Volume 58 , issue 3$-$4 ( 2006 ) | back |

Valeurs des ondelettes aux bords d'un intervalle | 77$-$83 |

**Abstract**

The aim of this work is to find an approximative method for computing the scaling functions constructed on an interval at edges, using the inner product in ${L}^{2}([0,1])$ between the scaling function and its derivative.

**Keywords:** Wavelets with a compact support, multiresolute analysis, wavelets on an interval.

**MSC:** 65N30; 65D25, 65A15, 15A18

Common fixed point theorems for contractive maps | 85$-$90 |

**Abstract**

The aim of this paper is to obtain new common fixed point theorems under strict contractive conditions for three and four maps without continuity.

**Keywords:** Coincidence point, fixed point, $(E\;
A)$-property, tangent point, compatible maps, noncompatible maps, pointwise $R$-weak commutativity.

**MSC:** 54H25; 47H10

Properties of functions of generalized bounded variation | 91$-$96 |

**Abstract**

The class of functions of $\Lambda BV^{(p)}$ shares many properties of functions of bounded variation. Here we have shown that $\Lambda BV^{(p)}$ is a Banach space with a suitable norm, the intersection of $\Lambda BV^{(p)}$, over all sequences $\Lambda$, is the class of functions of BV$^{(p)}$ and the union of $\Lambda BV^{(p)}$, over all sequences $\Lambda$, is the class of functions having right- and left-hand limits at every point.

**Keywords:** Banach space, right and left-hand limits, p-$\Lambda$-bounded variation.

**MSC:** 26A45

Some generalizations of Littlewood-Paley inequality in the polydisc | 97$-$110 |

**Abstract**

The paper generalizes the well-known inequality of Littlewood-Paley in the polydisc. We establish a family of inequalities which are analogues and extensions of Littlewood-Paley type inequalities proved by Sh.\ Yamashita and D. Luecking in the unit disk. Some other generalizations of the Littlewood-Paley inequality are stated in terms of anisotropic Triebel-Lizorkin spaces. With the help of an extension of Hardy-Stein identity, we also obtain area inequalities and representations for quasi-norms in weighted spaces of holomorphic functions in the polydisc.

**Keywords:** Littlewood-Paley inequalities, polydisc, Triebel-Lizorkin spaces, weighted spaces.

**MSC:** 32A37; 32A36

Continuous representation of interval orders by means of decreasing scales | 111$-$117 |

**Abstract**

We characterize the representability of an interval order on a topological space through a pair of continuous real-valued functions which in addition represent two total preorders associated to the given interval order. Such a continuous representation is obtained by using the notion of a decreasing scale.

**Keywords:** Interval order; continuous numerical representation; decreasing scale.

**MSC:** 06A06; 91B16

On certain new subclass of close-to-convex functions | 119$-$124 |

**Abstract**

In the present paper, the authors introduce a new subclass ${\Cal K}^{(k)}_s(\alpha,\beta)$ of close-to-convex functions. The subordination and inclusion relationship, and some coefficient inequalities for this class are provided. The results presented here would provide extensions of those given in earlier works.

**Keywords:** Starlike functions, close-to-convex functions, differential subordination.

**MSC:** 30C45

Spaces related to $\gamma$-sets | 125$-$129 |

**Abstract**

We characterize Ramsey theoretically two classes of spaces which are related to $\gamma$-sets.

**Keywords:** Selection principles, Ramsey theory, game theory, $\omega$-cover, $k$-cover, $\gamma$-cover,
$\gamma_k$-cover, $\gamma$-set, $k$-$\gamma$-set, $\gamma_k^\prime$-set.

**MSC:** 54D20; 05C55, 03E02, 91A44

Metrizable groups and strict $o$-boundedness | 131$-$138 |

**Abstract**

We show that for metrizable topological groups %Tkachenko's notion of being a strictly o-bounded group is equivalent to being a Hurewicz group. In [5] Hernandez, Robbie and Tkachenko ask if there are strictly $o$-bounded groups $G$ and $H$ for which $G\times H$ is not strictly $o$-bounded. We show that for metrizable strictly $o$-bounded groups the answer is no. In the same paper the authors also ask if the product of an $o$-bounded group with a strictly $o$-bounded group is again an $o$-bounded group. We show that if the strictly $o$-bounded group is metrizable, then the answer is yes.

**Keywords:** Metrizable topological group, strict $o$-boundedness, product groups.

**MSC:** 54H11