A counterexample for a sup theorem in normed spaces

An example is given of a normed linear space that is not complete but for which each continuous linear functional attains its supremum on the unit ball.     

A counterexample to the Bishop-Phelps Theorem in complex spaces

The Bishop-Phelps Theorem asserts that the set of functionals which attain the maximum value on a closed bounded convex subsetS of a real Banach spaceX is norm dense inX ^*. We show that this statement cannot be extended to general complex Banach spaces by constructing a closed bounded convex set with no support points.     

A counterexample for boundedness of pseudo-differential operators on modulation spaces

We prove that pseudo-differential operators with symbols in the class ( ) are not always bounded on the modulation space ().

     

A consistent counterexample in the theory of collectionwise Hausdorff spaces

It is shown to be consistent that there is a normal first countable locally countable space which is not collectionwise Hausdorff and in which there is a closed discrete non-G _δ set which provides the counterexample to collectionwise Hausdorffness. This answers a question of P. Nyikos. Publication 349, partially supported by the BSF.     

Almost orthogonal operators on the bitorus

The operators S _p f (x, y), for the sum of which we prove an L ²-estimate, act as a kind of Fourier coefficients on one variable and a kind of truncated Hilbert transforms with a phase N(x, y) on the other variable. This result is an extension to two-dimensions of an argument of almost orthogonality in Fefferman’s proof of a.e. convergence of Fourier series, under the basic assumption N(x, y) “mainly” a function of y and the additional assumption N(x, y) non-decreasing in x, for every y fixed.     

A counterexample to a weak-type estimate for potential spaces and tangential approach regions

We show that for every potential space , there exists an approach region for which the associated maximal function is of weak-type, but the boundedness for the completed region is false, which is in contrast with the nontangential case.

     

Weakly compact operators into sequence spaces: A counterexample

In a recent paper Gutiérrez and Villanueva have used, without giving a detailed proof, an analogue of a well-known result of Ryan characterizing the weakly compact operators from a Banach space into the space of null sequences in a Banach space . In this note a counterexample is given showing that in the statement of Gutiérrez and Villanueva an additional condition is needed.

     

Center manifolds and contractions on a scale of Banach spaces

We give a new proof for the existence of a C^k-center manifold at a nonhyperbolic equilibrium point of a finite-dimensional vector field of class C^k. The problem is reduced to a fixed point problem on a scale of Banach spaces; these Banach spaces consist of mappings with a certain maximal exponential growth at infinity. We give conditions under which there is a unique fixed point depending differentiably on the parameters; the main difficulty is that the mappings under consideration become only differentiable after composition with appropriate embeddings on the scale of Banach spaces.     

Almost orthogonal operators on the bitorus II

We prove that the operator ${Tf(x,y)=\int^\pi_{-\pi}\int_{|x^{\prime}|<|y^{\prime}|} \frac{e^{iN(x,y) x^{\prime}}}{x^{\prime}}\frac{e^{iN(x,y) y^{\prime}}}{y^{\prime}}f(x-x^{\prime}, y-y^{\prime}) dx^{\prime} dy^{\prime}}$ , with ${x,y \in[0,2\pi]}$ and where the cut off ${|x^{\prime}|<|y^{\prime}|}$ is performed in a smooth and dyadic way, is bounded from L ² to weak- ${L^{2-\epsilon}}$ , any ${\epsilon > 0 }$ , under the basic assumption that the real-valued measurable function N(x, y) is “mainly” a function of y and the additional assumption that N(x, y) is non-decreasing in x, for every y fixed. This is an extension to 2D of C. Fefferman’s proof of a.e. convergence of Fourier series of L ² functions.     

A counterexample to uniqueness and regularity for harmonic maps between hyperbolic spaces

We construct a harmonic diffeomorphism from the Poincaré ballH ⁿ⁼¹ to itself, whose boundary value is the identity on the sphereS ⁿ, and which is singular at a boundary point, as follows: The harmonic map equations between the corresponding upper-half-space models reduce to a nonlinear o.d.e. in the transverse direction, for which we prove the existence of a solution on the whole R₊ that grows exponentially near infinity and has an expansion near zero. A conjugation by the inversion brings the singularity at the origin, and a conjugation by the Cayley transform and an isometry of the ball moves the singularity at any point on the sphere.     

Positive matrix functions on the bitorus with prescribed Fourier coefficients in a band

Let S be a band in Z² bordered by two parallel lines that are of equal distance to the origin. Given a positive definite ¹ sequence of matrices {c_j}_jS we prove that there is a positive definite matrix function f in the Wiener algebra on the bitorus such that the Fourier coefficients equal c_k for k S. A parameterization is obtained for the set of all positive extensions f of {c_j}_jS. We also prove that among all matrix functions with these properties, there exists a distinguished one that maximizes the entropy. A formula is given for this distinguished matrix function. The results are interpreted in the context of spectral estimation of ARMA processes.     

Cauchy problem in a scale of Banach spaces

The concept of quasidifferential operator in a scale of Banach spaces is formulated. A theorem of existence and uniqueness of a solution to the Cauchy problem for the equation with a nonlinear quasidifferential operator is proved. As an example of application of the theorem, the correctness of the nonlinear nonlocal problem of plane-parallel unsteady potential motion of a liquid with free boundary is proved.     

Elliptic pseudodifferential operators in the improved scale of spaces on a closed manifold

We study linear elliptic pseudodifferential operators in the improved scale of functional Hilbert spaces on a smooth closed manifold. Elements of this scale are isotropic Hörmander-Volevich-Paneyakh spaces. We investigate the local smoothness of a solution of an elliptic equation in the improved scale. We also study elliptic pseudodifferential operators with parameter.     

On the proximinality of the unit ball of proximinal subspaces in Banach spaces: A counterexample

A known, and easy to establish, fact in Best Approximation Theory is that, if the unit ball of a subspace of a Banach space is proximinal in , then itself is proximinal in . We are concerned in this article with the reverse implication, as the knowledge of whether the unit ball is proximinal or not is useful in obtaining information about other problems. We show, by constructing a counterexample, that the answer is negative in general.

     

Growth conditions and regularity,a counterexample

It is shown that the so-called growth conditions are necessary for the local regularity of minimizers.This paper has been written while the author was visiting the Mathematisches Institut der Universität Bonn under the support of the Sonderforschungsberich 256.     

Elliptic boundary-value problem in a two-sided improved scale of spaces

We study a regular elliptic boundary-value problem in a bounded domain with smooth boundary. We prove that the operator of this problem is a Fredholm one in a two-sided improved scale of functional Hilbert spaces and that it generates there a complete collection of isomorphisms. Elements of this scale are Hörmander-Volevich-Paneyakh isotropic spaces and some their modi.cations. An a priori estimate for a solution is obtained and its regularity is investigated.     

Methods of evaluation and scale of functional spaces

Linear equations of mathematical physics with constant coefficients have fed calculational mathematics since the 18th century. The area of nonlinear equations with variable coefficients arose due to gas-hydrodynamic problems in the 20th century. Now, one of the methods of research for properties of solutions of such equations and, accordingly, applied problems is the use of calculations on modern computers. The capacities of computers and their efficiency have increased in the 21st century and allowed progress to be made in solving applied problems, except for cases of methodical errors in calculations. One of the basic sources of such methodical errors is the “uncontrollable” machine accuracy of calculations. One of the methods of solving such numerical problems is a suitable localization of the problem and the choice of an adequate basis of the necessary functional space. Below, we state new results in this area of mathematical and applied research.