Some functional inequalities and their Baire category properties

Summary In the paper we consider, from a topological point of view, the set of all continuous functionsf:I I for which the unique continuous solution:I – [0, ) of(f(x)) (x, (x)) and(x, (x)) (f(x)) (x, (x)), respectively, is the zero function. We obtain also some corollaries on the qualitative theory of the functional equation(f(x)) = g(x, (x)). No assumption on the iterative behaviour off is imposed.     

Estimation of density functionals

Given a sequence of independent random variables with densityf we estimate quantities of the form = (f(x))dx, a known function, by inserting histograms and kernel density estimators for the unknownf. We obtain conditions for consistency and asymptotic normality and discuss the choice of cell size and bandwidth.     

Decay of Concentration Functions for Adapted Probabilities on Discrete Groups

Given a probability measure, , on a locally compact group, necessary and sufficient support conditions which ensure that the concentration functions associated with converge to zero have previously been determined. In this note the rate of this convergence when is adapted on a discrete group G is shown to depend on the volume growth rate of N , the smallest normal subgroup a coset of which contains the support of .     

Product integration with the Clenshaw-Curtis points: Implementation and error estimates

Summary This paper is concerned with the practical implementation of a product-integration rule for approximating , wherek is integrable andf is continuous. The approximation is , where the weightsw _ni are such as to make the rule exact iff is any polynomial of degree n. A variety of numerical examples, fork(x) identically equal to 1 or of the form |–x| with >–1 and ||1, or of the form cosx or sinx, show that satisfactory rates of convergence are obtained for smooth functionsf, even ifk is very singular or highly oscillatory. Two error estimates are developed, and found to be generally safe yet quite accurate. In the special casek(x)1, for which the rule reduces to the Clenshaw-Curtis rule, the error estimates are found to compare very favourably with previous error estimates for the Clenshaw-Curtis rule.     

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.     

Posterior Sensitivity to the Sampling Distribution and the Prior: More than One Observation

Sensitivity of a posterior quantity (f, P) to the choice of the sampling distribution f and prior P is considered. Sensitivity is measured by the range of (f, P) when f and P vary in nonparametric classes ^f and ^P respectively. Direct and iterative methods are described which obtain the range of (f, P) over f ^f when prior P is fixed, and also the overall range over f ^f and P ^P . When multiple i.i.d. observations X ₁,...,X _k are observed from f, the posterior quantity (f, P) is not a ratio-linear function of f. A method of steepest descent is proposed to obtain the range of (f, P). Several examples illustrate applications of these methods.     

Finite difference methods and their convergence for a class of singular two point boundary value problems

Summary We discuss the construction of three-point finite difference approximations and their convergence for the class of singular two-point boundary value problems: (x y)=f(x,y), y(0)=A, y(1)=B, 0<<1. We first establish a certain identity, based on general (non-uniform) mesh, from which various methods can be derived. To obtain a method having order two for all (0,1), we investigate three possibilities. By employing an appropriate non-uniform mesh over [0,1], we obtain a methodM ₁ based on just one evaluation off. For uniform mesh we obtain two methodsM ₂ andM ₃ each based on three evaluations off. For =0,M ₁ andM ₂ both reduce to the classical second-order method based on one evaluation off. These three methods are investigated, theirO(h ²)-convergence established and illustrated by numerical examples.     

On discrete and continuous norms in Paley-Wiener spaces and consequences for exponential frames

It is well known that for certain sequences {t_n}_n the usual L^p norm ·_p in the Paley-Wiener space PW ^p is equivalent to the discrete norm f_p,{tn}:=( _n=– |f(t_n)|^p)^1/p for 1 p = < and f_,{tn}:=sup _n|f(t_n| for p=). We estimate f_p from above by Cf_p, _n and give an explicit value for C depending only on p, , and characteristic parameters of the sequence {t_n}_n. This includes an explicit lower frame bound in a famous theorem of Duffin and Schaeffer.     

A generalization of a theorem of M. Riesz to the case of functions of several variables

The following theorem was proved by M. Riesz: Iff(x) L(–,),f(x) 0 and the conjugate functionf (x) is also integrable on [-, ], thenf(x) L log⁺L. The analog of this theorem for functions of several variables is established.Translated from Matematicheskie Zametki, Vol. 4, No. 3, pp. 269–280, November, 1968.     

On the order of approximation of a continuous 2π-periodic function by Fejer and Poisson means of its Fourier series

Let _n (f) and P_r (f) be, respectively, the Fejer and Poisson means of the Fourier series of the functionf. The present work considers problems associated with the rapidity of approximation of a continuous 2-periodic function by means of Fejer and Poisson processes, and gives, in particular, an upper bound to the deviation of the Fejer and Poisson processes from the function in terms of moduli of continuity, and a lower bound to _n (f)–f in terms of functionals composed of best approximations to the functionf; in addition, some relationships among the quantities P_r (f)–f and _n (f)–f are established.Translated from Matematicheskie Zametki, Vol. 4, No. 1, pp. 21–32, July, 1968.     

SharpL ∞-error estimates for semilinear elliptic problems with free boundaries

Summary A numerical scheme to approximate a semilinear PDE involving a (singular) maximal monotone graph is analyzed inL . A preliminary regularization is combined with piecewise linear finite elements defined on a triangulation which is not assumed to be acute; the discrete maximum principle is thus avoided. Sharp pointwise error estimates are derived for both the smoothing and the discretization procedures. An optimal choice of the regularization parameter as a function of the mesh size leads to a sharp global rate of convergence. These error estimates for solutions, in conjunction with nondegeneracy properties of continuous problems, provide sharp interface error estimates. Two model examples are discussed: the obstacle problem and a combustion equation.This work was partially supported by Consiglio Nazionale delle Ricerche of Italy while the author was in residence at the Istituto di Analisi Numerica del C.N.R. di Pavia     

A remark on Nehari's problem

An example of a strickly contractive Hankel matrix is given such that the central contractive extensionf _c of satisfies f _c =1. This way we answer a problem raised by Ciprian Foias.Partially supported by a Georgia State University Research Grant.     

Numerical application of Euler's series transformation and its generalizations

Summary A nonlinear generalizationÊ _z of Euler's series transformation is compared with the (linear) Euler-Knopp transformationE _z and a twoparametric methodE . It is shown how to applyE orE _, to compute the valuef(z_o) of a functionf from the power series at 0 iff is holomorphic in a half plane or in the cut plane. BothE andE _, are superior toÊ _z . A compact recursive algorithm is given for computingE andE _,.     

Regularity of ∫Ω|∇u|2 + λ∫Ω|u−f{2 and some gap phenomenon

We study the regularity of the minimizer u for the functional F (u,f)=|u|² + |u–f{² over all maps uH ¹(, S ²). We prove that for some suitable functions f every minimizer u is smooth in if ₀ and for the same functions f, u has singularities when is large enough.

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Résumé On étudie la régularité des minimiseurs u du problème de minimisation min_ueH ¹(,S²)(|u|² + |u–f{². On montre que pour certaines fonctions f, u est régulière lorsque ₀ et pour les mêmes f, si est assez grand, alors u possède des singularités.

     

Riemann sums for stochastic integrals and L p moduli of continuity

Summary The Brownian stochastic integral for a deterministic function f can be approximated by Riemann sums . Here is a partion of [0,1], and f is obtained by replacing f by its mean value in each interval of . In an earlier paper [3], the author determined which Lipschitz conditions on f imply f db f db a.s. as becomes fine. This is now done for L ^p Lipschitz conditions. The results are stated in terms of Besov spaces. Nonanticipating are considered when p = l. Further, we determine on which L ^p Lipschitz spaces f db defines a bounded linear functional.     

Best approximation by splines on classes of periodic functions in the metric of L

We have obtained the exact value of the upper bound on the best approximations in the metric of L on the classes W^rH of functionsf C ₂ ^r for which ¦f ^(r) (x)-f ^(r) (x)) ¦ <(¦ x-xf) [ (t) is the upwards-convex modulus of continuity] by subspaces of r-th order polynomial splines of defect 1 with respect to the partitioning k/n.Translated from Matematicheskie Zametki, Vol. 20, No. 5, pp. 655–664, November, 1976.     

On a class of composite functional equations in a single variable

Summary We prove a general result on continuous functions of the typef: (0, ) (0, ) which satisfy the functional equationf(x[f(x)]^p) = (f(x))^{p + 1}, wherep is an arbitrary fixed real number. Applying this result we determine all continuous solutionsf: [0, ) [0, ) forp > 0, as well as all the continuous solutionsf: for a positive integerp. Forp = 1 this equation is relevant to a division model of population.     

Measures not charging semipolars and equations of schrödinger type

SupposeX is a Borel right process andm is a -finite excessive measure forX. Given a positive measure not chargingm-semipolars we associate an exact multiplicative functionalM(). No finiteness assumptions are made on . Given two such measures and ,M()=M() if and only if and agree on all finely open measurable sets. The equation (q–L)u+u=f whereL is the generator of (a subprocess of)X may be solved for appropriatef by means of the Feynman-Kac formula based onM(). Both uniqueness and existence are considered.Supported in part by NSF Grant DMS 92-24990.     

L ∞ stability of finite element approximations to elliptic gradient equations

Summary We examine theL stability of piecewise linear finite element approximationsU to the solutionu to elliptic gradient equations of the form –·[a(x)u]+f(x, u)=g(x) wheref is monotonically increasing inu. We identify a prioriL bounds for the finite element solutionU, which we call reduced bounds, and which are marginally weaker than those for the original differential equations. For the general,N-dimensionai, case we identify new conditions on the mesh, such that under the assumption thatf is Lipschitz continuous on a finite interval,U satisfies the reducedL bounds mentioned above. The new,N-dimensional regularity conditions preclude quasi-rectangular meshes.Moreover, we show thatU is stable inL in two dimensions for a discretization mesh on which –·[a(x)u] gives rise to anM-matrix, whileU is stable for any mesh in one dimension. The condition that the discretization of –·[a(x)u] has to be anM-matrix, still allows the inclusion of the important case of triangulating in a quasi-rectangular fashion.The results are valid for either the pure Neumann problem or the general mixed Dirichlet-Neumann boundary value problem, while interfaces may be present. The boundary conditions forU are obtained by use of (nonexpansive) pointwise projection operators.The first author is supported by the National Science Foundation under grant EET-8719100Research of the second author supported by National Science Foundation grant DMS.8420192     

On the equation of similar profiles

Summary Considerf+ ff+ (1–f²)+ f=0 together with the boundary conditionsf(0)=f(0)=0,f ()=1. If=–1,>0, arbitrary there is at least one solution which satisfies 0<f<1 on (0, ). By the additional conditionf>0 on (0, ) or, alternately 0<1, the uniqueness of the solution is demonstrated.If=1,<0, arbitrary the existence of solutions for which –1<f<0 in some initial interval (0,t) and satisfying generallyf>1 is established. In both problems, bounds forf (0) and qualitative behavior of the solutions are shown.

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Sommario Si consideri il problema definito dall'equazionef+ f f+ (1–f²)+ f=0 e dalle condizioni al contornof(0)=f (0)=0,f()=1. Assumendo=–1,>0, arbitrario si dimostra che esiste almeno una soluzione che soddisfa 0<f<1 nell'intervallo (0, ). Se in aggiunta si ipotizzaf>0 in (0, ), oppure 0<=1, l'unicità délia soluzione è assicurata.Successivamente si considéra il problema di valori al contorno con=1,<0, arbitrario. In questo caso esiste un'intera classe di soluzioni che soddisfano –1<f<0 in un intorno dell'origine e tali chef>1, in generale.Di detti problemi viene studiato il comportamento délle soluzioni e vengono determinate dalle maggiorazioni e minorazioni del valoref(0).