Approximation by rational functions in integral metrics and differentiability in the mean

The paper deals with approximations of a functionf of space L_p[0, 1] by rational functions in the metric of this same space (0n(f, p) of functionf of rational functions of degree no higher than n is evidence of the presence inf of derivatives and differentials of a definite order if differentiation is understood as differentiation in the metric of space L_q[0, 1], with 0相似文献   

Weak convergence of linear forms in D[0, 1]

Convergence in probability of the linear forms Σ_k=1^∞a_nkX_k is obtained in the space D[0, 1], where (X_k) are random elements in D[0, 1] and (a_nk) is an array of real numbers. These results are obtained under varying hypotheses of boundedness conditions on the moments and conditions on the mean oscillation of the random elements (X_n) on subintervals of a partition of [0, 1]. Since the hypotheses are in general much less restrictive than tightness (or convex tightness), these results represent significant improvements over existing weak laws of large numbers and convergence results for weighted sums of random elements in D[0, 1]. Finally, comparisons to classical hypotheses for Banach space and real-valued results are included.     

On pairs of commuting derivations of the polynomial ring in one or two variables

It is well known that each pair of commuting linear operators on a finite dimensional vector space over an algebraically closed field has a common eigenvector. We prove an analogous statement for derivations of k[x] and k[x,y] over any field k of zero characteristic. In particular, if D₁ and D₂ are commuting derivations of k[x,y] and they are linearly independent over k, then either (i) they have a common polynomial eigenfunction; i.e., a nonconstant polynomial f∈k[x,y] such that D₁(f)=λf and D₂(f)=μf for some λ,μ∈k[x,y], or (ii) they are Jacobian derivations

     

Dependence of variables construed as an atomic formula

We define a logic D capable of expressing dependence of a variable on designated variables only. Thus D has similar goals to the Henkin quantifiers of [4] and the independence friendly logic of [6] that it much resembles. The logic D achieves these goals by realizing the desired dependence declarations of variables on the level of atomic formulas. By [3] and [17], ability to limit dependence relations between variables leads to existential second order expressive power. Our D avoids some difficulties arising in the original independence friendly logic from coupling the dependence declarations with existential quantifiers. As is the case with independence friendly logic, truth of D is definable inside D. We give such a definition for D in the spirit of [11] and [2] and [1].     

Divergence of the greedy algorithm in the Faber-Schauder system on a continuum cardinality set

The paper establishes that there exist a continuum cardinality set E ₀ ? [0, 1] and a function f ₀(x) ∈ C _[0,1], such that the greedy algorithm of f ₀(x) with respect to the Faber-Schauder system converges to +∞ at all points of E ₀.     

A note on approximation properties of q-Durrmeyer operators

In this paper, the approximation properties of q-Durrmeyer operators D_n,q(f;x) for f∈C[0,1] are discussed. The exact class of continuous functions satisfying approximation process lim_n→∞D_n,q(f;x)=f(x) is determined. The results of the paper provide an elaboration of the previously-known ones on operators D_n,q.     

A Two-Dimensional Bisection Envelope Algorithm for Fixed Points

In this paper we present a new algorithm for the two-dimensional fixed point problem f(x)=x on the domain [0, 1]×[0, 1], where f is a Lipschitz continuous function with respect to the infinity norm, with constant 1. The computed approximation x satisfies 6f(x)−x6_∞⩽ε for a specified tolerance ε<0.5. The upper bound on the number of required function evaluations is given by 2⌈log₂(1/ε)⌉+1. Similar bounds were derived for the case of the 2-norm by Z. Huang et al. (1999, J. Complexity15, 200–213), our bound is the first for the infinity norm case.     

Outer Γ-Convexity in Vector Spaces

A subset S of some vector space X is said to be outer Γ-convex w.r.t. some given balanced subset Γ ? X if for all x ₀, x ₁ ? S there exists a closed subset Λ ? [0,1] such that {x _λ | λ ? Λ} ? S and [x ₀, x ₁] ? {x _λ | λ ? Λ} + 0.5 Γ, where x _λ: = (1 ? λ)x ₀ + λ x ₁. A real-valued function f:D → ? defined on some convex D ? X is called outer Γ-convex if for all x ₀, x ₁ ? D there exists a closed subset Λ ? [0,1] such that [x ₀, x ₁] ? {x _λ | λ ? Λ} + 0.5 Γ and f(x _λ) ≤ (1 ? λ)f(x ₀) + λ f(x ₁) holds for all λ ? Λ. Outer Γ-convex functions possess some similar optimization properties as these of convex functions, e.g., lower level sets of outer Γ-convex functions are outer Γ-convex and Γ-local minimizers are global minimizers. Some properties of outer Γ-convex sets and functions are presented, among others a simplex property of outer Γ-convex sets, which is applied for establishing a separation theorem and for proving the existence of modified subgradients of outer Γ-convex functions.     

Positive solution of singular Dirichlet boundary value problems for second order differential equation system

This paper investigates the existence of positive solutions of singular Dirichlet boundary value problems for second order differential system. A necessary and sufficient condition for the existence of C[0,1]×C[0,1] positive solutions as well as C¹[0,1]×C¹[0,1] positive solutions is given by means of the method of lower and upper solutions and the fixed point theorems. Our nonlinearity fi(t,x₁,x₂) may be singular at x₁=0, x₂=0, t=0 and/or t=1, i=1,2.     

Traces of functions with majorizable derivatives

We study the limiting values (y→+0) of functionsf (x, y), x ε R_n, y > 0, for which ¦?f/?y¦≤M?(y), ¦?f/?x_k¦≤Mψk(y), M=M [f], in the case of arbitrary weight functions. It is shown that the space of traces can be described as the set of all functionsf (x, 0) which satisfy a Lipschitz condition in some metricω(x, x) associated with the weights.     

Potential theory associated with Uhlenbeck-Ornstein process

Some parallel results of Gross' paper (Potential theory on Hilbert space, J. Functional Analysis1 (1967), 123–181) are obtained for Uhlenbeck-Ornstein process U(t) in an abstract Wiener space (H, B, i). Generalized number operator $\text{N}$ is defined by $\text{N}$f(x) = ?lim_∈←0{E[f($\text{U}$(τ^∈_ξ))] ? f(x)}/E[τ^∈_ξ, where τ_x^? is the first exit time of U(t) starting at x from the ball of radius ? with center x. It is shown that $\text{N}$f(x) = ?trace D²f(x)+〈Df(x),x〉 for a large class of functions f. Let r_t(x, dy) be the transition probabilities of U(t). The λ-potential G_λf, λ > 0, and normalized potential Rf of f are defined by G_λf(X) = ∫₀^∞e^?λtr_tf(x) dt and Rf(x) = ∫₀^∞ [r_tf(x) ? r_tf(0)] dt. It is shown that if f is a bounded Lip-1 function then trace D²G_λf(x) ? 〈DG_λf(x), x〉 = ?f(x) + λG_λf(x) and trace D²Rf(x) ? 〈DRf(x), x〉 = ?f(x) + ∫_Bf(y)p₁(dy), where p₁ is the Wiener measure in B with parameter 1. Some approximation theorems are also proved.     

Realcompactness of hyperspaces and extensions of open-closed maps

In response to questions of Ginsburg [9, 10], we prove that if cf(c)>ω₁, then there exists an open-closed, continuous map f from a normal, realcompact space X onto a space Y which is not realcompact. By his result the hyperspace 2^x of closed subsets of X is then not realcompact, and the extension μf(vf) of f to the topological completion (the Hewitt realcompactification) of X is not onto. The latter fact solves problems raised by Morita [16] and by Isiwata [12] both negatively. We also consider the problem whether or not the hyperspace of a hereditarily Lindelöf space is hereditarily realcompact.     

Derivations and right ideals of algebras

Let R be a K-algebra acting densely on V_D, where K is a commutative ring with unity and V is a right vector space over a division K-algebra D. Let ρ be a nonzero right ideal of R and let f(X₁,…,X_t) be a nonzero polynomial over K with constant term 0 such that μR≠0 for some coefficient μ of f(X₁,…,X_t). Suppose that d:R→R is a nonzero derivation. It is proved that if rankd(f(x₁,…,x_t))?m for all x₁,…,x_t∈ρ and for some positive integer m, then either ρ is generated by an idempotent of finite rank or d=ad(b) for some b∈End(V_D) of finite rank. In addition, if f(X₁,…,X_t) is multilinear, then b can be chosen such that rank(b)?2(6t+13)m+2.     

Factorization of a convolution-type operator

Three convolution-type equations are considered in the space of entire functions with topology ofd uniform convergence: $$\begin{gathered} M{_{\mu}{_1}} [f] \equiv \smallint _C f(z + t)d\mu _1 = 0, \hfill \\ M{_\mu{_1}} [f] \equiv \smallint _C f(z + t)d\mu _2 = 0, \hfill \\ M_\mu [f] \equiv \smallint _C f(z + t)d\mu = 0 \hfill \\ \end{gathered}$$ with respective characteristic functions L₁(λ), L₂(λ), L(λ)=L₁(λ)· L₂(λ), suppμ ?c, suppμ ₁ ?c, suppμ ₂ ?c. The necessary and sufficient conditions are found that every solutionf(z) of the equation M_μ[f[ can be written as a sumf ₁(z) +f ₂(z), wheref ₁(z) is the solution of the equation \(M{_\mu{_1}} [f] = 0\) ,f ₂(z) is the solution of the equation \(M{_\mu{_2}} [f] = 0\) .     

Nonemptiness of Approximate Subdifferentials of Midpoint δ-Convex Functions

ABSTRACT

For some given positive δ, a function f:D ? X → ? is called midpoint δ-convex if it satisfies the Jensen inequality f[(x ₀ + x ₁)/2] ≤ [f(x ₀) + f(x ₁)]/2 for all x ₀, x ₁ ∈ D satisfying ‖x ₁ ? x ₀‖ ≥ δ (Hu, Klee, and Larman, SIAM J. Control Optimiz. Vol. 27, 1989). In this paper, we show that, under some assumptions, the approximate subdifferentials of midpoint δ-convex functions are nonempty.     

On the mean oscillation of the Hessian of solutions to the Monge-Ampère equation

We give interior a priori estimates for the mean oscillation of second derivatives of solutions to the Monge-Ampère equation detD²u=f(x) with zero boundary values, where f(x) is a non-Dini continuous function. If the modulus of continuity of f(x) is φ(r) such that limr→0φ(r)log(1/r)=0, then D²u∈VMO.     

A Characterization of the free n-generated MV-algebra

An MV-algebra A=(A,0,¬,⊕) is an abelian monoid (A,0,⊕) equipped with a unary operation ¬ such that ¬¬x=x,x⊕¬0=¬0, and y⊕¬(y⊕¬x)=x⊕¬(x⊕¬y). Chang proved that the equational class of MV-algebras is generated by the real unit interval [0,1] equipped with the operations ¬x=1?x and x⊕y=min(1,x+y). Therefore, the free n-generated MV-algebra Free _n is the algebra of [0,1]-valued functions over the n-cube [0,1] ⁿ generated by the coordinate functions ξ _i ,i=1, . . . ,n, with pointwise operations. Any such function f is a McNaughton function, i.e., f is continuous, piecewise linear, and each piece has integer coefficients. Conversely, McNaughton proved that all McNaughton functions f: [0,1] ⁿ →[0,1] are in Free _n . The elements of Free _n are logical equivalence classes of n-variable formulas in the infinite-valued calculus of ?ukasiewicz. The aim of this paper is to provide an alternative, representation-free, characterization of Free _n .     

Schauder estimates for a degenerate second order elliptic operator on a cube

In the present article we are concerned with a class of degenerate second order differential operators LA,b defined on the cube d[0,1], with d?1. Under suitable assumptions on the coefficients A and b (among them the assumption of their Hölder regularity) we show that the operator LA,b defined on C²(d[0,1]) is closable and its closure is m-dissipative. In particular, its closure is the generator of a C₀-semigroup of contractions on C(d[0,1]) and C²(d[0,1]) is a core for it. The proof of such result is obtained by studying the solvability in Hölder spaces of functions of the elliptic problem λu(x)−LA,bu(x)=f(x), x∈d[0,1], for a sufficiently large class of functions f.     

Theorems on asymptotics for singular Sturm-Liouville operators with various boundary conditions

We consider the Sturm-Liouville operator L(y) = ?d ² y/dx ² + q(x)y in the space L ₂[0, π], where the potential q(x) is a complex-valued distribution of the first order of singularity; i.e., q(x) = u′(x), u ∈ L ²[0, π]. (Here the derivative is understood in the sense of distributions.) We obtain asymptotic formulas for the eigenvalues and eigenfunctions of the operator in the case of the Neumann-Dirichlet conditions [y ^[1](0) = 0, y(π) = 0] and Neumann conditions [y ^[1](0) = 0, y ^[1](π) = 0] and refine similar formulas for all types of boundary conditions. The leading and second terms of asymptotics are found in closed form.