A Phragmén–Lindelöf Theorem via Proximate Orders, and the Propagation of Asymptotics

We prove that, for asymptotically bounded holomorphic functions in a sector in $$\mathbb {C},$$ C , an asymptotic expansion in a single direction towards the vertex with constraints in terms of a logarithmically convex sequence admitting a nonzero proximate order entails asymptotic expansion in the whole sector with control in terms of the same sequence. This generalizes a result by Fruchard and Zhang for Gevrey asymptotic expansions, and the proof strongly rests on a suitably refined version of the classical Phragmén–Lindelöf theorem, here obtained for functions whose growth in a sector is specified by a nonzero proximate order in the sense of Lindelöf and Valiron.

Non-equidistance DGM(1,1) Model Based on the Concave Sequence and Its Application to Predict the China’s Per Capita Natural Gas Consumption

Although the grey forecasting model has been successfully adopted in various fields and demonstrated promising results, the literatures show its performance could be further improved, such as for the DGM(1,1) model, based on a concave sequence, the modeling error will be larger. In this paper, firstly the definition of sequence convexity is given out, and it is proved that the output sequence of DGM(1,1) model is a convex sequence. Next, the residual change law of DGM(1,1) model based on the concave sequence is discussed, and the non-equidistance DGM(1,1) model is proposed. Finally, by introducing the symmetry transformation, a concave sequence is transformed into a convex sequence, called the symmetric sequence of the concave sequence, and then construct the non-equidistance DGM(1,1) model based on the convex sequence. The example results show that the novel method is more accurate than the direct modeling for a concave sequence.

Convex sequences of higher order

In this paper we study the class of convex sequences of higher order defined using the difference operators and investigate their properties. The notion of the convex sequence of order r ? N will be extended for r a real number. Some necessary and sufficient conditions such that a real sequence belongs to the class of convex sequences of higher order r ? R are introduced. Using different types of means we will investigate the convexity of higher order for real sequences.

A general matrix application of convex sequences to Fourier series

By using a convex sequence Bor [H. Bor, Local properties of factored Fourier series, Appl. Math. Comp. 212 (2009) 82-85] has obtained a result dealing with local property of factored Fourier series for weighted mean summability. The purpose of this paper is to extend this result to more general cases by taking normal matrices in place of weighted mean matrices.

Improved GM(1, 1) model based on concave sequences and its applications

Purpose The purpose of this paper is to improve the GM(1, 1) model based on concave sequences. Design/methodology/approach First, the restored sequence of the GM(1, 1) model is proved to be convex, and the residual characters of the GM(1, 1) model for concave sequences are analyzed. Second, two symmetry transformations are introduced to transform an original concave sequence into a convex sequence, and then the GM(1, 1) model is established based on the convex sequence. Findings Compared with the traditional modeling method, the new method has high accuracy and is applicable for all concave sequence modeling. Practical implications Two cases are used to illustrate the superiority of this modeling method. Case A is to predict China’s per capita natural gas consumption, and case B is to predict the annual output of an oilfield. Originality/value The application scope of GM (1, 1) model is greatly extended.

Pathological Phenomena in Denjoy–Carleman Classes

Let M denote a Denjoy–Carleman class of ∞ functions (for a given logarithmically-convex sequence M = (Mn)). We construct: (1) a function in M((−1, 1)) that is nowhere in any smaller class; (2) a function on ℝ that is formally M at every point, but not in M (ℝ); (3) (under the assumption of quasianalyticity) a smooth function on ℝp (p ≥ 2) that is M on every M curve, but not inM (ℝp).

Dual pairs of sequence spaces

The paper aims to develop for sequence spacesEa general concept for reconciling certain results, for example inclusion theorems, concerning generalizations of the Köthe-Toeplitz dualsE×(×∈{α,β})combined with dualities(E,G),G⊂E×, and theSAK-property (weak sectional convergence). TakingEβ:={(yk)∈ω:=𝕜ℕ|(ykxk)∈cs}=:Ecs, wherecsdenotes the set of all summable sequences, as a starting point, then we get a general substitute ofEcsby replacingcsby any locally convex sequence spaceSwith sums∈S′(in particular, a sum space) as defined by Ruckle (1970). This idea provides a dual pair(E,ES)of sequence spaces and gives rise for a generalization of the solid topology and for the investigation of the continuity of quasi-matrix maps relative to topologies of the duality(E,Eβ). That research is the basis for general versions of three types of inclusion theorems: two of them are originally due to Bennett and Kalton (1973) and generalized by the authors (see Boos and Leiger (1993 and 1997)), and the third was done by Große-Erdmann (1992). Finally, the generalizations, carried out in this paper, are justified by four applications with results around different kinds of Köthe-Toeplitz duals and related section properties.