CMV Matrices and Little and Big ?1 Jacobi Polynomials

We introduce a new map from polynomials orthogonal on the unit circle to polynomials orthogonal on the real axis. This map is closely related to the theory of CMV matrices. It contains an arbitrary parameter ?? which leads to a linear operator pencil. We show that the little and big ?1?Jacobi polynomials are naturally obtained under this map from the Jacobi polynomials on the unit circle.     

D_1 Spaces and Their Metrization

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1 — Consistency and faithful interpretations

Archive for Mathematical Logic -     

Hamiltonian Oscillators in 1—1—1 Resonance: Normalization and Integrability

Received July 16, 1998; accepted August 7, 1999     

Precipitous ideals and Σ 1 4 setssets

We prove under the assumption of the existence of a measurable, cardinal and precipitous ideal onw ¹ that every Σ ₁ ³ set is Lebesgue measurable, has the Baire property and is either countable or contians a perfect subset. We get similar results for Σ ₁ ⁴ sets, if we add the additional assumptions of C. H. and that carries a normal precipitous ideal.     

Univalent Functions and Diff（S^1）/S^1

The relation between Diff(S ¹)/S ¹ and the space of univalent analytic functions on the disk is elucidated and shown to provide upper bounds for the volumes of exhaustive approximations to an analytic submanifold of an infinite-dimensional space. The maximum magnitudes of the coefficients in the series expansions of univalent superanalytic functions on the superdisk are inferred.     

Frobenius Partitions and the Combinatorics of Ramanujan's 1ψ1 Summation

We examine the combinatorial significance of Ramanujan's famous summation. In particular, we prove bijectively a partition theoretic identity which implies the summation formula.     

Consistency statements and iterations of computable functions in IΣ1 and PRA

In this paper we will state and prove some comparative theorems concerning PRA and IΣ₁. We shall provide a characterization of IΣ₁ in terms of PRA and iterations of a class of functions. In particular, we prove that for this class of functions the difference between IΣ₁ and PRA is exactly that, where PRA is closed under iterations of these functions, IΣ₁ is moreover provably closed under iteration. We will formulate a sufficient condition for a model of PRA to be a model of IΣ₁. This condition is used to give a model-theoretic proof of Parsons’ theorem, that is, IΣ₁ is Π₂-conservative over PRA. We shall also give a purely syntactical proof of Parsons’ theorem. Finally, we show that IΣ₁ proves the consistency of PRA on a definable IΣ₁-cut. This implies that proofs in IΣ₁ can have non-elementary speed up over proofs in PRA.     

Gibbs–butzer derivatives and their applications 1

A series of results on Gibbs derivatives and their applications made by the group of modern harmonic analysts and approximation theorists in Nanjing University is presented in this paper. It is a survey of these results concerned with certain locally compact groups obtained by that group. Moreover, some new results are also included.     

Hardy and Bellman Transformations in Spaces Close to L ∞ and L 1

The replacement of coefficients of a trigonometric series by their arithmetic averages gives rise to the Hardy operator. The Bellman operator is its adjoint. The spaces _Lp with p[1,) are invariant under the Hardy transformation. This result was proved by Hardy. On the other hand, the space L is not invariant under the Hardy transformation, and the space _L1 is not invariant under the Bellman transformation. Golubov proved that the space BMO is not invariant under the Hardy transformation and is not invariant under the Bellman operator. In the present paper, the exact ``shift' of the domain under the action of these operators is described for certain Orlicz, Lorenz, and Marcinkiewicz spaces and the spaces BMO and . For the Hardy operator, this shift occurs if the domain is close to L , and for the Bellman operator the same happens if the domain is close to L ₁. Bibliography: 15 titles.     

Convex polynomial and spline approximation in C[−1, 1]

We prove that a convex functionf C[–1, 1] can be approximated by convex polynomialsp _n of degreen at the rate of ₃(f, 1/n). We show this by proving that the error in approximatingf by C² convex cubic splines withn knots is bounded by ₃(f, 1/n) and that such a spline approximant has anL third derivative which is bounded by n³₃(f, 1/n). Also we prove that iff C²[–1, 1], then it is approximable at the rate ofn ^–2 (f, 1/n) and the two estimates yield the desired result.Communicated by Ronald A. DeVore.     

P_1-Compact Mappings and Leary-Schauder Boundary Condition

Let X be a Banach space and let Γ_a={X_n,P_n} be a projectionally complete scheme. Let D be a bounded open subset of X and T:D→X an P_lcompact mapping undel the condition weaker than Leary Schauder boundarycondition, we show several fixed point theorems of T and consider he relation between the fixed points and eigenvectors of T.     

Shadowing and Inverse Shadowing for C^1 Endomorphisms

In this paper, we consider the shadowing and the inverse shadowing properties for C^1 endomorphisms. We show that near a hyperbolic set a C^1 endomorphism has the shadowing property, and a hyperbolic endomorphism has the inverse shadowing property with respect to a class of continuous methods. Moreover, each of these shadowing properties is also ＂uniform＂ with respect to C^1 perturbation.     

On the Volume Formulas of Cones and Orthogonal Multi-cones in S~n(1) and H~n(-1)

In the study of n-dimensional spherical or hyperbolic geometry, n ≥3, the volume of various objects such as simplexes, convex polytopes, etc. often becomes rather difficult to deal with. In this paper, we use the method of infinitesimal symmetrization to provide a systematic way of obtaining volume formulas of cones and orthogonal multiple cones in Sn(l) and Hn(-1).     

Special representations of the groups U(∞, 1) and O(∞, 1) and the associated representations of the current groups U(∞, 1) X and O(∞, 1) X in quasi-poisson spaces

A method for constructing representations of the current groups O(n, 1) ^X and U(n, 1) ^X , n ε ℕ, developed in the previous papers by the authors is generalized to the case of infinite n. This leads to an interesting difference in construction (absent for finite n) between the cases of the orthogonal and unitary groups, which is due to the different character of special representations of the groups of coefficients.     

On the Volume Formulas of Cones and Orthogonal Multi-cones in S^n（1） and H^n（-1）

Abstract In the study of n-dimensional spherical or hyperbolic geometry, n≥ 3, the volume of various objects such as simplexes, convex polytopes, etc. often becomes rather difficult to deal with. In this paper, we use the method of infinitesimal symmetrization to provide a systematic way of obtaining volume formulas of cones and orthogonal multiple cones in Sⁿ(1) and Hⁿ(—1). (Dedicated to the memory of Shiing-Shen Chern)     

Fractional 0–1 programming: applications and algorithms

We consider a class of nonlinear integer optimization problems commonly known as fractional 0–1 programming problems (also, often referred to as hyperbolic 0–1 programming problems), where the objective is to optimize the sum of ratios of affine functions subject to a set of linear constraints. Such problems arise in diverse applications across different fields, and have been the subject of study in a number of papers during the past few decades. In this survey we overview the literature on fractional 0–1 programs including their applications, related computational complexity issues and solution methods including exact, approximation and heuristic algorithms.     

Legendre polynomials and Ramanujan-type series for 1/π

We resolve a family of recently observed identities involving 1/π using the theory of modular forms and hypergeometric series. In particular, we resort to a formula of Brafman which relates a generating function of the Legendre polynomials to a product of two Gaussian hypergeometric functions. Using our methods, we also derive some new Ramanujan-type series.