Complex Interpolation Method Research Articles

Complex matrix interpolation model of the S-transform for electric load movement forecast

Electric load movement forecast is increasingly importance for the industry. This study addresses the load movement forecast modeling based on complex matrix interpolation of the S-transform (ST). In complex matrix of time-frequency representation of the ST, each row follows conjugate symmetric property and each column appears a certain degree of similarity. Based on these characteristics, a complex matrix interpolation method for the time-frequency representation of the ST is proposed to interpolate each row of the complex matrix based on the conjugate symmetric property, and then to perform nearest-neighbor interpolation on each column. Then with periodic extension for daily and yearly electric load movement, a forecast model employing the complex matrix interpolation of the ST is introduced. The forecast approach is applied to predict daily load movement of the European Network on Intelligent Technologies (EUNITE) load dataset and annual electric load movement of State Gird Corporation of China and its branches in 2005 and 2006. Result analysis indicates workability and effectiveness of the proposed method.

Sharp coincidences for absolutely summing multilinear operators

In this paper we prove the optimality of a family of known coincidence theorems for absolutely summing multilinear operators. We connect our results with the theory of multiple summing multilinear operators and prove the sharpness of similar results obtained via the complex interpolation method.

On the interpolation of analytic mappings

Let (E0,E1) and (H0,H1) be two pairs of complex Banach spaces densely and continuously embedded into each other, E1 ↪ E0 and H1 ↪ H0 and also let \(\left\| x \right\|_{E_0 } \leqslant \left\| x \right\|_{E_1 } \). By Eθ = [E0, E1]θ and Hθ = [H0, H1]θ we denote the spaces obtained by the complex interpolation method for θ ∈ [0, 1], and by Bθ(0,R) we denote an open ball of radius R in the space Eθ. Let Φ: B0(0,R) → H0 be an analytic mapping taking B1(0,R) into H1, and let the estimates $$\left\| {\Phi (x)} \right\|_{H_\theta } \leqslant C_\theta \left\| x \right\|_{H_\theta } for allx \in B_\theta (0,R)$$ hold for θ = 0, 1. Then, for all θ ∈ (0, 1), the mapping Φ takes the ball Bθ(0,r) of radius r ∈ (0,R) in the space Eθ into Hθ and $$\left\| {\Phi (x)} \right\|_{H_\theta } \leqslant C_0^{1 - \theta } C_1^\theta \frac{R} {{R - r}}\left\| x \right\|_{E_\theta } ,x \in B_\theta (0,r). $$ .

Application of optical force measurement to mode characterization of atomic force microscopy nanomachining

Atomic force microscopy is often used not only to acquire the sample surface topography at the subnanometer scale but also to measure forces on the surface during imaging. This study aims to develop a practical measurement scheme of the actual scratching forces exerted during atomic force microscopy nanoscratching using a simple optical microscope setup. The measurement results are utilized to analyze the mode characteristics of atomic force microscopy nanomachining. Unlike typical atomic force microscopy force measurement methods using position-sensitive detector signal analysis, the optically measured atomic force microscopy cantilever deformation data using varying input (normal) forces along with information on the cantilever stiffness were used to estimate the components of the actual force exerted during nanomachining without using complex data interpolation methods. Scratching experiments were performed on Si(100) workpieces, and the estimated real force values were compared with the experimental data from atomic force microscopy nanomachining for each input force and corresponding scratch depth. Frictional coefficients and in-process acoustic emission monitoring results were also used to conduct an in-depth analysis of the actual force results. It is shown that the estimation results are meaningfully close to both the theoretical evaluations and the scratching experimental data as the scratch depth changes. Moreover, the force data are shown to have the ability to detect mode transitions such as the elastic–plastic and the plowing–cutting transitions during nanomachining, which validates the utility of this approach.

The periodic complex method in interpolation spaces

In this paper we consider the “periodic” variant of the complex interpolation method, apparently first studied by Peetre (Rend Sem Mat Univ Padova 46:173–190, 1971). Cwikel showed (Indiana Univ Math J 27:1005–1009, 1978) that using functions with a given period $$i\lambda $$ in the complex method construction introduced and studied by Calderón (Studia Math 24:113–190, 1964), one may construct the same interpolation spaces as in the “regular” complex method, up to equivalence of norms. Cwikel also showed that one of the constants of this equivalence will, in some cases, “blow up” as $$\lambda \rightarrow 0$$ . (The other constant is obviously bounded by 1.) We show that this same equivalence constant approaches $$1$$ as $$\lambda \rightarrow \infty $$ . Intuitively, this means that when applying the complex method of Calderón, it makes a very small difference if one restricts oneself to periodic functions, provided that the period is very large.

The Complex Interpolation Method Preserves Compactness of Linear Operators

We obtain that for Banach couples (X0,X1),(Y0,Y1) and a linear operator T on X0+X1 to Y0+Y1 such that either T:X0→Y0 is compact or T:X1→Y1 is compact, the operators T:Xj→Yj being bounded, it holds that the interpolation operator T:[X0,X1]θ→[Y0,Y1]θ is compact.

Interpolation of Banach Lattices and Factorization of p-Convex and q-Concave Operators

We extend a result of Sestakov to compare the complex interpolation method [X0, X1]θ with Calderon-Lozanovskii’s construction \({{{{X^{1-\theta}_{0}X^{\theta}_{1}}}}}\), in the context of abstract Banach lattices. This allows us to prove that an operator between Banach lattices T : E → F which is p-convex and q-concave, factors, for any \({{{{\theta \in (0, 1)}}}}\), as T = T2T1, where T2 is (\({{\left({\frac{p}{{\theta + (1 - \theta)p}}} \right)}}\)-convex and T1 is \({{\left({\frac{q}{{1 - \theta }}} \right)}}\)-concave.

Commutator Estimates for Interpolation Scales with Holomorphic Structure

The main result of this paper is some quantitative estimates for nonlinear commutators under the complex interpolation methods and more general interpolation scales with holomorphic structures. We also investigate the spectral behaviour of bounded linear operators under this kind of interpolation methods.

Effect of spatial trends on interpolation of river bathymetry

Continuous surface of river bathymetry (bed topography) is typically produced by spatial interpolation of discrete point or cross-section data. Several interpolation methods that do not account for spatial trend in river bathymetry produce inaccurate surfaces, thus requiring complex interpolation methods such as anisotropic kriging. Although isotropic methods are unsuitable for interpolating river bathymetry, issues that limit their use remain unaddressed. This paper addresses the issue of effect of spatial trend in river bathymetry on isotropic interpolation methods. It is hypothesized that if the trend is removed from the data before interpolation, the results from isotropic methods should be comparable with anisotropic methods. Data from six river reaches in the United States are used to: (i) interpolate bathymetry data using seven spatial interpolation methods; (ii) separate trend from bathymetry; (iii) interpolate residuals (bathymetry minus trend) by using all seven interpolation methods to get residual surfaces, (iv) add the trend back to residual surfaces; and (v) compare resulting surfaces from (iv) with surfaces created in (i). Quantitative and qualitative comparison of results through root mean square error (RMSE), semi-variograms, and cross-section profiles show that significant improvement (as much as 60% in RMSE) can be accomplished in spatial interpolation of river bathymetry by separating trend from the data. Although this paper provides a new simple way for interpolating river bathymetry by using (otherwise deemed inappropriate) isotropic methods, the choice of trend function and spatial arrangement of discrete bathymetry data play a vital role in successful implementation of the proposed approach.

Geometric interpolation in symmetrically-normed ideals

The aim of this work is to apply the complex interpolation method to norms of n -tuples of operators in a symmetrically-normed ideal J ϕ ⊆ B ( H ) defined by a ϕ symmetric norming function (s.n.f.). The norms considered define Finsler metrics in a certain manifold of positive operators, and can be regarded as weighted ϕ -norms, the weight being a positive invertible operator.

Geometric interpolation in p-Schatten class

The aim of this work is to apply the complex interpolation method to norms of n-tuples of operators in the p-Schatten classes of a Hilbert space H. The norms considered define Finsler metrics in a certain manifold of positive operators, and can be regarded as weighted p-norms, the weight being a positive invertible operator. As a by-product, we obtain Clarkson's type inequalities.

Compactness Problems in Interpolation Theory and Function Spaces

Compactness is undoubtedly one of the central and most relevant notions in mathematics. The present mini-workshop was centered around some important compactness problems connected to interpolation theory and to the theory of function spaces, two very closely related areas. Our aim was to bring together leading experts and active younger researchers in these fields. The mini-workshop was attended by 16 participants from Germany (5), Spain (5), Israel (2), Poland (2), Sweden (1) and United States (1), and the main problems we dealt with can, more specifically, be grouped as follows. 1.1. Complex interpolation of compact operators 1.2. Real interpolation of compactness and similar properties 2.1. Entropy numbers of such embeddings and applications to spectral theory of differential operators 2.2. Entropy techniques in sequence spaces Let us shortly describe these topics. 1.1. An outstanding problem in interpolation theory is the question, whether the complex interpolation method preserves compactness of operators. This is open since 40 years, and by now only partial answers are known. Recently a new and promising general approach has been proposed. We discussed this approach. 1.2. For the real interpolation method, however, it is well-known that it does preserve compactness. So it is quite natural to ask for quantitative versions of this purely qualitative result, for instance in terms of the measure of non-compactness, or in terms of entropy numbers. Another natural question is to study whether similar properties, like weak compactness, for example, are stable under real interpolation as well. These problems were addressed in some talks. 2.1. The sequence of entropy numbers (e_k(T))_{k=1}^\infty of a bounded linear operator T between quasi-Banach spaces can be considered as a quantification of compactness, since T is compact if and only if \lim_{k\to\infty} e_k(T)=0 . The basis for applications to spectral theory is the famous Carl–Triebel inequality, which relates entropy numbers of Riesz operators to its eigenvalues. Many concrete problems lead to the investigation of compact embeddings of certain function spaces, e.g. Sobolev or Besov spaces. In the talks both a survey on the general framework as well as new entropy estimates for specific embeddings were given. 2.2. Using various methods, for instance wavelet or atomic decompositions, the function space embeddings can very often be reduced to embeddings of (fairly complicated) sequence spaces. For the estimation of their entropy numbers one needs many different techniques, some of them quite new. Such techniques also were the subject of talks. Finally, several other aspects of interpolation were treated in talks, e.g. approximation spaces, bilinear interpolation, relation to eigenvalues and operator ideals. We list the abstracts of all talks in chronological order. The scientific program started with two survey lectures by Triebel and Cwikel, leading experts in their fields, followed by two more survey-style talks of the organisers. Then all other participants reported on own recent research results. In addition to this “official” program, which was already scheduled in advance, there was a number of further activities. Several participants offered a second talk, on another topic of common interest, or continued their respective talks in order to explain some technicalities in greater detail. The remaining time was used for many intensive discussions in smaller groups, and on Friday a problem session was held. The aim was to summarize the results of the mini-workshop and to discuss and collect several relevant problems, thus pointing out possible directions for further research in our field. Concerning social activities, one should mention the traditional hiking tour to St. Roman on Wednesday afternoon and the, maybe less traditional, joint session of all three parallel mini-workshops. The aim of this informal interdisciplinary session was to explain very briefly the kind of problems and ideas of our respective areas. Before this meeting there were serious doubts whether the intended goal would be achievable in only a few minutes, but afterwards it was general opinion that we have had a surprisingly inspiring and interesting evening, giving in fact a rough impression of the other two research areas. Last but not least, the organisers would like to express their gratitude to the director and the authorities of the Mathematisches Forschungsinstitut Oberwolfach for making this mini-workshop possible and for the constant support in its organisation and preparation. On behalf of all participants we thank all members of the staff for creating the unique working atmosphere, which made our stay so pleasant and which contributed substantially to the success of our mini-workshop.

New function spaces of BMO type, the John‐Nirenberg inequality, interpolation, and applications

AbstractIn this paper, we introduce and develop some new function spaces of BMO (bounded mean oscillation) type on spaces of homogeneous type or measurable subsets of spaces of homogeneous type. The new function spaces are defined by variants of maximal functions associated with generalized approximations to the identity, and they generalize the classical BMO space. We show that the John‐Nirenberg inequality holds on these spaces and they interpolate withLpspaces by the complex interpolation method. We then give applications onLp‐boundedness of singular integrals whose kernels do not satisfy the Hörmander condition. © 2005 Wiley Periodicals, Inc.

On Interpolation of Vector - Valued Banach Lattices and Calderón - Lozanovskii Construction

We establish the connection between the variants of Schechter's complex interpolation methods, Peetre – Gustavsson's interpolation methods, and the Calderón – Lozanovskii construction on vector – valued Banach lattices. As applications, we show that the uniform convexity and the UMD property are stable by interpolation.

Scales of ε-uniform a priori estimates for nonstationary convection-dominated diffusion problems

Convection-diffusion problems of practical interest are often convection-dominated in the sense that the diffusion parameter e satisfies 0<e≪1. Standard error estimates for numerical approximations of such problems mostly contain constants which depend either reciprocally on e or at least on higher order norms of the solution. These norms possibly increase to infinity in the hyperbolic limit e→0. For the Lagrange-Galerkin scheme such error estimates have been proven by Douglas and Russell, Pironneau or Suli. In the particular case of the Navier-Stokes equations Johnson, Rannacher and Boman have observed that some of those constants which arise in existing analyses depend exponentially on the Reynolds number. Consequently, standard error analyses have no real meaning in the case of convection-dominance. Recently, Bause and Knabner have developed a rigorous e-uniform convergence theory for finite element and Lagrange-Galerkin approximations of convection-dominated diffusion problems. The error analysis is heavily based on e-uniform a priori estimates for the solution of the continuous problems. In their work such e-uniform estimates are established in the Sobolev spacesL2(Ω) andH2(Ω) in a Lagrangian framework by transforming the convection-diffusion problem into subcharacteristic coordinates. To obtain the e-uniformH2-bound, strong conditions about the right-hand side which cannot realistically be assumed in practice are supposed. In this paper we demonstrate how the method of complex interpolation of Banach spaces can be employed to derive elegantly e-uniform a priori estimates in intermediate Sobolev spaces of fractional order. We will observe that in particularHα-regularity of the data has to be supposed to get an e-uniformHα-bound for the solution. To derive the desired estimates we use several complex interpolation results which we provide first.

NATURAL BILINEAR FORMS, NATURAL SESQUILINEAR FORMS AND THE ASSOCIATED DUALITY ON NON-COMMUTATIVE Lp-SPACES

In the author's previous paper, he constructed a complex one-parameter family of non-commutative Lp-spaces [Formula: see text], [Formula: see text], 1 &lt; p &lt; ∞, for a von Neumann algebra ℳ with respect to a fixed faithful normal semi-finite weight φ on ℳ by using Calderón's complex interpolation method. In this paper, we will construct bounded non-degenerate bilinear forms &lt; , &gt;p,(α) on [Formula: see text], [Formula: see text], 1 &lt; p &lt;∞, 1/p + 1/q = 1, and bounded non-degenerate sesquilinear forms ( | )p,(α) on [Formula: see text], [Formula: see text], 1 &lt; p &lt; ∞, 1/p + 1/q = 1, and by using general theory of the complex interpolation method we show the reflexivity of [Formula: see text] and the duality between [Formula: see text] and [Formula: see text] via &lt; , &gt;p,(α) (or the duality between [Formula: see text] and [Formula: see text] via ( | )p,(α)). Moreover, we discuss bimodule properties of [Formula: see text].

Constructions of Non-Commutative Lp-Spaces with a Complex Parameter Arising from Modular Actions

For a von Neumann algebra ℳ and a weight φ on ℳ, we will construct a complex one-parameter family [Formula: see text] of non-commutative Lp-spaces by using Calderón's complex interpolation method. This is a simultaneous and complete extension of the construction of non-commutative Lp-spaces by H. Kosaki and M. Terp. Moreover, we will show that for each p, all the parametrized Lp-spaces are mutually isometrically isomorphic as Banach spaces via natural maps.

Interpolation of families $\{L^{p(\gamma)}_{\mu(\gamma)}, \gamma\in\Gamma\}$

We identify the intermediate space of a complex interpolation family -in the sense of Coifman, Cwikel, Rochberg, Sagher and Weiss- of Lp spaces with change of measure, for the complex interpolation method associated to any analytic functional.

One-sided compactness results for Aronszajn-Gagliardo functors

We show that if T: A 0 → B 0 is compact and T: A 1 → B 1 is bounded, then T: G( A ̄ ) → H( B ̄ ) is also compact. Here G (resp. H) stands for certain minimal (resp. maximal) functors in the sense of Aronszajn and Gagliardo. Applications to the real method, the “±” method, and the complex interpolation method are given. We also establish a compactness result of the type mentioned above for N-tuples.

Complemented Subspaces of Spaces Obtained by Interpolation

If Z is a quotient of a subspace of a separable Banach space X, and V is any separable Banach space, then there is a Banach couple (A0, A1) such that A0 and A1 are isometric to X ⊕ V , and any intermediate space obtained using the real or complex interpolation method contains a complemented subspace isomorphic to Z. Thus many properties of Banach spaces, including having non-trivial cotype, having the Radon-Nikodym property, and having the analytic unconditional martingale difference sequence property, do not pass to intermediate spaces.