Wavelet decompositions of L 2-functionals

Based on distribution-theoretical definitions of L ₂ and Sobolev spaces given by Werner in [P. Werner (1970). A distribution-theoretical approach to certain Lebesgue and Sobolev spaces. J. Math. Anal. Appl., 29, 19–78.] real interpolation, Besov type spaces and approximation spaces with respect to multiresolution approximations are considered. The key for the investigation are generalized moduli of smoothness introduced by Haf in [H. Haf (1992). On the approximation of functionals in Sobolev spaces by singular integrals. Applicable Analysis, 45, 295–308.]. Those moduli of smoothness allow to connect the concept of L ₂-functionals with more recent developments in multiscale analysis, see e.g. [W. Dahmen (1995). Multiscale analysis, approximation, and interpolation spaces. In: C.K. Chui and L.L. Schumaker (Eds.), Approximation Theory VIII, Vol. 2: Wavelets and Multilevel Approximation, pp. 47–88.]. In particular, we derive wavelet characterizations for the Sobolev spaces introduced by Werner and establish stable wavelet decompositions of L ₂-functionals. Generalizations to more general spaces of functionals and applications are also mentioned.     

Spaces of Ideal Convergent Sequences of Bounded Linear Operators

Abstract

The notation of I–convergence was introduced and studied by Kostyrko, Macaj, Salat, and Wilczynski. Recently, the concept of I–convergent for a sequence of bounded linear operators has been studied by Khan and Shafiq. This has motivated us to introduce and study some new spaces of double sequences of bounded linear operators and their basic topological and algebraic properties of these spaces. And we study some of their basic topological and algebraic properties of these spaces. We prove some inclusion relations on these spaces.     

Iterative approximation to common fixed points of a countable family of nonexpansive mappings

We proved several strong convergence results by using the conception of a uniformly asymptotically regular sequence {T _n } of nonexpansive mappings in a reflexive Banach space which admits a weakly continuous duality mapping J _?(l ^p (1?p?t)?=?t ^p?1. The results presented develop and complement the corresponding ones by Song, Y. and Chen, R., 2007 [Iterative approximation to common fixed points of nonexpansive mapping sequences in reflexive Banach spaces. Nonlinear Analysis, 66, 591–603], Song, Y., Chen, R. and Zhou, H., 2007 [Viscosity approximation methods for nonexpansive mapping sequences in Banach spaces. Nonlinear Analysis, 66, 1016–1024] and O'Hara, J.G., Pillay, P. and Xu, H.K., 2006 [Iterative approaches to convex feasibility problem in Banach Space. Nonlinear Analysis, 64, 2022–2042], O'Hara, J.G., Pillay, P. and Xu, H.K., 2003 [Iterative approaches to fineding nearest common fixed point of nonexpansive mappings in Hilbert spaces. Nonlinear Analysis, 54, 1417–1426] and Jung, J.S., 2005 [Iterative approaches to common fixed points of nonexpansive mappings in Banach spaces. Journal of Mathematical Analysis and Applications, 302, 509–520] and many other existing literatures.     

Boundedness of Monge–Ampère singular integral operators on Besov spaces

ABSTRACT

The purpose of this paper is to establish a theory of Besov spaces associated with sections under only the doubling condition on the measure and prove that Monge–Ampère singular integral operators are bounded on these spaces.     

Müntz spaces and special Bloch type inequalities

Abstract

In this paper, we are interested in classical Müntz spaces and their properties. We first reprove Müntz theorem and obtain the Clarkson–Erdös–Schwartz expansion in the nondense case for some general abstract Müntz spaces without invoking any blow up property, but using classical complex variables tools. We introduce new Bloch type Müntz spaces and investigate the validity of Bloch type inequalities in this framework.     

Regularity of the Local Time for the d-dimensional Fractional Brownian Motion with N-parameters

ABSTRACT

We give the Wiener–Ito? chaotic decomposition for the local time of the d-dimensional fractional Brownian motion with N-parameters and study its smoothness in the Sobolev–Watanabe spaces.     

Littlewood–Paley decomposition in quantum calculus

ABSTRACT

We introduce Littlewood–Paley decomposition related to q-Rubin's operator, this allows us to provide a dyadic characterization of Sobolev, Hölder and Lebesgue spaces associated with the q-Rubin's operators and to establish some embedding results for these spaces. We construct the paraproduct operators associated with the q-Rubin's operators and we establish its action on the Sobolev and Hölder spaces.     

Optional decomposition of optional supermartingales and applications to filtering and finance

ABSTRACT

The classical Doob–Meyer decomposition and its uniform version the optional decomposition are stated on probability spaces with filtrations satisfying the usual conditions. However, the comprehensive needs of filtering theory and mathematical finance call for their generalizations to more abstract spaces without such technical restrictions. The main result of this paper states that there exists a uniform Doob–Meyer decomposition of optional supermartingales on unusual probability spaces. This paper also demonstrates how this decomposition works in the construction of optimal filters in the very general setting of the filtering problem for optional semimartingales. Finally, the application of these optimal filters of optional semimartingales to mathematical finance is presented.     

Golusin–Krylov formulas in complex analysis

Abstract

This is a brief survey of a constructive technique of analytic continuation related to an explicit integral formula of Golusin and Krylov (1933). It goes far beyond complex analysis and applies to the Cauchy problem for elliptic partial differential equations as well. As started in the classical papers, the technique is elaborated in generalised Hardy spaces also called Hardy–Smirnov spaces.     

Paley–Wiener properties for spaces of power series expansions

ABSTRACT

We extend Paley–Wiener results in the Bargmann setting deduced in Nabizadeh et al. [Paley-Wiener properties for spaces of entire functions, (preprint), arXiv:1806.10752.] to larger classes of power series expansions. At the same time, we deduce characterizations of all Pilipovi? spaces and their distributions (and not only of low orders as in Nabizadeh et al. [Paley-Wiener properties for spaces of entire functions, (preprint), arXiv:1806.10752.]).     

Walsh Multipliers for Dyadic Hardy Spaces

In this article we are interested in conditions on the coefficients of a Walsh multiplier operator that imply the operator is bounded on certain dyadic Hardy spaces H ^p , 0 < p < ∞. In particular, we consider two classical coefficient conditions, originally introduced for the trigonometric case, the Marcinkiewicz and the Hörmander–Mihlin conditions. They are known to be sufficient in the spaces L ^p , 1 < p < ∞. Here we study the corresponding problem on dyadic Hardy spaces, and find the values of p for which these conditions are sufficient. Then, we consider the cases of H ¹ and L ¹ which are of special interest. Finally, based on a recent integrability condition for Walsh series, a new condition is provided that implies that the multiplier operator is bounded from L ¹ to L ¹, and from H ¹ to H ¹. We note that existing multiplier theorems for Hardy spaces give growth conditions on the dyadic blocks of the Walsh series of the kernel, but these growth are not computable directly in terms of the coefficients.     

On the convergence of some Procrustean averaging algorithms

Euclidean “(size-and-)shape spaces” are spaces of configurations of points in R ^N modulo certain equivalences. In many applications one seeks to average a sample of shapes, or sizes-and-shapes, thought of as points in one of these spaces. This averaging is often done using algorithms based on generalized Procrustes analysis (GPA). These algorithms have been observed in practice to converge rapidly to the Procrustean mean (size-and-)shape, but proofs of convergence have been lacking. We use a general Riemannian averaging (RA) algorithm developed in [Groisser, D. (2004) “Newton's method, zeroes of vector fields, and the Riemannian center of mass”, Adv. Appl. Math. 33, pp. 95–135] to prove convergence of the GPA algorithms for a fairly large open set of initial conditions, and estimate the convergence rate. On size-and-shape spaces the Procrustean mean coincides with the Riemannian average, but not on shape spaces; in the latter context we compare the GPA and RA algorithms and bound the distance between the averages to which they converge.     

Osculating Degeneration of Curves

Abstract

The main objects of this paper are osculating spaces of order mto smooth algebraic curves, with the property of meeting the curve again. We prove that the only irreducible curves with an infinite number of this type of osculating spaces of order mare curves in P ^m+1whose degree nis greater than m + 1. This is a generalization of the result and proof of Kaji (Kaji, H. (1986). On the tangentially degenerate curves. J. London Math. Soc.33(2):430–440) that corresponds to the case m = 1. We also obtain an enumerative formula for the number of those osculating spaces to curves in P ^m+2. The case m = 1 of it is a classical formula proved with modern techniques by Le Barz (Le Barz, P. (1982). Formules multisécantes pour les courbes gauches quelconques. In: Enumerative Geometry and Classical Algebraic Geometry. Prog. in Mathematics 24, Birkhäuser, pp. 165–197).     

Schur-Finiteness and Endomorphisms Universally of Trace Zero Via Certain Trace Relations

We provide a sufficient condition that ensures the nilpotency of endomorphisms universally of trace zero of Schur-finite objects in a category of homological type, i.e., a ?-linear ?-category with a tensor functor to super vector spaces. This generalizes previous results about finite-dimensional objects, in particular by Kimura in the category of motives. We also present some facts which suggest that this might be the best generalization possible of this line of proof. To get the result we prove an identity of trace relations on super vector spaces which has an independent interest in the field of combinatorics. Our main tool is Berele–Regev's theory of Hook Schur functions. We use their generalization of the classic Schur–Weyl duality to the “super” case, together with their factorization formula.     

Modified Tseng's extragradient methods for solving pseudo-monotone variational inequalities

ABSTRACT

We propose two modified Tseng's extragradient methods (also known as Forward–Backward–Forward methods) for solving non-Lipschitzian and pseudo-monotone variational inequalities in real Hilbert spaces. Under mild and standard conditions, we obtain the weak and strong convergence of the proposed methods. Numerical examples for illustrating the behaviour of the proposed methods are also presented     

On the solution existence of implicit quasivariational inequalities with discontinuous multifunctions

In this article, we established some solution existence theorems for implicit quasivariational inequalities. We first established some results in finite-dimensional spaces and then a solution existence result in infinite-dimensional spaces was derived. Our theorems are proved for discontinuous mappings and sets which may be unbounded. The results presented in this article are improvements of results in Cubiotti, and Yao (Cubiotti, P. and Yao, J.C., 1997, Discontinuous implicit quasi-variational inequalities with applications to fuzzy mappings. Mathematical Methods of Operations Research, 46, 213–328; Cubiotti, P. and Yao, J.C., 2007, Discontinuous implicit generalized quasi-variational inequalities in Banach spaces. Journal of Global Optimization (To appear))     

A cyclic iterative method for solving Multiple Sets Split Feasibility Problems in Banach Spaces

In this paper, we construct an iterative scheme and prove strong convergence theorem of the sequence generated to an approximate solution to a multiple sets split feasibility problem in a p-uniformly convex and uniformly smooth real Banach space. Some numerical experiments are given to study the efficiency and implementation of our iteration method. Our result complements the results of F. Wang (A new algorithm for solving the multiple-sets split feasibility problem in Banach spaces, Numerical Functional Anal. Optim. 35 (2014), 99–110), F. Scho¨pfer et al. (An iterative regularization method for the solution of the split feasibility problem in Banach spaces, Inverse Problems 24 (2008), 055008) and many important recent results in this direction.     

A NOTE ON A PAPER BY NG PENG-NUNG AND LEE PENG-YEE

Abstract

In the paper “Convergence in normed Köthe spaces” (J. Singapore National Academy of Science, 4, 146–148 (1975) M.R. 52 # 11568) Ng Peng-Nung and Lee Peng-Yee obtained a convergence result in the general setting of Banach funcation spaces providing conditions in order that pointwise and weak convergence imply norm convergence. They claim this result to be a generalization of a corresponding well known result in the Lebesgue space L₁ (X, u). To substantiate their claim it is necessary to show that the class of Banach function spaces for which their theorem holds is larger than the class of L₁-spaces. This, we shall show, is unfortunately not the case.     

Approach Theory in Merotopic, Cauchy and Convergence Spaces. II

We define the notions of approach-Cauchy structure and ultra approach-Cauchy structure. We study the categorical properties of ACHY and uACHY and show that in these schemes Cauchy spaces and extended pseudo-(ultra)metric spaces are regarded as entities of the same kind. Furthermore, we investigate the relation with convergence-approach spaces and obtain a relationship similar to that of CONV and CHY.