Approximate first-order and second-order directional derivatives of a marginal function in convex optimization

Given a convex functionf: ^p × ^q (–, +], the marginal function is defined on ^p by (x)=inf{f(x, y)|y ^q }. Our purpose in this paper is to express the approximate first-order and second-order directional derivatives of atx ₀ in terms of those off at (x ₀,y ₀), wherey ₀ is any element for which (x ₀)=f(x ₀,y ₀).The author is indebted to one referee for pointing out an inaccuracy in an earlier version of Theorem 4.1.     

Weighted Composition Operators on Bergman and Dirichlet Spaces

Let H() denote a functional Hilbert space of analytic functions on a domain . Let w : C and : be such that w f is in H() for every f in H(). The operator wC given by f w f is called a weighted composition operator on H(). In this paper we characterize such operators and those for which (wC )* is a composition operator. Compact weighted composition operators on some functional Hilbert spaces are also characterized. We give sufficient conditions for the compactness of such operators on weighted Dirichlet spaces.     

Nonlinear programming,approximation, and optimization on infinitely differentiable functions

A nonnegative, infinitely differentiable function defined on the real line is called a Friedrichs mollifier function if it has support in [0, 1] and ₀ ¹ (t)dt=1. In this article, the following problem is considered. Determine _k =inf ₀ ¹ |^(k)(t)|dt,k=1, 2, ..., where ^(k) denotes thekth derivative of and the infimum is taken over the set of all mollifier functions , which is a convex set. This problem has applications to monotone polynomial approximation as shown by this author elsewhere. The problem is reducible to three equivalent problems, a nonlinear programming problem, a problem on the functions of bounded variation, and an approximation problem involving Tchebycheff polynomials. One of the results of this article shows that _k =k!2^2k–1,k=1, 2, .... The numerical values of the optimal solutions of the three problems are obtained as a function ofk. Some inequalities of independent interest are also derived.This research was supported in part by the National Science Foundation, Grant No. GK-32712.     

Fast Summation of Power Series with Coefficients Analytic at Infinity

We propose a fast summation algorithm for slowly convergent power series of the form _{j=j 0} z ^j _j j _i=1 ^s (j+ _i )^– _i , where R, _i 0 and _i C, 1is, are known parameters, and _j =(j), being a given real or complex function, analytic at infinity. Such series embody many cases treated by specific methods in the recent literature on acceleration. Our approach rests on explicit asymptotic summation, started from the efficient numerical computation of the Laurent coefficients of . The effectiveness of the resulting method, termed ASM (Asymptotic Summation Method), is shown by several numerical tests.     

Algorithmes de calcul du maximum des formes quadratiques sur la boule unité de la norme du maximum

Résumé Certaines méthodes directes et indirectes pour le calcul de Max {x ^t Ax, (x)1} sont étudiées.Les méthodes directes sont basées sur les propriétés particulières des normes ₁, ₂ et . Ces méthodes sont très simples mais ne s'appliquent qu'à certaines familles de matrices.La méthode indirecte est la méthode autoduale introduite dans [25, 26] avec = ₁. Dans ce cas, le choix du vecteur initial pour qu'il y ait convergence vers une solution optimale est largement discuté.

---

Some methods for computing the maximum of quadratic from on the unit ball of the maximum norm

Summary Some direct and indirect methods are studied for computing Max {x ^t Ax, (x)1} whereA is symmetric definite positive.Direct methods are constructed using particular properties of ₁, ₂, norms. These methods are very simple, but uniquely suitable to certains families of matrices.The indirect method is the autodual method, introduced in [25, 26, 29] with = ₁. In this case the problem of choosing an initial vector so that convergence of the iterative sequence occurs to an optimal solution is largely discussed.

     

Surjectivity of quotient maps for algebraic (ℂ,+)-actions and polynomial maps with contractible fibers

In this paper we establish two results concerning algebraic (,+)-actions on ⁿ . First, let be an algebraic (,+)-action on ³. By a result of Miyanishi, its ring of invariants is isomorphic to [t ₁,t ₂]. Iff ₁,f ₂ generate this ring, the quotient map of is the mapF:³²,x(f ₁(x), f₂(x)). By using some topological arguments we prove thatF is always surjective. Secon, we are interested in dominant polynomial mapsF: ^{n n-1} whose connected components of their generic fibers are contractible. For such maps, we prove the existence of an algebraic (,+)-action on ⁿ for whichF is invariant. Moreover we give some conditions so thatF*([t ₁,...,t _n-1 ]) is the ring of invariants of .Dedicated to all my friends and my family     

Adjoints of slant Toeplitz operators

In this paper, we will use the Birkhoff's ergodic theorem to do some finer analysis on the spectral properties of slant Toeplitz operators. For example, we will show that if is an invertibleL function on the unit circle, then almost every point in (A ^* ) is not an eigenvalue ofA ^* . More specifically, we will show that the point spectrum ofA ^* is contained in a circle with positive radius.     

Characterizations of BMOφ(w)

In this paper we give two characterizations of functions with weighted mean oscillation over cubes controlled by a nonnegative function , that is, functions in BMO(w). The first characterization is formulated in terms of rearrangements, while the second one by means of Riesz transforms and -Lipschitz functions. These results extend those contained in [S] and [J].     

φ-Factors for Function Seminorms

Let (, A, ) be a measure space, a function seminorm on M, the space of measurable functions on , and M the space {f M : (f) < }. Every Borel measurable function : [0, ) [0, ) induces a function : M M by (f)(x) = (|f(x)|). We introduce the concepts of -factor and -invariant space. If is a -subadditive seminorm function, we give, under suitable conditions over , necessary and sufficient conditions in order that M be invariant and prove the existence of -factors for . We also give a characterization of the best -factor for a -subadditive function seminorm when is -finite. All these results generalize those about multiplicativity factors for function seminorms proved earlier.     

Interpolation theorems for a family of spanning subgraphs

Let G be a graph with order p, size q and component number . For each i between p – and q, let be the family of spanning i-edge subgraphs of G with exactly components. For an integer-valued graphical invariant if H H is an adjacent edge transformation (AET) implies |(H)-(H^')|1 then is said to be continuous with respect to AET. Similarly define the continuity of with respect to simple edge transformation (SET). Let M _j() and m _j() be the invariants defined by . It is proved that both M _p–() and m _p–(;) interpolate over , if is continuous with respect to AET, and that M _j() and m _j() interpolate over , if is continuous with respect to SET. In this way a lot of known interpolation results, including a theorem due to Schuster etc., are generalized.     

Universalities

We show that a quotient category of the category of all topological spaces and all open continuous mappings contains an isomorphic copy of every category as a full subcategory. We construct a functorF : K K universal in the following sense: for every functorH : H ₁ H ₂ (H ₁,H ₂ arbitrary) there exist full one-to-one functors _i :H _i K such thatF o ₁ = ₂ oH (the construction proceeds in a more general setting of enriched categories).     

On the decomposition theorem for meromorphic Fredholm resolvents

In this note results of B. Gramsch and W. Kaballo [8] on the decomposition of meromorphic (semi-) Fredholm resolvents are sharpened. A condition on an Orlicz function is given, under which the singular part in this decomposition can be chosen meromorphic inN , the ideal of -nuclear operators. Then the necessity of this condition is studied. Moreover, it is shown that for the rather steep Orlicz functions relevant to this question,N equalsS , the ideal of -approximable operators.Dedicated to Professor Albert Schneider on the occasion of his 60 ^th birthdayresearch supported by a grant from DAAD     

Weighted Bergman Spaces and Carleson Measures on the Unit Ball of Cn

Let be a normal function on [0, 1), B _n the unit ball of C ⁿ , and A ^p (B _n ) the weighted Bergman spaces on B _n with weight . The purpose of this paper is to discuss some relations among A ^p (B _n ), weighted Bergman kernels, and Carleson measures on B _n .     

Continuity of Approximation by Neural Networks in Lp Spaces

Devices such as neural networks typically approximate the elements of some function space X by elements of a nontrivial finite union M of finite-dimensional spaces. It is shown that if X=L ^p () (1<p< and R ^d ), then for any positive constant and any continuous function from X to M, f–(f)>f–M+ for some f in X. Thus, no continuous finite neural network approximation can be within any positive constant of a best approximation in the L ^p -norm.     

Asymptotic formulas for Toeplitz and Wiener-Hopf operators

For certain analytic functions f, the expression trace {f(T_n[]) – f(T_n[]T_n[])} is computed asymptotically. Here T_n[] is the finite Toeplitz matrix generated by the function . The analogous expression for Wiener-Hopf operators is also computed asymptotically. These results in turn yield information concerning the asymptotic behavior of determinants of finite Toeplitz and Wiener-Hopf operators with discontinuous generating function.     

A functional equation of Ih-Ching Hsu

Summary A recent note of Ih-Ching Hsu poses an unsolved problem, to wit, the general solution of the functional equation g(x₁, x₂) + g(₁(x₁), ₂(x₂)) = g(x₁, ₂(x₂)) + g(₁(x₁),x₂), where the _i are given functions. This short paper obtains the general solution. It gives conditions which imply that anycontinuous solution has form g₁(x₁) + g₂(x₂).     

Uniqueness of Nelson's Diffusions II: Infinite Dimensional Setting and Applications

Under mild condition on the modulus = of the time independent wave function , we prove that the generalized Schrödinger operator = + 2 (, ·)/ (or the generator of Nelson's diffusion) defined on a good space of test-functions on a general Polish space, generates a unique semigroup of class (C _o) in L ¹. This result reinforces the known results on the essential Markovian self-adjointness in different contexts and extends our previous works in the finite dimensional Euclidean space setting. In particular it can be applied to the ground or excited state diffusion associated with an usual Schr\"odinger operator , and to stochastic quantization of several Euclidean quantum fields.     

On some properties of functions of generalized variation

We show that the spaceV ^* generated by a -variation (or -variation of Riesz, or -variation of Waterman) forms a commutative Banach algebra with respect to the pointwise multiplication under the appropriate choice of norms.     

Some homological properties of commutative semitrivial ring extensions

LetR be a commutative ring with 1 andM anR-module. If:M _R MR is anR-module homomorphism satisfying(mm)=(mm) and(mm)m=m(mm), the additive abelian groupRM becomes a commutative ring, if multiplication is defined by (r,m)(r,m)=(rr+(mm),rm+rm). This ring is called the semitrivial extension ofR byM and and it is denoted byR M. This generalizes the notion of a trivial extension and leads to a more interesting variety of examples. The purpose of this paper is to studyR M; in particular, we are interested in some homological properties ofR M as that of being Cohen-Macaulay, Gorenstein or regular. A sample result: Let (R,m) be a local Noetherian ring,M a finitely generatedR-module and Im() m. ThenR M is Gorenstein if and only if eitherRM is Gorenstein orR is Gorenstein,M is a maximal Cohen-Macaulay module andMM ^*, where the isomorphism is given by the adjoint of.