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This paper deals with Gabor representation of (generalized) functions. By this we mean that a given function is to be expanded in a series involving Gabor functions that are located in the points of a given lattice in the time-frequency plane. We shall mainly consider lattices in which the area of the elementary cells of the lattice equals 1. In 1946, Gabor used these expansions for the simultaneous analysis of signals in time and frequency (cf. 181). Gabor stated that the above mentioned expansions exist for every reasonable signal, and that the coefficients in the expansion are uniquely determined by the signal. This statement is true when interpreted carefully, and the aim of this paper is to find out what kind of (generalized) functions are sufficiently well-behaved to allow a development in a Gabor series. We shall also consider the question of the uniqueness of the coefficients in the expansions, as well as questions concerning convergence. It turns out that the uniqueness questions can be handled with the aid of the main results of [IO] (in particular 2.12 and 2.13); the questions on the existence of expansions are harder to answer, and require special care depending on the particular function. We consider Lpfunctions (1

Let G be a locally compact Abelian group with Haar measure 1. For 1 < r < co, L,(G) will denote the usual Lebesgue space defined on G with respect to 1. In recent years a number of papers have been devoted to the study of multipliers from L,(G) to a function space defined on G. For example, Burnham and Goldberg [ 1 ] and Goldberg and Seltzer [3] have studied the multipliers from L,(G) to a Segal algebra S(G); Feichtinger [2] has studied the multipliers from L,(G) to a homogeneous Banach space B(G); in [6] we have characterized the space of multipliers from L,(G) to a Lipschitz space Lip(a,p, G) (or lipp G) (or Ll,(a,p; G)), where G is a metrizable locally compact Abelian group (with additional hypothesis imposed on G in some of our results). In this note we present two results (see Theorems 1 and 2) on the multipliers from one Lipschitz space yip(a,p; G) to another Lipschitz space Uip(/.?, q; G), where G is a Vilenkin group (i.e., G is an infinite, compact, O-dimensional, metrizable, Abelian group). Zygmund [ 111 and Mizuhara [ 51 have obtained similar results for Lipschitz spaces defined on the circle group. For the remainder of this note G will denote a Vilenkin group. We shall need some facts about G; the relevant facts, notation and terminology can be found in the first two pages of our paper [7]. The following definitions are slight extensions of Definitions 2.2 and 2.3 in

One of the most interesting and specific instruments of investigating a large class of problems pertaining especially to functional equations theory is, undoubtedly, that represented by the so-called Hilbert’s and Thompson’s projective metrics. The first of them, introduced by Hilbert in his 1903 paper [ 151 and subsequently developed (under a “modern” form) by Birkhoff [7] and Bushel1 [lo], appears to be a very adequate tool for solving some delicate questions about linear functional equations (see, in this direction, the well-known Jentzsch’s theorem on integral equations with positive kernels (discussed, especially, in the above quoted Birkhoff paper), the Perron-Frobenius theorem on positive matrices (Samuelson [27]) as well as the multiplicative processes theory (Birkhoff (8,9])), but a somehow inadequate one in the treatment of the nonlinear versions of these problems (as typical examples of this kind being Theorem 1.2 of Bushel1 [ 111 and Theorem 4.1 of Turinici [38]). On the contrary, the second of these metrics-introduced by Thompson in his 1963 paper [33]-may be considered as a complementary instrument with respect to the preceding one since it was demonstrated that a large number of nonlinear functional equations (such as, e.g., those generated by (uniformly) “concave” operators in Bakhtin’s sense [2] (cf. also Krasnoselskii and Stetsenko [20])) may be treated by this procedure. Under these lines, it is, the main aim of the present paper to introduce a class of projective metrics on ordered linear spaces comprising Thompson’s metric as a particular case and, subsequently, to investigate-under this perspective-some specific examples of (nonlinear) Volterra functional equations-not reductible, in general, to other “nonprojective” techniques-including, among others, a “vectorial” andogue of an interesting mathematical model for metabolic growth processes due to Bertalanffy [5, Chapter VII]. It’s not without importance to specify at this moment that the basic instrument employed in our construction is the monotone semigroups theory on ordered linear spaces; some 211 0022-247X/84 $3.00

The concept of weak contraction from the case of single-valued mappings is extended to multi-valued mappings and then corresponding convergence theorems for the Picard iteration associated to a multi-valued weak contraction are obtained. The main results in this paper extend, improve and unify a multitude of classical results in the fixed point theory of single and multi-valued contractive mappings and also improve recent results from the paper [P.Z. Daffer, H. Kaneko, Fixed points of generalized contractive multi-valued mappings, J. Math. Anal. Appl. 192 (1995), 655–666].

In this paper, the equivalence of the strong convergence between the modified Mann and Ishikawa iterations with errors in two different schemes by Xu [Y.G. Xu, Ishikawa and Mann iteration process with errors for nonlinear strongly accretive operator equations, J. Math. Anal. Appl. 224 (1998) 91–101] and Liu [L.S. Liu, Ishikawa and Mann iterative process with errors for nonlinear strongly accretive mappings in Banach spaces, J. Math. Anal. Appl. 194 (1995) 114–125] respectively is proven for the generalized strongly successively Φ-pseudocontractive mappings without Lipschitzian assumption. Our results generalize the recent results of the papers [Zhenyu Huang, F. Bu, The equivalence between the convergence of Ishikawa and Mann iterations with errors for strongly successively pseudocontractive mappings without Lipschitzian assumption, J. Math. Anal. Appl. 325 (1) (2007) 586–594; B.E. Rhoades, S.M. Soltuz, The equivalence between the convergences of Ishikawa and Mann iterations for an asymptotically nonexpansive in the intermediate sense and strongly successively pseudocontractive maps, J. Math. Anal. Appl. 289 (2004) 266–278; B.E. Rhoades, S.M. Soltuz, The equivalence between Mann–Ishikawa iterations and multi-step iteration, Nonlinear Anal. 58 (2004) 219–228] by extending to the most general class of the generalized strongly successively Φ-pseudocontractive mappings and hence improve the corresponding results of all the references given in this paper by providing the equivalence of convergence between all of these iteration schemes for any initial points u 1 , x 1 in uniformly smooth Banach spaces.

In recent years the study of set-valued functions has been developed extensively by many authors, with applications to mathematical economics and control theory; see Refs. [5, 11, 15, 161. In those papers, three approaches can be distinguished according to whether the range space (values of setvalued functions) is .Z”, a Banach space, or a locally convex topological space. The purpose of this paper is to establish properties of Aumann’s integrals of set-valued functions, F: T+ 2’, whose values are nonempty subsets of a real separable reflexive Banach space X, and to continue the work due to Aumann [2] and Datko [7-81. While previous analysis has always treated the case of special finite nonatomic measure spaces, we focus here on the case of general a-finite nonatomic measure spaces. In this last situation, moreover, the analogous results we establish hold under less stringent hypotheses. More precisely, all through the paper we consider a measure space (T, C, p), where p is supposed to be positive, nonatomic and u-finite, and we give the following statements.

In a Polish group G, a property is said to hold for a generic g∈G if the set on which it does not hold is meager in G; a property holds for almost every (ae) g∈G if the set on which it does not hold is Haar null. In this paper we study the properties of generic and ae elements of the Baer–Specker Group ZZ, the space of all self-maps of Z. We find that with respect to most of the properties under consideration in this paper, the properties of a generic element and ae element are complementary. Our results are similar in flavor to results by Dougherty and Mycielski for the group S∞ of all permutations of N.

This paper continues the joint investigation by Bennett et al. (2001) of the extent to which sequence spaces are determined by the sequences of 0's and 1's that they contain. The first main result gives a negative answer to Question 6 in their paper: There exists a sequence space E such that each matrix domain containing all of the sequences of zeros and ones in E contains all of E, but such that this statement fails, if we replace matrix domains by separable FK-spaces. The second main result goes on from Hahn's theorem that tells us that each matrix domain including χ, the set of all sequences of 0's and 1's, contains all of the bounded sequences: It is shown that there exists a really ‘small’ subset χ of χ such that Hahn's theorem remains true when χ is replaced with it. The proofs of both results have in common that, by identifying sequence spaces and double sequence spaces, the constructions and the required investigations are done in double sequence spaces that allow the description of finer structures.

In a proper setting, the problem of the title amounts to the determination of all pairs (d, T) for which the function ƒ(z): = (z + 1)exp(zT) + d and the functions fn(z)≔(z+1)(1+zTn)n+d, n∈N, d, T⩾0, have all their zeros in Re(z) < 0 of the complex z-plane. Correct versions of results and proofs found in papers of K. L. Cooke and J. M. Grossman [J. Math. Anal. Appl. 86 (1982), 592–627] and S. P. Blythe, R. M. Nisbeth, W. S. C. Gurney, and N. MacDonald [J. Math. Anal. Appl. 109 (1985), 388–396] are given. The argumentation in the proofs of the present paper differs from that of the studies mentioned. In contrast to Blythe et al., the stability behaviours of both models are seen as similar. Stability charts and stability conditions for some other classes of systems are also given. In addition to the gamma distribution for the lags, other distributions which are similar in shape but differ greatly in their stability behaviour are also considered.

In the paper we obtain an ergodic decomposition generated by a positive operator T:E → E (E being a Banach lattice having the projection property). If T is a power bounded operator and if E is a KB-space, then the decomposition permits us to describe sufficient conditions for the existence of nonzero T-subinvariant elements in E. We also study the relationship between the Hopf decomposition in Banach lattices and the one defined in the paper in order to obtain sufficient conditions for the existence of a T-invariant weak order unit.

L -fuzzy sets and, subsequently, L -fuzzy topological spaces are considered in this paper, where L is a complete and completely distributive non-atomic Boolean algebra. A few separation properties as well as subspace L -fuzzy topology are defined here, in the light of an earlier paper. These are more similar to ordinary topological spaces than to fuzzy topological spaces.

In the present paper we derive a number of key formulas involving fractional derivatives for the H-function of several variables, which was introduced and studied in a series of papers by H. M. Srivastava and R. Panda [cf., e.g., J. Reine Angew. Math. 283/284 (1976), 265–274; J. Reine Angew. Math. 288 (1976), 129–145; Comment. Math. Univ. St. Paul. 24 (1975), fasc. 2, 119–137; ibid. 25 (1976), fasc. 2, 167–197; Nederl. Akad. Wetensch. Proc. Ser. A 81 = Indag. Math. 40 (1978), 118–131 and 132–144; Nederl. Akad. Wetensch. Proc. Ser. A 82 = Indag. Math. 41 (1979), 353–362; see also Bull. Inst. Math. Acad. Sinica 9 (1981), 261–277].We make use of the generalized Leibniz rule for fractional derivatives in order to obtain one of the aforementioned results, which involves a product of two multivariable H-functions. Each of these results is shown to apply to yield interesting new results for certain multivariable hypergeometric functions and, in addition, several known results due, for example, to J. L. Lavoie, T. J. Osler and R. Tremblay [SIAM Rev. 18 (1976), 240–268], H. L. Manocha and B. L. Sharma [J. Austral. Math. Soc. 6 (1966), 470–476; J. Indian Math. Soc. (N.S.) 38 (1974), 371–382] and R. K. Raina and C. L. Koul [Jñānābha 7 (1977), 97–105].

1. The most commonly used generalization of the Bellman-Gronwall inequality is due to I. Bihari [ 11. See also [2] where related integral inequalities are discussed and more references given. The integral equation associated with Bihari’s inequality is nonlinear in the function of interest (see Section 2), but linear as a function of the integral involved. Recent special uses of integral inequalities in which the associated integral equation is nonlinear in the integral have been considered in papers by D. Willett [3] and H.E. Gollwitzer [4]. In Section 2 of this paper, Bihari’s inequality is generalized. Some results of Muldowney and Wong in [5] are improved, while it is shown that in some situations results of Willett and Gollwitzer are considerable sharpened. Results in terms of maximal solutions and uniqueness theorems are anticipated in a later paper. Fred Brauer considered such extensions of Bihari’s inequality in [6]. S C. Chu and F. T. Metcalf in [7] produced an explicit maximal solution for an integral inequality involving a general Volterra integral.

The purpose of this paper is to revise the definitions of Ishikawa and Mann iterative processes with errors, by using a new inequality to study the unique solution of the nonlinear strongly accretive operator equationTx = fand the convergence problem of Ishikawa and Mann iterative sequences for strongly pseudo-contractive mappings without the Lipschitz condition. The results presented in this paper improve and extend the corresponding results in [ 4 , 5 , 7 – 10 , 12 , 15 , 16 ] in the more general setting. In particular, the open problem mentioned by Chidume in [ 5 ] has been given an affirmative answer.

This paper deals with a generating function of the Jacobi polynomials that satisfies the following properties (I) and (II). (I) The generating function is the kernel of an integral operator that is unitary. (II) The image of the unitary operator is a reproducing kernel Hilbert space of analytic functions and the reproducing kernel is given as a special value of the generating function above. A generating function that satisfies (I) is given in Watanabe (1998) [11] . The purpose of this paper is to give a generating function that satisfies (I) and (II). From a group theoretical point of view, a similar construction for zonal spherical functions is given in Watanabe (2006) [12] .

The aim of this paper is to find those pairs of generalized quasi-arithmetic means on an open real interval I for which the arithmetic mean is invariant, i.e., to characterize those continuous strictly monotone functions φ , ψ : I → R and Borel probability measures μ , ν on the interval [ 0 , 1 ] such that φ −1 ( ∫ 0 1 φ ( t x + ( 1 − t ) y ) d μ ( t ) ) + ψ −1 ( ∫ 0 1 ψ ( t x + ( 1 − t ) y ) d ν ( t ) ) = x + y ( x , y ∈ I ) holds. Under at most fourth-order differentiability assumptions and certain nondegeneracy conditions on the measures, the main results of this paper show that there are three classes of the solutions φ , ψ : they are equivalent either to linear, or to exponential or to power functions.

We study the Pettis integral for multi-functions F : Ω → cwk ( X ) defined on a complete probability space ( Ω , Σ , μ ) with values into the family cwk ( X ) of all convex weakly compact non-empty subsets of a separable Banach space X. From the notion of Pettis integrability for such an F studied in the literature one readily infers that if we embed cwk ( X ) into l ∞ ( B X ∗ ) by means of the mapping j : cwk ( X ) → l ∞ ( B X ∗ ) defined by j ( C ) ( x ∗ ) = sup ( x ∗ ( C ) ) , then j ○ F is integrable with respect to a norming subset of B l ∞ ( B X ∗ ) ∗ . A natural question arises: When is j ○ F Pettis integrable? In this paper we answer this question by proving that the Pettis integrability of any cwk ( X ) -valued function F is equivalent to the Pettis integrability of j ○ F if and only if X has the Schur property that is shown to be equivalent to the fact that cwk ( X ) is separable when endowed with the Hausdorff distance. We complete the paper with some sufficient conditions (involving stability in Talagrand's sense) that ensure the Pettis integrability of j ○ F for a given Pettis integrable cwk ( X ) -valued function F.

In this paper we generalize an old result of Littlewood and Hardy about bilinear forms defined in a class of sequence spaces. Historically, Littlewood [Quart. J. Math. 1 (1930)] first proved a result on bilinear forms on bounded sequences and this result was then generalized by Hardy and Littlewood in a joint paper [Quart. J. Math. 5(1934)] to bilinear forms on a class of lp spaces. Later Davie and Kaijser proved Littlewood's results for multilinear forms. In this paper, Theorems A and B generalize the results to multilinear forms on lp spaces. All the results are stated at the end of Section 1. Theorems A and B are proved, respectively, in Sections 2 and 3.

Consider the linear delay difference equation xn+1 − xn = anxnk, where an′s are real numbers and k is a positive integer. In this paper we establish asymptotic formulae for the solutions of the above equation assuming Σ∞n=0 |an|p < ∞ for some p ≥ 1. The asymptotic formulae are given by an appropriate recursive sequence. For p ∈ {1, 2, 3, 4} we present explicit formulae. We also construct an example which shows that some comparison results in a recent paper of Yan and Qian are incorrect.

The notion of convexity was generalized by many means. Some of these notions are useful in questions of topology and analysis, for example, see [S, 12, 14-161. In this paper our aim is to give simultaneously generalizations of these concepts and to show that many classical theorems, in which the word “convex” appears remain valid by replacing this word by “simpiicial convex.” In Section 1 we define a simplicial convexity by assignation with each n elements x, ,..., x, of a topological space X and each element (A, ,..., A,) of the (n l)-dimensional simplex a formal “linear combination” @[xi ,..., x,](A, ,..., A,,)E X, which depends in a continuous way on (A, ,..., A,), but not necessarily on x, ,..., x,*. In Section 2 we give a characterization of absolute retracts in terms of simplicial convexity. In Section 3 we look for continuous selections of lower semi-continuous maps. In the last section, using the technique of Knaster-KuratowskiMazurkiewicz maps, we obtain generalizations of some theorems concerning fixed point theory and its applications [3-6, 13, 171. I am very grateful to Professor Lech Gorniewicz for his valuable remarks and aid in the preparation of this paper.

In this paper, we will introduce and compare a number of definitions of oscillations for vectorial delay differential equations. Oscillatory properties of equations have received the most attention in the scalar case. For this case, there have been a great many contributions. Without any attempt to be complete, we can quote notably [ 1, 3, 9, 12, 14, 151. Recently, a book was devoted to this subject [16]. Let us also mention a contribution by G. Seifert [ 181. For the vector case, several generalizations were undertaken, notably by J. M. Ferreira and I. Gyori [S]. We can also mention a recent paper by J. Wiener and K. L. Cooke [21] for delay equations with piecewise constant arguments, However, no systematic study of the very notion of oscillation seems to have been done in the vector case, despite the fact that we can envision there a number of possible choices. Our purpose in this paper is to present some of them, among which are a weak notion and a strong notion (Definition 6). A generate will be performed in a restricted class of equations, those which generate an order preserving semi-flow. In such a class, we will prove that both notions are equivalent to each other. The significance of oscillations in monotone semiflows was recently underlined. In [2], it has been shown that oscillatory properties characterize the asymptotic behavior. We refer to [3,4, 63 for other developments on this subject.

In this paper, two methods are discussed for defining fractional powers of the nth order ordinary differential expression L = x a 1 Dx a 2 D ··· x a n Dx a n + 1 (D = d dx ) , where each ai (i = 1, 2, …, n + 1) is a complex number and x is real and positive. An important role is played by the number m = ¦∑ i = 1 n + 1 a i − n¦ . When m = 0, spectral theory is used to obtain a suitable formula for Lα (α ϵ C ) while the case m>0, considered in a previous paper by A. C. McBride (Proc. London Math. Soc. (3) 45 (1982), 519–546), is dealt with by expressing Lα in terms of Erdelyi-Kober operators. We give accounts of each approach and go on to consider how the results for m = 0 can be obtained as limiting cases of the results for m>0.

Convergence theory is a primary topic in topology. In fact, topology and so-called convergence class are characterized by each other. In fuzzy topology ( L -fuzzy topology), more than 40 papers published in the last ten years were concerned with convergence theory. Among these papers, the problem of convergence class was solved for the case of L = [0, 1] [ 7 ]. Since the neighbor structure, so called “quasi-coincident neighborhood system,” of an L -fuzzy point in an L -fuzzy topological space is in general not directed under the inclusion order, the conditions of convergence class in [0, 1]-fuzzy topology will not be valid any longer in the case of lattice. Moreover, quite different from the cases of {0, 1}-fuzzy topology (i.e., ordinary topology) and [0, 1]-fuzzy topology, the so called Bolzano–Weierstrass property does not hold, i.e., a net with a cluster point in an L -fuzzy topological space is not still necessary to have a subnet converging to the point. In this paper, a necessary and sufficient condition for the Bolzano–Weierstrass property is produced, the result is also used in a satisfactory theory of convergence classes in L -fuzzy topological spaces, and the associated characterization theorem between L -fuzzy topologies and convergence classes is established.

have important applications to the theory of orthogonal polynomials. For most of the applications of (1.1) and (1.2) mentioned in these papers, one does not need to have explicit formulas for the kernels (measures and coefficients, respectively) since only their non-negativity was needed. By concentrating on the special case in which &(x) and qn( x are the classical orthogonal ) polynomials (Jacobi polynomials and their limits) the above-mentioned authors were able to develop methods for determining when the kernels are non-negative. In this paper we further extend these methods to obtain many results on when projection formulas of the above types hold for the discrete orthogonal polynomials of Hahn, Krawtchouk, Meixner, and Charlier [lo, 14, 171. Since Jacobi polynomials are limits of Hahn polynomials (see (1.5) below), non-negativity results for Jacobi polynomials will also follow from our work. For 01, /3 > 1, N a non-negative integer and n = 0, l,..., N we shall define the Hahn polynomials by

In recent papers [2,6), the authors have established existence and comparison theorems for the well known Cauchy problem for ordinary diferential equations without using the monotone property on the given system. These results are obtained under the conditions of the type which have been considered in the classical paper of Miiller [ 8 1. The interesting feature of the results established in (2,6] is the fact that the solutions of the Cauchy problem remain in the given sector. In this paper, we shall first establish existence and comparison theorems for a class of more general functional differential systems without using monotone property. Further we develop a monotone iterative technique to establish the existence of -minimal and maximal solutions.

In the last few years, the mean value theorem for various classes of nondifferentiable functions has been the subject of many investigations. In particular, Hiriart-Urruty [4, Sect. II] proved some variants of this theorem for extended-real-valued functions defined on locally convex spaces. In this paper we present two other theorems which are, to some extent, generalizations of the above-mentioned results. More precisely, we introduce the notion of an upper convex approximation for a function (being a modification of the first-order convex approximation of Ioffe [ 51) and formulate the mean value theorems in terms of subdifferentials of these approximations. We first consider the class of functions f’: X-t R (where X is a locally convex space) such that the restriction offto the closed line segment [a, 61 is finite and lower semicontinuous. Our result for this case (Theorem 4.2) has a weaker form than the classical mean value theorem: the difference f(h) -f(a) is expressed by means of the subdifferential of an upper convex approximation for f at a point CE [a, b]. If f is discontinuous, c may be equal to a or h. However, if the restriction ,fl [a, h] is continuous, we obtain a stronger result (Theorem 4.3) stating that c belongs to the open line segment ]a, h[. This theorem is a generalization of Theorem 1 of [9] and, in part, of Theorem 2 of [4]. Finally, we obtain, as a corollary from Theorem 4.3, a sufficient condition for a strict local minimum of a continuous function on a convex subset of X (Theorem 5.1). This theorem extends the result of [lo] to the case of functions which may not satisfy the Lipschitz condition. In the paper we make use of some notions and theorems of convex analysis which can be found in Chapter 6 of [6].

In this paper, we generalize Hilbert's integral inequality and its equivalent form by introducing three parameterst,a, andb. Iff, g ∈ L2[0, ∞), then[formula]where π is the best value. The inequality (1) is well known as Hilbert's integral inequality, and its equivalent form is[formula]where π2is also the best value (cf. [ 1 , Chap. 9]). Recently, Hu Ke made the following improvement of (1) by introducing a real functionc(x),[formula]wherek(x) = 2/π∫∞0(c(t2x)/(1 + t2)) dt − c(x), 1 − c(x) + c(y) ≥ 0, andf, g ≥ 0 (cf. [ 2 ]). In this paper, some generalizations of (1) and (2) are given in the following theorems, which are other than those in [ 2 ].

In two recent papers, "A Simpler Proof of a Theorem of Steinmetz" (1989, J. Math. Anal. Appl. 143 , 290-294) and "An Extension of a Theorem of Steinmetz" (1991, J. Math. Anal. Appl. 156 , 287-292), the present authors gave a new proof of Steinmetz′s theorem that avoided use of the Second Fundamental Theorem of Nevanlinna. It was also shown in the second paper how to generalize Steinmetz′s theorem to handle the case where the inner function is not precisely the same in all cases. In this paper we shall extend these results even further.

In an earlier paper we extended Lai's (1974) law of the single logarithm for delayed sums to a class of delayed sums of random fields. In this paper we allow for more general windows.

The largely singular operators that are defined in this paper are a sharp contrast to invertible operators or Fredholm operators. Nevertheless, we discover that the set has a surprising amount of nice structure. Without assuming that the underlying Hilbert space is separable, we are able to compute the distance from an arbitrary operator to the set. The distance formula involves the modulus of invertibility, which was introduced in an earlier paper.

Let S(n) be the set of all polynomials of degree n with all roots in the unit disk, and define d(P) to be the maximum of the distances from each of the roots of a polynomial P to that root's nearest critical point. In this notation, Sendov's conjecture asserts that d(P)=d(Q) for all nearby Q in S(n), and note that maximizing d(P) over all locally extremal polynomials P would settle the Sendov conjecture. Prior to now, the only polynomials known to be locally extremal were of the form P(z)=c(z^n+a) with |a|=1. In this paper, we determine sufficient conditions for real polynomials of a different form to be locally extremal, and we use these conditions to find locally extremal polynomials of this form of degrees 8, 9, 12, 13, 14, 15, 19, 20, and 26., 10 pages, AMS-LaTeX, no figures. The Maple code and results used in this paper are included in the source files. We constructed an unexpected locally extremal polynomial of degree 8 in version 1, then added degrees 12, 14, 20 and 26 in version 2, and degrees 9, 13, 15 and 19 in version 3

Bojanic [Publ. Inst. Math. (Beograd) (N.S.) 26 (40) (1979), 57-60] gave an estimate of the rate of convergence for Fourier series of functions of bounded variation. Cheng [J. Approx. Theory39 (1983), 259-274] gave the result of this type for the Bernstein operator. Recently, Guo [J. Approx. Theory56 (1989), 245-255] used the results of probability theory to arrive at an estimate for the rate of convergence of the nth integrated Meyer-Konig and Zeller operator Mn, n∈N (Maier et al. [J. Approx. Theory32 (1981), 27-31]) for a real-valued Lebesgue integrable function ƒ of bounded variation defined on I = [0, 1]. We have been able to correct and improve Guo′s estimate, while using parts of his work. This paper was in the first place motivated by some misprints in S. Guo′s paper.

This paper is a continuation of [4]. It contains our results on resultant operators for a family of matrix functions that are analytic in a finitely connected domain. The proofs are based on the results about the Bezout equation obtained in [4]. It is assumed that the reader is familiar with the concepts and results from [4]. Let f denote a Cauchy contour in the complex plane consisting of several non intersecting simple smooth closed contours which form the positively oriented boundary of a finitely connected bounded domain A + Throughout this chapter we assume for convenience that 0 E A + The general case can be obtained by a shift in the complex plane. Let A, ,..., A, be regular n x n matrix functions which are analytic in A + and continuous in A + u r, and assume that C(A,) n r= $3. The paper consists of two sections. In Section 1 we show that the role of the resultant operator of the matrix functions A,,..., A, is played by the operator R,: L;;(T) + L;(/‘) (1

In 1940 S.M. Ulam proposed the famous Ulam stability problem. In 1941 D.H. Hyers solved the well-known Ulam stability problem for additive mappings subject to the Hyers condition on approximately additive mappings. The first author of this paper investigated the Hyers–Ulam stability of Cauchy and Jensen type additive mappings. In this paper we generalize results obtained for Jensen type mappings and establish new theorems about the Hyers–Ulam stability for general additive functional equations in quasi- β -normed spaces.

Let E be a subset of R . We say that f : R → R is an E-Polya frequency function if the kernel f ( u − x ) is totally positive on E × R and f is integrable on R . In this paper, we show that under some conditions on E, an E-Polya frequency function is a Polya frequency function. This strengthens Schoenberg's result on Polya frequency functions.

This paper investigates solutions of the general recurrence M (0) = g (0), M ( n + 1) = g ( n + 1) + min 0⩽ k ⩽ n ( αM ( k ) + βM ( n − k )) for various choices of α, β, and g ( n ). In a large number of cases it is possible to prove that M ( n ) is a convex function whose values can be computed much more efficiently than would be suggested by the defining recurrence. The asymptotic behavior of M ( n ) can be deduced using combinatorial methods in conjunction with analytic techniques. In some cases there are strong connections between M ( n ) and the function H ( x ) defined by H(x) = 1 for x (x − 1) α ) + H( (x − 1) β ) for x ⩾ 1 . Special cases of these recurrences lead to a surprising number of interesting problems involving both discrete and continuous mathematics.

In this paper we prove the existence of continuous order-preserving functions on subsets of ordered Banach spaces using a constructive approach.

Let Ω denote a simply connected domain in the complex plane and let K[Ω] be the collection of all entire functions of exponential type whose Laplace transforms are analytic on Ω′, the complement of Ω with respect to the sphere. Define a sequence of functionals {L n } on K[Ω] by L n (f) = 1 2πi ∝ Γ g n (ζ) F(ζ) dζ , where F denotes the Laplace transform of f, Γ ⊂ Ω is a simple closed contour chosen so that F is analytic outside and on Ω, and gn is analytic on Ω. The specific functionals considered by this paper are patterned after the Lidstone functions, L2n(f) = f(2n)(0) and L2n + 1(f) = f(2n)(1), in that their sequence of generating functions {gn} are “periodic.” Set gpn + k(ζ) = hk(ζ) ζpn, where p is a positive integer and each hk (k = 0, 1,…, p − 1) is analytic on Ω. We find necessary and sufficient conditions for f ∈ k[Ω] with L n (f) = 0 (n = 0, 1,…) . DeMar previously was able to find necessary conditions [7]. Next, we generalize {Ln} in several ways and find corresponding necessary and sufficient conditions.

INTRODUCTION In this paper we prove a g2obaZ existence theorem for classical solutions of second order semilinear parabolic systems of the form g + A(x, t, D) u =f(x, t, u, Du) in Q x (0, Tl, B(x, D) u = 0 on aJ x (0, q, u(., 0) = 240 on 0. Here u = (ul,..., @-‘): fi

In this paper we study the behavior of general integral operators on weighted Lp spaces. Particular cases include the main index transforms and the operators with complex Gaussian kernels. We also extend some previous results established in [E.R. Negrin, Proc. Amer. Math. Soc. 123 (1995) 1185–1190].

By now the L q -spectra of self-conformal measures satisfying the so-called open set condition are well understood. However, if the open set condition is not satisfied, then almost nothing is known. In this paper we compute the mixed L q -spectra for self-conformal measures, which have bound distortion property and satisfy the weak separation condition.

In this paper, first we present a cl-semigroup structure on L-fuzzy ideals, and conclude that the set of all L-fuzzy quasi-ideals is a cl-monoid with zero. Then we obtain a Galois connection between two cl-semigroups of L-fuzzy ideals of homomorphic rings. Finally, by using Galois connections, we give the concepts of contracted and extended L-fuzzy ideals, and some related results are given.

In this paper we study convex games with an infinite countable set of agents and provide characterizations of this class of games. To do so, and in order to overcome some shortcomings related to the difficulty of dealing with infinite orderings, we need to use a continuity property. Infinite sequencing situations where the number of jobs is infinite countable can be related to convex cooperative TU games. It is shown that some allocations turn out to be extreme points of the core of an infinite sequencing game.

In this paper we present a sandwich-type theorem for Carathéodory multifunctions. We obtain our result by means of the Marczewski function.

The functions ur(x), u2(x) characterize the optimal cost of a stochastic control problem, whose decision variables are switching points. The results on existence, uniqueness and regularity of (l), (2) are recalled in Section 1, where we also describe the stochastic control problem which corresponds to it. We are interested here in obtaining regions which belong to the set of points where ur = u2 + k, or u2 = ur + k, , denoted by S, , S, respectively, and called the supports of the solution of (l), (2). Th’ IS p ro bl em is the analog of the problem on the support of solutions of elliptic variational inequalities (v.i.) considered by Brezis [6]. For the q.v.i. arising in the impulse control problem, studied by Bensoussan and Lions [3], the analogous question of support has been considered recently by Bensoussan, Brezis and Friedman [5]. The general methodology that we apply relies on the results on support for vi. However, we also use here the special characteristic of (l), (2) in order to obtain sharp estimates on the support, which is the main objective of the paper. We notice that if g, = g, = g then ur - u2 = w is a solution of the v.i.

The semiatom is a basic concept in the non-additive measure theory, or the fuzzy measure theory, and has been used for applications of the theory (T. Murofushi et al., 1997, Internat. J. Uncertain. Fuzziness Knowledge-Based Systems5, 563–585; and T. Murofushi and M. Sugeno, 2000, ibid.8, 385–415). This paper shows several properties of semiatoms on set operations: union, intersection, difference, symmetric difference, countable union, and countable intersection. Characteristic consequences are as follows: if S and T are semiatoms, and if S∩T is non-null, then S∪T and S∩T are semiatoms; moreover, if S\T and T\S are non-null, then S\T, T\S, S▵T also are semiatoms.

This paper offers an alternative to the standard method of applying the Schauder-Tychonoff theorem to establish the existence of solutions to mixed initial and final value problems for a system x ′ = Fx ( t > t 0 ) of functional differential equations. The main result reduces the application of the Schauder-Tychonoff theorem to merely verifying that the functional F satisfies four conditions, three of which are trivially verifiable in most instances. The fourth condition has to do with integrability properties of F , and is considerably less stringent than the requirements usually imposed, since it allows conditional convergence of some or all of the improper integrals that occur. The conclusions are also sharper than those usually obtained.

In this paper, we discuss the Lipschitz equivalence of a class of general Sierpinski carpets in which all non-trivial connected components are line segments. We define a bijection between two link-separated sets with same type by pairing off the basic sets using the indexing by the corresponding symbol space and get a sufficient condition that two general Sierpinski carpets are Lipschitz equivalent. Several examples will be given to illustrate our idea.

In this paper, we first establish the fundamental framework of multiresolution on local field in wavelet analysis. Accurately, orthonormal systems consisting of integer translations {φ(·−u(k))} k∈ P of a single function φ∈L2(K) are characterized (Theorems 1 and 2), the concept of multiresolution analysis on local field (Definition 3) and construction of corresponding wavelet vectors (Theorem 3) are given. The definition of integral periodicity of a function on local field (Definition 4) is also given. An example is finally presented.

LetM = {μs:s ∈ S} be a family of scalar bounded finitely additive measures defined on a σ-algebra A . The Nikodym–Grothendieck boundedness theorem states that ifMis simply bounded in A thenMis uniformly bounded in A . In this paper we prove that if V = {Ascr;n1, n2,…,np:p, n1, n2…np ∈ N } is an increasing web in A , then there is a strand {An1n2…ni:i ∈ N } such that ifMis simply bounded in one A n1n2…nithenMis uniformly bounded in A (Theorem 3.1). This result is deduced from the fact that if W = {En1n2…np:p, n1, n2,…,np ∈ N } is a linear increasing web inl0∞(X, A ), then there exists a strand {En1n2…ni:i ∈ N } such that everyEn1n2…niis barrelled and dense inl0∞(X, A ) (Theorem 2.7). From this strong barrelledness condition previous results of the author jointly with J. C. Ferrando are improved here. These results are related to the classical result of Diestel and Faires in vector measures.