Regularized spectral distributions

For C a bounded, injective operator with dense image, we define a C-regularized spectral distribution. This produces a functional calculus, f f(B), from C() into the space of closed densely defined operators, such that f(B)C is bounded when f has compact support. As an analogue of Stone's theorem, we characterize certain regularized spectral distributions as corresponding to generators of polynomially bounded C-regularized groups. We represent the regularized spectral distribution in terms of the regularized group and in terms of the C-resolvent. Applications include the Schrödinger equation with potential, and symmetric hyperbolic systems, all on L^p(ⁿ) (1p<), C _o(ⁿ), BUC(ⁿ), or any space of functions where translation is a bounded strongly continuous group.     

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.     

Schatten class hankel operators on the Bergman space

In this paper we characterize Hankel operatorsH _f andH _f on the Bergman spaces of bounded symmetric domains which are in the Schatten p-class for 2p< and f inL ² using a Jordan algebra characterization of bounded symmetric domains and properties of the Bergman metric.     

A note on Davis type error bounds

Bounds for the error of the approximation of the value of a bounded linear functionalUf by the weighted sumU _n f= _k ^=0/n–1 a _k f (x _k ) in Hilbert function spacesH(B) based upon the Davis method are considered. Taking into account that functionsf H (B) withUf=U _n f are known, improved error bounds are computed. Applying the explicit formulas numerical examples are treated and comparisons are made.     

On the Decay of Infinite Products of Trigonometric Polynomials

We consider infinite products of the form f(=_k=1 m _k(2^-k), where {m _k} is an arbitrary sequence of trigonometric polynomials of degree at most n with uniformly bounded norms such that m _k(0)= 1 for all k. We show that f() can decrease at infinity not faster than O(^-n) and present conditions under which this maximal decay is attained. This result can be applied to the theory of nonstationary wavelets and nonstationary subdivision schemes. In particular, it restricts the smoothness of nonstationary wavelets by the length of their support. This also generalizes well-known similar results obtained for stable sequences of polynomials (when all m _k coincide). By means of several examples, we show that by weakening the boundedness conditions one can achieve exponential decay.     

Strong Cesàro summability of double Fourier integrals

We prove the following theorem. Assume f ∈ L ^∞(R ²) with bounded support. If f is continuous at some point (x ₁,x ₂) ∈ R ², then the double Fourier integral of f is strongly q-Cesàro summable at (x ₁,x ₂) to the function value f(x ₁,x ₂) for every 0 < q < ∞. Furthermore, if f is continuous on some open subset of R ², then the strong q-Cesàro summability of the double Fourier integral of f is locally uniform on . Research partially supported by the Australian Research Council and the Hungarian National Foundation for Scientific Research under Grant T 046 192.     

Sukzessive Minima mit und ohne Nebenbedingungen

Given semi-normsf andg on ⁿ and a real number >0. Then the successive minima off under the constraintg are defined by _j : = inf {: there existj linear independent vectors inZ ⁿ withf andg}. The main theorem of this paper (Lagrange multiplier theorem) states that the successive minima of a certainnorm h on ⁿ (without constraints) coincide with the _j 's up to bounded factors. Moreover, this norm is constructed explicitly. Using Minkowski's wellknown theorem on successive minima and our result certain inequalities on simultaneous Diophantine approximations are derived.     

Integral Operators in Sobolev Spaces on Domains with Boundary

Properties of integral operators with weak singularities arc investigated. It is assumed that G ? ?ⁿ is a bounded domain. The boundary δG should be smooth concerning the Sobolev trace theorem. It will be proved that the integral operators $\int {_G \frac{{f\left(\Theta \right)}}{{x - y|^{n - 1} }}u\left(\nu \right)d\partial G_\nu }$ and $ \int {_{\partial G} \frac{{f\left(\Theta \right)}}{{|x - y|^{n - 1} }}u\left(y \right)d\partial G_y }$ maps W_p^k(G) into W_p^k+1(G) and W_p^k?1(G) into W_p^k/p(G), respectively, and are bounded. Here θ ∈ S ? ?ⁿ, where S is the unit sphere. Furthermore, f possesses bounded first order derivatives and is bounded on S. Then applications to first order systems are discussed.     

On monotone and convex approximation by algebraic polynomials

The following results are obtained: If >0, 2, [3, 4], andf is a nondecreasing (convex) function on [–1, 1] such thatE _n (f) n ^– for any n>, then E _n ⁽¹⁾ (f)Cn ^– (E _n ⁽²⁾ (f)Cn ^–) for n>, where C=C(), E_n(f) is the best uniform approximation of a continuous function by polynomials of degree (n–1), and E _n ⁽¹⁾ (f) (E _n ⁽²⁾ (f)) are the best monotone and convex approximations, respectively. For =2 ( [3, 4]), this result is not true.Published in Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 9, pp. 1266–1270, September, 1994.     

Optimal Constant in Approximation by Bernstein Operators

We obtained that for any n N, C = 1 is the smallest constant for which the inequality ||B _n (f) - f|| C ₂(f, 1/n) holds on the class of continuous functions f, as well as on the class of bounded functions f, where B _n is the Bernstein operators of degree n, ₂ is the second order modulus and || || is the sup-norm.     

Upper bounds for theL p -norms of martingales

Summary Let (f _n ) be a martingale. We establish a relationship between exponential bounds for the probabilities of the typeP(|f _n |>·T(f _n )) and the size of the constantC _p appearing in the inequality f ^* _p C _p T ^*(f) _p , for some quasi-linear operators acting on martingales.This research was supported in part by NSF Grant, no. DMS-8902418On leave from Academy of Physical Education, Warsaw, Poland     

On the greatest length of a dead-end disjunctive normal form for almost all Boolean functions

The maximal numberl(f) of conjunctions in a dead-end disjunctive normal form (d.n.f.) of a Boolean functionf and the number (f) of dead-end d.n.f. are important parameters characterizing the complexity of algorithms for finding minimal d.n.f. It is shown that for almost all Boolean functionsl(f)2^n–1, log₂ (f)2^n–1log₂nlog₂log₂n (n).Translated from Matematicheskie Zametki, Vol. 4, No. 6, pp. 649–658, December, 1968.     

Some Convergence Properties of Descent Methods

In this paper, we discuss the convergence properties of a class of descent algorithms for minimizing a continuously differentiable function f on R ⁿ without assuming that the sequence { x _k } of iterates is bounded. Under mild conditions, we prove that the limit infimum of is zero and that false convergence does not occur when f is convex. Furthermore, we discuss the convergence rate of { } and { f(x _k )} when { x _k } is unbounded and { f(x _k )} is bounded.     

Approximation by Reciprocals of Polynomials with Positive Coefficients in L p Spaces

We prove that, if f(x) L ^p _[0,1], 1 < p < , f(x) 0, x [0,1], f 0, then there is a polynomial p(x) ⁺ _n such that f - 1/p _LP C(p)(f,n ^-1/2) _LP where ⁺ _n indicates the set of all polynomials of degree n with positive coeficients (see the definition (1) in the text).     

On the best approximation in the metric of L to certain classes of functions by Haar-system polynomials

Let H, H ^L be classes of functionsf(x) whose modulus of continuity (f; t) and, respectively, integral modulus of continuity(f; t)_L do not exceed a given modulus of continuity(t), while H_v is a class of functionsf(x) whose variation fdoes not exceed a given number V > 0. Bounds are obtained for the upper limit of the best approximations in the metric of L by Haar-system polynomials on the classes just introduced (on the class H ^L only when (t)=Kt). These bounds are exact for class H_V and, in case(t) is convex, also for the classes H and H ^L .Translated from Matematicheskie Zametki, Vol. 6, No. 1, pp. 47–54, July, 1969.The author wishes to thank N. P. Korneichuk for having posed the problem and for his constant attention to this work.     

On problems with bounded state variables

A set of first-order necessary conditions is obtained for the general control problem of Bolza with bounded state constraints of the form (t, x)0, =1,...,m. With the solution required to satisfy the vector differential equationsx=f(t, x, u), whereu is control, an important feature of this paper is in relaxing the assumption on the rank of the matrix _x f _u generally made in attacking problems of this type. This is accomplished even though the solution may have an infinite number of intervals satisfying (t, x)=0 for various .The preparation of this paper was sponsored in part by the US Army Research Office, Grant No. DA-31-214-ARO(D)-355.     

On the Problem of Describing Sequences of Best Trigonometric Rational Approximations

For a strictly decreasing sequence a_{n n=0} of nonnegative real numbers converging to zero, we construct a continuous 2-periodic function f such that R^T _n(f) = a_n, n=0,1,2,..., where R^T _n(f) are best approximations of the function f in uniform norm by trigonometric rational functions of degree at most n.     

The Lê varieties,I

Summary For a complex polynomial,f:( ⁿ⁺¹ ,0) (, 0), with a singular set of complex, dimensions at the origin, we define a sequence of varieties—the Lê varieties, _f ^(k) , off at 0. The multiplicities of these varieties, _f ^(k) , generalize the Milnor number for an isolated singularity. In particular, we show that ifsn-2, the Milnor, fibre off is obtained fromB ²ⁿ by successively attaching _f ^{(n – k)} k-handles, wheren-skn Ifs=n-1, the Milnor fibre off is obtained from a2n-manifold with the homotopy type of a bouquet of _f ^{(n – 1)} circles by successively attaching _f ^{(n – k)} k-handles, where 2kn.The author is a National Science Foundation, Postdoctoral Research Fellow supported by grant # DMS-8807216     

Differentiability of functions: approximate,global, and differentiability along curves over non-archimedean fields

This paper is devoted to the study of approximate and global smoothness and smoothness along curves of functions f(x ₁,...,x _m ) of variables x ₁,...,x _m in infinite fields with nontrivial non-Archimedean valuations and relations between them. Theorems on classes of smoothness C ⁿ or of functions with partial difference quotients continuous or bounded uniformly continuous on bounded domains up to order n are investigated. We prove that from f ○ u ∈ C ⁿ (K, K ^l) or f ○ u ∈ (K, K ^l) for each C ^∞ or curve u: K → K ^m it follows that f ∈ C ⁿ (K ^m , K ^l) or f ∈ (K ^m , K ^l), where m ≥ 2. Then the classes of smoothness C ^n,r and and more general in the sense of Lipschitz for partial difference quotients are considered and theorems for them are proved. Moreover, the approximate differentiability of functions relative to measures is defined and investigated. Its relations with the Lipschitzian property and almost everywhere differentiability are studied. Non-Archimedean analogs of classical theorems of Kirzsbraun, Rademacher, Stepanoff, and Whitney are formulated and proved, and substantial differences between two cases are found. Finally, theorems about relations between approximate differentiability by all variables and along curves are proved. Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 52, Functional Analysis, 2008.     

On the monotonicity of sequences of quadrature formulae

Summary LetI(f)L(f)= _k=0 ^r ₌₀ ^vk–1 a _k f ⁽⁾(X _k ) be a quadrature formula, and let {S _n (f)} _n=1 be successive approximations of the definite integralI(f)= ₀ ¹ f(x)dx obtained by the composition ofL, i.e.,S _n(f)=L( _n ), where .We prove sufficient conditions for monotonicity of the sequence {S _n (f)} _n=1 . As particular cases the monotonicity of well-known Newton-Cotes and Gauss quadratures is shown. Finally, a recovery theorem based on the monotonicity results is presented