) c ₀ n, then the set A has asymptotic density .     

On a hypothesis on Poincaré series

Let F(x₁,..., x_m) (m1) be a polynomial with integral p-adic coefficients, and let N, be the number of solutions of the congruence F(x₁,..., X_m)=0 mod A proof is given that the Poincaré series (t) = ₀ N t is rational for a class of isometrically-equivalent polynomials of m variables (m2) containing a form of degree n2 of two variables.Translated from Matematicheskie Zametki, Vol. 14, No. 3, pp. 453–463, September, 1973.The author wishes to thank N. G. Chudakov for discussing this paper and for his helpful advice.     

On a global descent method for polynomials

Summary Given a complex polynomialp we determine a functionf _p : such that |p(f _p (z))||p(z)|,z withk<1. This result is used to introduce a global root-finding algorithm for polynomials.     

Continued Fractions and Szegö Polynomials in Frequency Analysis and Related Topics

This paper is an expository survey of recent research on the application of Szegö polynomials and PPC-continuedfractions to the frequency analysis problem described as follows: We want to determine the unknown frequencies ₁, ₂, ..., _I from a sample of N observed values x _N (m), m = 0, 1, ..., N– 1, arising from a continuous waveform that is the superposition of a finite number of sinusoidal waves with frequencies ₁,₂, ..., _I . The method is based on the property that certain zerosof the Szegö polynomials (and poles of the PPC-fraction approximants) converge (as N ) to the frequency points e ⁱ _j , j = ±1, ± 2, ..., ± I. The remaining zeros are bounded away from the unit circle |z|=1, asN . The Levinson algorithm is used to construct the Szegö polynomials and PPC-fractions from the values x _N (m). A discussion is given on connections between the topics: Carathéodory functions,the trigonometric moment problem, Szegö polynomials and PPC-fractions. We also describe applications to Doppler radar, medicine, speech processing, speech therapy, meteorology and ocean tides.     

Compact interval spaces in which all closed subsets are homeomorphic to clopen ones,I

A topological space X whose topology is the order topology of some linear ordering on X, is called an interval space. A space in which every closed subspace is homeomorphic to a clopen subspace, is called a CO space. We regard linear orderings as topological spaces, by equipping them with their order topology. If L and K are linear orderings, then L ^*, L+K, L·K denote respectively the reverse orderings of L, the ordered sum of L and K and the lexicographic order on L×K (so ·2=+ and 2·=). Ordinals are considered as linear orderings, and cardinals are initial ordinals. For cardinals , 0, let L(, )= + 1 + ^* . Main theorem. Let X be a compact interval space. Then X is a CO space if and only if X is homeomorphic to a space of the form + 1 + _i L( _i , _i ), where is any ordinal, n, for every ii, _i are regular cardinals and _{i i}, and if n>0, then max({ _i: i}) · . This first part is devoted to show the following result. Theorem: If X is a compact interval CO space, then X is a scattered space (that means that every subspace of X has an isolated point).Supported by the Université Claude-Bernard (Lyon-1), the Ben Gurion University of the Negev, and the C.N.R.S.: UPR 9016Supported by the City of Lyon     

Moduli of smoothness and best approximation in the spacesL p , 0<p<1

L ^p , 0<<1, .     

Coproducts in the Category <Emphasis Type="Bold">M</Emphasis>κ<Emphasis Type="Bold">Frm</Emphasis>

The purpose of this paper is to study coproducts in the category MFrm (resp. M_cFrm), of metric -frames and uniform (resp. contractive) -frame maps. First, by applying the same technic that was used to find coproducts in Frm, we construct coproducts in the category Frm of -frames and -frame maps. Then, we define a metric diameter on the coproduct in Frm of a family of metric -frames and show that coproduct in Frm preserves metrizability.Mathematics Subject Classifications (2000) 06B23, 06D22, 18A30.     

Solving systems of polynomial inequalities over a real closed field in subexponential time

Let _m= (₁,..., _m) denote an ordered field, where _i+1>0 is infinitesimal relative to the elements of _i, 0 < –i < m (by definition, ₀= ). Given a system of inequalities f₁ > 0, ..., f_s > 0, f_s+1 0, ..., f_k 0, where f_{j m} [X₁,..., X_n] are polynomials such that, and the absolute value of any integer occurring in the coefficients of the f_js is at most 2^M. An algorithm is constructed which tests the above system of inequalities for solvability over the real closure of _m in polynomial time with respect to M, ((d)ⁿd₀)^n+m. In the case _m=, the algorithm explicitly constructs a family of real solutions of the system (provided the latter is consistent). Previously known algorithms for this problem had complexity of the order ofM(d d ₀ ^{m 2U(n)} .Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Maternaticheskogo Instituta im. V. A. Steklova Akad. Nauk SSSR, Vol. 174, pp. 3–36, 1988.     

Tangential Limits of Potentials on Homogeneous Trees

Let T be a homogeneous tree of homogeneity q+1. Let denote the boundary of T, consisting of all infinite geodesics b=[b ₀,b ₁,b ₂,] beginning at the root, 0. For each b, 1, and a0 we define the approach region _,a (b) to be the set of all vertices t such that, for some j, t is a descendant of b _j and the geodesic distance of t to b _j is at most (–1)j+a. If >1, we view these as tangential approach regions to b with degree of tangency . We consider potentials Gf on T for which the Riesz mass f satisfies the growth condition _T f ^p (t)q ^–|t|<, where p>1 and 0<<1, or p=1 and 0<1. For 11/, we show that Gf(s) has limit zero as s approaches a boundary point b within _,a (b) except for a subset E of of -dimensional Hausdorff measure 0, where H (E)=sup_>0inf _i q ^{–|t i|}:E a subset of the boundary points passing through t _i for some i,|t _i |>log _q (1/).     

Homogenization of a Singularly Perturbed Parabolic Problem in a Thick Periodic Junction of the Type 3:2:1

We prove a convergence theorem and obtain asymptotic (as 0) estimates for a solution of a parabolic initial boundary-value problem in a junction that consists of a domain ₀ and a large number N ² of -periodically located thin cylinders whose thickness is of order = O(N ^–1).     

The existence of probability measures with specified projections

Let X and Y be locally compact-compact topological spaces, F X×Y is closed, and P(F) is the set of all Borel probability measures on F. For us to find, for the pair of probability measures (_x, _y P (X)×P(Y), a probability measure P(F) such that _X = _X ^–1 , _Y = _Y ^–1 it is necessary and sufficient that, for any pair of Borel sets A X, B Y for which (A× B) F=Ø, the condition _XA+ _YB 1 holds.Translated from Matematicheskie Zametki, Vol. 14, No. 4, pp. 573–576, October, 1973.     

Laguerre Polynomials, Restriction Principle, and Holomorphic Representations of SL(2,R)

The restriction principle is used to implement a realization of the holomorphic representations of SL(2,R) on L ² (R ⁺,t dt) by way of the standard upper half plane realization. The resulting unitary equivalence establishes a correspondence between functions that transform according to the character e^–i(2n++1); under rotations and the Laguerre polynomials. The standard recursion relations amongst Laguerre polynomials are derived from the action of the Lie algebra.     

Orthogonality properties of linear combinations of orthogonal polynomials

Let {P _n } be a sequence of orthogonal polynomials with respect to the measured on the unit circle and letP _n =P _n + _j ^=1l _nj P _n–j fornl, where _n,j . It is shown that the sequence of linear combinations {P _n },n2l, is orthogonal with respect to a positive measured if and only ifd is a Bernstein-Szegö measure andd is the product of a unique trigonometric polynomial and the Bernstein-Szegö measured. Furthermore for a given sequence ofP _n 's an algorithm for the calculation of the _n,j 's is provided.Supported by Dirección General de Investigación Cientifica y Técnica (DGICYT) of Spain and Österreichischer Akademischer Austauschdienst of Austria with grant 4B/1995.Also supported by the Austrian Fonds zur Förderung der wissenschaftlichen Forschung, project-number P9267-PHY.     

Upper bounds of best one-sided approximations of the classes WrLΨ in the metric of L

The lowest upper bound is obtained for best one-sided approximations of classes (r=1,2 ...) by trigonometric polynomials and splines of minimum deficiency with equidistant knots, in the metric of space L, where W^rL={f:f(x+2)=f(x), f^(r–1)(x) is absolutely continuous, f ^(r)L 1} and L is an Orlicz space.Translated from Matematicheskie Zametki, Vol. 22, No. 2, pp. 257–267, August, 1977.     

Das 0-1-Gesetz der terminalen σ-Algebra bei Harrisirrfahrten

Für homogene M.K., bestehend aus einer aperiodischen rekurrenten Klasse, ist bekannt, da\ die terminale -Algebra das 0–1-Gesetz erfüllt. Der erste Teil dieser Arbeit behandelt die Gültigkeit des 0–1-Gesetzes für transiente Ketten, für die eine Folge ₀, ₁, ₂, ₃,... von ZustÄnden existiert mit P(nX_n= _l+1¦X₀= _l) = 1, N. Das herzuleitende Kriterium wird nur von dieser Folge und den Stoppzeiten, von _l nach _l+1 zu gelangen, abhÄngen. Eine Anwendung auf Harrisirrfahrten wird uns dort konkrete Aussagen liefern.