A space <Emphasis Type="Italic">C</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>(<Emphasis Type="Italic">X</Emphasis>) is dominated by irrationals if and only if it is<Emphasis Type="Italic"> K</Emphasis>-analytic

Summary We prove that, for any Tychonoff X, the space C_p(X) is K-analytic if and only if it has a compact cover {K_p: p } such that K_p subset K_q whenever p,q and p q. Applying this result we show that if C_p(X) is K-analytic then C_p(X) is K-analytic as well. We also establish that a space C_p(X) is K-analytic and Baire if and only if X is countable and discrete.     

A Lower Bound of <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript> Norms in the CLT for Strongly Mixing Random Variables

We derive a lower bound of L _p norms, 1 ⩽ p ⩽ ∞, in the central limit theorem for strongly mixing random variables X ₁,..., X _n with under the boundedness condition ℙ{|X _i | ⩽ M} = 1 with a nonrandom constantM > 0 and condition ∑ _r⩾1 r ²α(r) < ∞, where α(r) are the Rosenblatt strong mixing coefficients. __________ Translated from Lietuvos Matematikos Rinkinys, Vol. 45, No. 4, pp. 587–602, October–December, 2005.     

Summability Kernels for <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> Multipliers

In this article we study the problem of extending Fourier Multipliers on L ^p (T) to those on L ^p (R) by taking convolution with a kernel, called a summability kernel. We characterize the space of such kernels for the cases p = 1 and p = 2. For other values of p we give a necessary condition for a function to be a summability kernel. For the case p = 1, we present properties of measures which are transferred from M(T) to M(R) by summability kernels. Furthermore it is shown that every l _p sequence can be extended to some L ^q (R) multipliers for certain values of p and q.     

The <Emphasis Type="Italic">s</Emphasis>-energy of spherical designs on <Emphasis Type="Italic">S</Emphasis><Superscript>2</Superscript>

This paper investigates the s-energy of (finite and infinite) well separated sequences of spherical designs on the unit sphere S ². A spherical n-design is a point set on S ² that gives rise to an equal weight cubature rule which is exact for all spherical polynomials of degree ≤n. The s-energy E _s (X) of a point set of m distinct points is the sum of the potential for all pairs of distinct points . A sequence Ξ = {X _m } of point sets X _m ⊂S ², where X _m has the cardinality card(X _m )=m, is well separated if for each pair of distinct points , where the constant λ is independent of m and X _m . For all s>0, we derive upper bounds in terms of orders of n and m(n) of the s-energy E _s (X _m(n)) for well separated sequences Ξ = {X _m(n)} of spherical n-designs X _m(n) with card(X _m(n))=m(n).      

Smooth noncompact operators from <Emphasis Type="Italic">C</Emphasis>(<Emphasis Type="Italic">K</Emphasis>), <Emphasis Type="Italic">K</Emphasis> scattered

Let X be a Banach space, K be a scattered compact and T: B _C(K) → X be a Fréchet smooth operator whose derivative is uniformly continuous. We introduce the smooth biconjugate T**: B _C(K)** → X** and prove that if T is noncompact, then the derivative of T** at some point is a noncompact linear operator. Using this we conclude, among other things, that either is compact or that ℓ₁ is a complemented subspace of X*. We also give some relevant examples of smooth functions and operators, in particular, a C ^1,u -smooth noncompact operator from B _{c O} which does not fix any (affine) basic sequence. P. Hájek was supported by grants A100190502, Institutional Research Plan AV0Z10190503.     

Maximal regularity for evolution equations in weighted <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>-spaces

Let X be a Banach space and let A be a closed linear operator on X. It is shown that the abstract Cauchy problem enjoys maximal regularity in weighted L _p -spaces with weights , where , if and only if it has the property of maximal L _p -regularity. Moreover, it is also shown that the derivation operator admits an -calculus in weighted L _p -spaces. Received: 26 February 2003     

Two variable <Emphasis Type="Italic">p</Emphasis>-adic <Emphasis Type="Italic">L</Emphasis> functions attached to eigenfamilies of positive slope

Consider a classical cusp eigenform f= _n=1 a _n (f)q ⁿ of weight k2 for ₀(N) with a Dirichlet character mod N, and let L _f (s,)= _n=1 (n)a _n (f)n ^-s denote the L-function of f twisted with an arbitrary Dirichlet character . For a prime number p5, consider a family of cusp eigenforms f _(k) of weight k , k {f _(k)= _n=1 a _n (f _(k))q ⁿ } containing f=f _(k), such that the Fourier coefficients a _n (f _(k)) are given by certain p-adic analytic functions k a _n (f _(k)). The purpose of this paper is to construct a two variable p-adic L function attached to Colemans family {f _(k)} of cusp eigenforms of a fixed positive slope =v _p ( _p )>0 where _p = _p (k ) is an eigenvalue (which depends on k ) of the Atkin operator U=U _p . Our p-adic L-function interpolates the special values L _f(k)(s,) at points (s,k ) with s=1,2,...,k -1. We give a construction using the Rankin-Selberg method and the theory of p-adic integration on a profinite group Y with values in an affinoid K-algebra A, where K is a fixed finite extension of Q _p . Our p-adic L-functions are p-adic Mellin transforms of certain A-valued measures. In their turn, such measures come from Eisenstein distributions with values in certain Banach A-modules M =M (N;A) of families of overconvergent forms over A. To Robert Alexander Rankin in memoriam     

Approximation of sine-shaped functions by constants in the spaces <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>, <Emphasis Type="Italic">p</Emphasis> < 1

We investigate the best approximations of sine-shaped functions by constants in the spaces L_p for p < 1. In particular, we find the best approximation of perfect Euler splines by constants in the spaces L_p for certain p(0,1).Translated from Ukrainskyi Matematychnyi Zhurnal, Vol. 56, No. 6, pp. 745–762, June, 2004.     

Point sets with low <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>-discrepancy

In this paper we study the L _p -discrepancy of digitally shifted Hammersley point sets. While it is known that the (unshifted) Hammersley point set (which is also known as Roth net) with N points has L _p -discrepancy (p an integer) of order (log N)/N, we show that there always exists a shift such that the digitally shifted Hammersley point set has L _p -discrepancy (p an even integer) of order which is best possible by a result of W. Schmidt. Further we concentrate on the case p = 2. We give very tight lower and upper bounds for the L ₂-discrepancy of digitally shifted Hammersley point sets which show that the value of the L ₂-discrepancy of such a point set mostly depends on the number of zero coordinates of the shift and not so much on the position of these. This work is supported by the Austrian Research Fund (FWF), Project P17022-N12 and Project S8305.     

Convexity and <Emphasis Type="Italic">w</Emphasis><Superscript><Emphasis Type="Italic">*</Emphasis></Superscript>-compactness in Banach spaces

Suppose X. We construct examples of bounded sets MX, such that These examples show that the previous results of the authors on quantitative versions of Kreins theorem are optimal.Mathematics Subject Classification (2000):46B20, 46B26A. S. Granero supported by DGICYT grant BFM2001-1284. P. Hájek supported by GAR 201/01/1198, A 101 92 05 and UPV grant PPI-02-02.     

<Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>-Boundedness of general index transforms

We establish the boundedness properties in L _p for a class of integral transformations with respect to an index of hypergeometric functions. In particular, by using the Riesz-Thorin interpolation theorem, we get the corresponding results in L _p (R ₊), 1 p 2, for the Kontorovich-Lebedev, Mehler-Fock, and Olevskii index transforms. An inversion theorem is proved for a general index transformation. The case p=2 is known as the Plancherel-type theory for this class of transformations.__________Published in Lietuvos Matematikos Rinkinys, Vol. 45, No. 1, pp. 127–147, January–March, 2005.     

The <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>-Approximation Order of Surface Spline Interpolation for 1 \leq <Emphasis Type="Italic">p</Emphasis> \leq 2

We show that the L_p-approximation order of surface spline interpolation equals m+1/p for p in the range 1 \leq p \leq 2, where m is an integer parameter which specifies the surface spline. Previously it was known that this order was bounded below by m + &frac; and above by m+1/p. With h denoting the fill-distance between the interpolation points and the domain , we show specifically that the L_p()-norm of the error between f and its surface spline interpolant is O(h^{m + 1/p}) provided that f belongs to an appropriate Sobolev or Besov space and that \subset R^d is open, bounded, and has the C^2m-regularity property. We also show that the boundary effects (which cause the rate of convergence to be significantly worse than O(h^2m)) are confined to a boundary layer whose width is no larger than a constant multiple of h |log h|. Finally, we state numerical evidence which supports the conjecture that the L_p-approximation order of surface spline interpolation is m + 1/p for 2 < p \leq \infty.     

On the Theory of <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>(<Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">q</Emphasis></Subscript>)-Banach Lattices

Given 1≤ p,q < ∞, let BL_pL_q be the class of all Banach lattices X such that X is isometrically lattice isomorphic to a band in some L_p(L_q)-Banach lattice. We show that the range of a positive contractive projection on any BL_pL_q-Banach lattice is itself in BL_pL_q. It is a consequence of this theorem and previous results that BL_pL_q is first-order axiomatizable in the language of Banach lattices. By studying the pavings of arbitrary BL_pL_q-Banach lattices by finite dimensional sublattices that are themselves in this class, we give an explicit set of axioms for BL_pL_q. We also consider the class of all sublattices of L_p(L_q)-Banach lattices; for this class (when p/q is not an integer) we give a set of axioms that are similar to Krivine’s well-known axioms for the subspaces of L_p-Banach spaces (when p/2 is not an integer). We also extend this result to the limiting case q = ∞.     

On the rate of convergence of <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript> norms in the CLT for poisson and gamma random variables

In the paper, we present upper bounds of L _p norms of order ( X)^-1/2 for all 1 ≤ p ≤ ∞ in the central limit theorem for a standardized random variable (X− X)/ √ X, where a random variable X is distributed by the Poisson distribution with parameter λ > 0 or by the standard gamma distribution Γ(α, 0, 1) with parameter α > 0. The research was partially supported by the Lithuanian State Science and Studies Foundation, grant No. T-70/09.     

On Linear Preservers of Two-Sided Gut-Majorization On <Emphasis Type="Bold">M</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis>,<Emphasis Type="Italic">m</Emphasis></Subscript>

For X, Y ∈ M_n,m it is said that X is gut-majorized by Y, and we write X ?_gutY, if there exists an n-by-n upper triangular g-row stochastic matrix R such that X = RY. Define the relation ~_gut as follows. X ~_gutY if X is gut-majorized by Y and Y is gut-majorized by X. The (strong) linear preservers of ?_gut on ?ⁿ and strong linear preservers of this relation on M_n,m have been characterized before. This paper characterizes all (strong) linear preservers and strong linear preservers of ~_gut on ?ⁿ and M_n,m.     

Spectral Exponent of Finite Sums of Weighted Positive Operators in <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript>-Spaces

In this paper we investigate the spectral exponent, i.e. logarithm of the spectral radius of operators having the form

<Emphasis Type="Italic">p</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis></Subscript>-sequences of algebras with one fundamental operation

We show that if (p₀, p₁, ...) is the p_n-sequence of a nontrivial algebra with one fundamental operation, then p₁ ≥ p₀. Moreover, if , then p₁ > 2p₀. Received April 21, 2003; accepted in final form November 28, 2005.     

Rationality of the vertex operator algebra <Emphasis Type="Italic">V</Emphasis><Subscript><Emphasis Type="Italic">L</Emphasis></Subscript>+ for a positive definite even lattice <Emphasis Type="Italic">L</Emphasis>

The lattice vertex operator algebra V_L associated to a positive definite even lattice L has an automorphism of order 2 lifted from –1-isometry of L. The fixed point set V_L+ of V_L for the automorphism is naturally a vertex operator algebra. We prove that any ₀-graded weak V_L⁺-module is completely reducible.Supported by JSPS Research Fellowships for Young Scientists.     

<Emphasis Type="Italic">q</Emphasis>-Fibonacci Polynomials and the Rogers-Ramanujan Identities

We derive some formulas for the Carlitz q-Fibonacci polynomials F_n(t) which reduce to the finite version of the Rogers-Ramanujan identities obtained by I. Schur for t = 1. Our starting point is a representation of the q-Fibonacci polynomials as the weight of certain lattice paths in contained in a strip along the x-axis. We give an elementary combinatorial proof by using only the principle of inclusion-exclusion and some standard facts from q-analysis.     

<Emphasis Type="Italic">M</Emphasis>-spherical <Emphasis Type="Italic">K</Emphasis>-modules of a rank one semisimple Lie group

Let G =K A N be an Iwasawa decomposition of a connected, noncompact real semisimple Lie group with finite center and let M be the centralizer of A in K . B. Kostant proved that for every irreducible M-spherical K-module V there exists a unique d (the Kostant degree of V) such that V can be realized as a submodule of the space of all -harmonic homogeneous polynomials of degree d on . Here is a Cartan decomposition of the complexification of the Lie algebra of G .In this paper we give an algorithm to obtain a highest weight vector from any M-invariant vector in an irreducible M-spherical K-module. This algorithm allows us to compute a sharp bound for the Kostant degree d(v) of any M-invariant vector v in a locally finite M-spherical K-module V. The method computes d(v) effectively for any V if G is locally isomorphic to SO(n,1) and for if G is locally isomorphic to SU(n,1).Partially supported by Agencia Córdoba Ciencia and CONICET Mathematics Subject Classification (2000):Primary 22E46, Secondary 43A85