Singular semi-linear equations inL 1(R)

Letg be a positive continuous function onR which tends to zero at −∞ and which is not integrable overR. The boundary-value problem −u″+g(u)=f, u′(±∞)=0, is considered forf∈L ¹(R). We show that this problem can have a solution if and only ifg is integrable at −∞ and if this is so then the problem is solvable precisely when ∫ _−∞ ^∞ . Some extensions of this result are also given. Sponsored by the United States Army under Contract No. DAAG29-75-C-0024 and by the National Science Foundation, Grant MPS 75-05501.     

On strong perturbations of the haar system inL 1(0,1)-space

In the paper we consider a Haar system perturbed in the sense of theL ₁(0,1)-metric. We prove that this perturbation is stronger than perturbations in the case of basis stability or stability of complete systems; moreover, the systems obtained as the result of a perturbation are complete inL ₁(0,1). An approximation algorithm inL ₁(0,1) for these systems is given. Translated fromMatematicheskie Zametki, Vol. 66, No. 4, pp. 596–602, October, 1999.     

Nonstandard finite-dimensional analogs of operators inL 2 (R n)

Gorkii. Translated from Sibirskii Matematicheskii Zhurnal, Vol. 29, No. 2, pp. 45–59, March–April, 1988.     

Semi-classical behaviour of the scattering phase for trapping perturbations of the laplacian

We study the influence of trapping trajectories on the semi-classical asymptotic of the scattering phase (spectral shift function), s_n(λ) associated to Schrodinger operators, if λ is a non critical energy level. Some results are known when the set of closed trajectories has Liouville measure 0 and in the case of a potential well. In this paper, for smooth and sufficiently decreasing potentials, we describe the behaviour of s_n(λ+rh) in terms of the continuity properties of a certain oscillating function Q(h,r). This function. intl.oduced by V. Petkov and G. Popov to study clustering of eigenvalues, is related to the periodic trajectories. If Q(h,r) is uniformly contiuuous in r for any h ε[0,h₀], we obtain a TVeyl type asynlptotic of s_n(λ+rh), with a second term containing Q(h,r). On the other hand, the point of discontinuity of Q(h, r) in r may give rise to a "clustering" phenomena for the scattering phase. Hence, in contrast to non-trapping case, the derivative d/dλ (s_h) can have no complete asymptotic expansion.     

Weak compactness inL 1(λ) and injective banach spaces

The isomorphic embedding of the Banach spacel ⁱ(Γ) into injective Banach spaces is investigated.     

The algebraic radical of a normed ideal inL 1(G)

LetA be a proper normed ideal (in the sense ofCigler) insideL ¹ (G), whereG is a non-discrete LCA group. This is proved: For each integern1 there existsfL ¹ (G) such thatf, f ² ,..., f ^nA whilef ⁿ⁺¹ A.     

Monotone subsequences in (0, 1)-matrices

A (0, 1)-matrix contains anS ₀(k) if it has 0-cells (i, j ₁), (i + 1,j ₂),..., (i + k – 1,j _k) for somei andj ₁ < ... < j_k, and it contains anS ₁(k) if it has 1-cells (i ₁,j), (i ₂,j + 1),...,(i _k ,j + k – 1) for somej andi ₁ < ... <i _k . We prove that ifM is anm × n rectangular (0, 1)-matrix with 1 m n whose largestk for anS ₀(k) isk ₀ m, thenM must have anS ₁(k) withk m/(k ₀ + 1). Similarly, ifM is anm × m lower-triangular matrix whose largestk for anS ₀(k) (in the cells on or below the main diagonal) isk ₀ m, thenM has anS ₁(k) withk m/(k ₀ + 1). Moreover, these results are best-possible.     

Polynomial approximation inL p (0<p<1)

We prove that forf∈L _p , 0<p<1, andk a positive integer, there exists an algebraic polynomialP _n of degree ≤n such that $$\left\| {f - P_n } \right\|_p \leqslant C\omega _k^\varphi \left( {f,\frac{1}{n}} \right)_p $$ whereω _k ^? (f,t)p is the Ditzian-Totik modulus of smoothness off inL _p , andC is a constant depending only onk andp. Moreover, iff is nondecreasing andk≤2, then the polynomialP _n can also be taken to be nondecreasing.     

Hereditary order convexity inL(X,Y)

LetX be a topological vector space,Y an ordered topological vector space andL(X,Y) the space of all linear and continuous mappings fromX intoY. The hereditary order-convex cover [K] ^h of a subsetK ofL(X,Y) is defined by [K] ^h ={A∈L(X,Y):Ax∈[Kx] for allx∈X}, where[Kx] is the order-convex ofKx. In this paper we study the hereditary order-convex cover of a subset ofL(X,Y). We show how this cover can be constructed in specific cases and investigate its structural and topological properties. Our results extend to the spaceL(X,Y) some of the known properties of the convex hull of subsets ofX ^*.     

A class of extreme contractions inL(l p )

Summary In this note we prove that: if T is a contraction in L(l^p) that maps elements of disjoint support to elements of disjoint support, then T is an extreme point of the unit ball of L(l^p), 1

i) form a p-orthonormal sequence or the nonzero elements of (y_i) form a p-orthonormal sequence for which supp (y_i)=N.     

Average σ-K width of convolution function class ofL pq R inL q R

In this paper, we study the average n−K width of the convolution classB (G) (orB which the kernel G(x) is a PF density, in the metricR (orR and obtain some exact results. The author was supported by the National Natural Science Found of China.     

Representation of measures with simultaneous polynomial denseness inL p (R,dμ), 1≤p<∞

We give characterisations of certain positive finite Borel measures with unbounded support on the real axis so that the algebraic polynomials are dense in all spacesL _p (R,dμ),p≥1. These conditions apply, in particular, to the measures satisfying the classical Carleman conditions. This work was completed while A. Bakan was visiting Würzburg University, supported by the German Academic Exchange Service (DAAD). S. Ruscheweyh received partial support from the German-Israeli Foundation (grant G-643-117.6/1999) and from INTAS (Project 99-00089).