Interpolation Spaces Research Articles

Some new results related to Lorentz GΓ-spaces and interpolation

We compute the K-functional related to some couple of spaces as small or classical Lebesgue space or Lorentz-Marcinkiewicz spaces completing the results of [11]. This computation allows to determine the interpolation space in the sense of Peetre for such couple. It happens that the result is always a GΓ-space, since this last space covers many spaces. The motivations of such study are various, among them we wish to obtain a regularity estimate for the so called very weak solution of a linear equation in a domain Ω with data in the space of the integrable function with respect to the distance function to the boundary of Ω.

New Young Inequalities and Applications

We establish upper bounds for the convolution operator acting between interpolation spaces. This gives new Young inequalities in the context of Lorentz–Karamata spaces, grand Lebesgue spaces and small Lebesgue spaces besides many other known results. Furthermore, we use this abstract Young inequality to prove a bilinear interpolation theorem for limit interpolation methods. Finally, we show applications to the study of bilinear multipliers.

Well-Posed Geoscientific Visualization Through Interactive Color Mapping

Scientific visualization aims to present numerical values, or categorical information, in a way that enables the researcher to make an inference that furthers knowledge. Well-posed visualizations need to consider the characteristics of the data, the display environment, and human capacity. In the geosciences, visualizations are commonly applied to spatially varying continuous information or results. In this contribution we make use of a suite of newly written computer applications which enable spatially varying data to be displayed in a performant graphics environment. We present a comparison of color-mapping using illustrative color spaces (RGB, CIELAB). The interactive applications display the gradient paths through the chosen color spaces. This facilitates the creation of color-maps that accommodate the non-uniformity of human color perception, producing an image where genuine features are seen. We also take account of aspects of a dataset such as parameter uncertainty. For an illustrative case study using a seismic tomography result, we find that the use of RGB color-mapping can introduce non-linearities in the visualization, potentially leading to incorrect inference. Interpolation in CIELAB color space enables the creation of perceptually uniform linear gradients that match the underlying data, along with a simply computable metric for color difference, ΔE. This color space assists accuracy and reproducibility of visualization results. Well-posed scientific visualization requires both visual literacy and visual numeracy on an equal footing with clearly written text. It is anticipated that this current work, with the included color-maps and software, will lead to wider usage of informed color-mapping in the geosciences.

Some properties of embeddings of rearrangement invariant spaces

Let and be rearrangement invariant spaces on , and let . This embedding is said to be strict if the functions in the unit ball of the space have absolutely equicontinuous norms in . For the main classes of rearrangement invariant spaces necessary and sufficient conditions are obtained for an embedding to be strict, and also the relationships this concept has with other properties of embeddings are studied, especially the property of disjoint strict singularity. In the final part of the paper, a characterization of the property of strict embedding in terms of interpolation spaces is obtained. Bibliography: 23 titles.

Assessing the Global Wind Atlas and local measurements for bias correction of wind power generation simulated from MERRA-2 in Brazil

NASA's MERRA-2 reanalysis is a widely used dataset in renewable energy resource modelling. The Global Wind Atlas (GWA) has been used to bias-correct MERRA-2 data before. There is, however, a lack of an analysis of the performance of MERRA-2 with bias correction from GWA on different spatial levels – and for regions outside of Europe, China or the United States. This study therefore evaluates different methods for wind power simulation on four spatial resolution levels from wind park to national level in Brazil. In particular, spatial interpolation methods and spatial as well as spatiotemporal wind speed bias correction using local wind speed measurements and mean wind speeds from the GWA are assessed. By validating the resulting timeseries against observed generation it is assessed at which spatial levels the different methods improve results – and whether global information derived from the GWA can compete with locally measured wind speed data as a source of bias correction. Results show that (i) bias correction with the GWA improves results on state, sub-system, and national-level, but not on wind park level, that (ii) the GWA improves results comparably to local measurements, and that (iii) complex spatial interpolation methods do not contribute in improving quality of the simulation.

A Riesz-Thorin type interpolation theorem in Euclidean Jordan algebras

In a Euclidean Jordan algebra V of rank n which carries the trace inner product, to each element a we associate the eigenvalue vector λ(a) in Rn whose components are the eigenvalues of a written in the decreasing order. For any p∈[1,∞], we define the spectral p-norm of a to be the p-norm of λ(a) in Rn. In a recent paper, based on the K-method of real interpolation theory and a majorization technique, we described an interpolation theorem for a linear transformation on V relative to the same spectral p-norm. In this paper, using complex function theory methods, we describe a Riesz-Thorin type interpolation theorem relative to two spectral p-norms. We illustrate the result by estimating the norms of certain special linear transformations such as Lyapunov transformations, quadratic representations, and positive transformations.

Functional Calculus on Real Interpolation Spaces for the Series Generators of C_0-groups

In this paper, discus functional calculus properties of C_0-groups on real interpolation spaces using transference principles. Obtain interpolation versions of the classical transference principle for bounded groups and of a recent transference principle for unbounded groups. Then showed in (Markus, H., & Jan, R. 2016) that all group sequence of generators on a Banach space has a bounded H_0^∞-calculus on real interpolation spaces. Additional results are derived from this.

Some equivalent definitions of Besov spaces of generalized smoothness

AbstractIn this paper, we present some alternative definitions of Besov spaces of generalized smoothness, defined via Littlewood–Paley‐type decomposition, involving weak derivatives, polynomials, convolutions and generalized interpolation spaces.

Singular limit problem for the Keller–Segel system and drift–diffusion system in scaling critical spaces

We consider a singular limit problem for the Cauchy problem of the Keller–Segel equation in a critical function space. We show that a solution to the Keller–Segel system in a scaling critical function space converges to a solution to the drift–diffusion system of parabolic–elliptic type (the simplified Keller–Segel model) in the critical space strongly as the relaxation time $$\tau \rightarrow \infty $$ . For the proof of singular limit problem, we employ generalized maximal regularity for the heat equation and use it systematically with the sequence of embeddings between the interpolation spaces $$\dot{B}^s_{q,\sigma }(\mathbb {R}^n)$$ and $$\dot{F}^s_{q,\sigma }(\mathbb {R}^n)$$ .

Jump inequalities via real interpolation

Jump inequalities are the $r=2$ endpoint of L\'epingle's inequality for $r$-variation of martingales. Extending earlier work by Pisier and Xu we interpret these inequalities in terms of Banach spaces which are real interpolation spaces. This interpretation is used to prove endpoint jump estimates for vector-valued martingales and doubly stochastic operators as well as to pass via sampling from $\mathbb{R}^{d}$ to $\mathbb{Z}^{d}$ for jump estimates for Fourier multipliers.

On the equivalence of the [formula omitted]-functional and the modulus of continuity on the Morrey spaces

In this note, the Morrey spaces and the Sobolev–Morrey spaces are considered. In particular, the K-functional with respect to these spaces is estimated from above and below. As an application, we characterize the Nikol’skii–Besov–Morrey spaces via real interpolation.

Functional inequalities for a class of nonlocal hypoelliptic equations of Hörmander type

We consider a class of second-order partial differential operators A of Hörmander type, which contain as a prototypical example a well-studied operator introduced by Kolmogorov in the ’30s. We analyse some properties of the nonlocal operators driven by the fractional powers of A, and we introduce some interpolation spaces related to them. We also establish sharp pointwise estimates of Harnack type for the semigroup associated with the extension operator. Moreover, we prove both global and localised versions of Poincaré inequalities adapted to the underlying geometry.

Multilevel approximation of parametric and stochastic PDES

We analyze the complexity of the sparse-grid interpolation and sparse-grid quadrature of countably-parametric functions which take values in separable Banach spaces with unconditional bases. Assuming a suitably quantified holomorphic dependence on the parameters, we establish dimension-independent convergence rate bounds for sparse-grid approximation schemes. Analogous results are shown in the case that the parametric families are obtained as approximate solutions of corresponding parametric-holomorphic, nonlinear operator equations as considered in [A. Cohen and A. Chkifa and Ch. Schwab: Breaking the curse of dimensionality in sparse polynomial approximation of parametric PDEs, J. Math. Pures Appl. 103 (2015) 400–428], for example by means of stable, finite-dimensional approximations. We discuss in detail nonlinear Petrov–Galerkin projections. Error and convergence rate bounds for constructive and explicit multilevel, sparse tensor approximation schemes combining sparse-grid interpolation in the parameter space and general, multilevel discretization schemes in the physical domain are proved. The present results unify and generalize earlier works in terms of the admissible multilevel approximations in the physical domain (comprising general stable Petrov–Galerkin and discrete Petrov–Galerkin schemes, collocation and stable domain approximations) and in terms of the admissible operator equations (comprising smooth, nonlinear locally well-posed operator equations). Additionally, a novel computational strategy to localize sequences of nested index sets for the anisotropic Smolyak interpolation in parameter space is developed which realizes best [Formula: see text]-term benchmark convergence rates. We also consider Smolyak-type quadratures in this general setting, for which we establish improved convergence rates based on cancellations in the integrands’ gpc expansions by symmetries of quadratures and the probability measure [J.Z̃ech and Ch.S̃chwab: Convergence rates of high dimensional Smolyak quadrature, Report 2017-27, SAM ETH Zürich (2017)]. Several examples illustrating the abstract theory include domain uncertainty quantification (UQ) for general, linear, second-order, elliptic advection–reaction–diffusion equations on polygonal domains, where optimal convergence rates of FEM are known to require local mesh refinement near corners. Further applications of the presently developed theory comprise evaluations of posterior expectations in Bayesian inverse problems.

Real interpolation method for transfer function approximation of distributed parameter system

Distributed parameter system (DPS) presents one of the most complex systems in the control theory. The transfer function of a DPS possibly contents: rational, nonlinear and irrational components. This thing leads that studies of the transfer function of a DPS are difficult in the time domain and frequency domain. In this paper, a systematic approach is proposed for linearizing DPS. This approach is based on the real interpolation method (RIM) to approximate the transfer function of DPS by rational-order transfer function. The results of the numerical examples show that the method is simple, computationally efficient, and flexible.

An Efficient Algorithm for Roadways Obstacles Detection using Smartphone in Infrastructure-less Environment

Apart from having smart systems in modern vehicles, roadways traffic positions in infrastructure-free environment are in various ways not suitable for driving which causes roadways accidents. Smart systems are costly, training and maintenance is not practiced by all, hence they are not useful for vehicles running in infrastructure-less environment. Modern days Smartphone are multifunctional, easily operated and consists of numerous sensors, can be utilized for roadways obstacles detection in infrastructure-less environment. In this study, we proposed an efficient algorithm depth learning approach for roadways obstacles detection using Smartphone which is useful for planned as well as unplanned roads. In this, road information is collected using Smartphone accelerometer sensors, data normalization take place using Euler angle method and location determination of obstacles take place using space interpolation method. In comparison to other same type of approach, our algorithm is efficient and cost-effective.

Strict density inequalities for sampling and interpolation in weighted spaces of holomorphic functions

Answering a question of Lindholm, we prove strict density inequalities for sampling and interpolation in Fock spaces of entire functions in several complex variables defined by a plurisubharmonic weight. In particular, these spaces do not admit a set that is simultaneously sampling and interpolating. To prove optimality of the density conditions, we construct sampling sets with a density arbitrarily close to the critical density.The techniques combine methods from several complex variables (estimates for ∂¯) and the theory of localized frames in general reproducing kernel Hilbert spaces (with no analyticity assumed). The abstract results on Fekete points and deformation of frames may be of independent interest.

Bounded and Compact Toeplitz Operators with Positive Measure Symbol on Fock-Type Spaces

In this note, we discuss the Bergman projection P and Toeplitz operators $$T_{\mu }$$ with positive measure symbol $$\mu $$ between $$F^p_{\Psi }(\mathbb {C}^n)$$ and $$F^q_{\Psi }(\mathbb {C}^n)$$ for $$1\le p, q\le \infty $$ . We first show that P is a bounded projection from $$L^p_{\Psi }$$ onto $$F^p_{\Psi }$$ when $$1\le p\le \infty $$ , and then apply it to obtain results on the complex interpolation and the duality of the Fock-type spaces. Furthermore, we obtain the equivalent conditions for the boundedness and compactness of $$T_{\mu }$$ in terms of the averaging function and the Berezin transform, which extend the main results about Toeplitz operators of Seip and Youssfi (J Geom Anal 23:170–201, 2013).

Complex interpolation and Calderón–Mityagin couples of Morrey spaces

We study interpolation spaces between global Morrey spaces and between local Morrey spaces. We prove that for a wide class of couples of these spaces the upper complex Calderon spaces are not described by the K-method of interpolation. A by-product of our results is that couples of Morrey spaces belonging to this class are not Calderon–Mityagin couples. A Banach couple (X0,X1) is said to have the universal K-property if all relative interpolation spaces from any Banach couple to (X0,X1) are relatively K-monotone. A couple of local Morrey spaces is proved to have the universal K-property once it is a Calderon–Mityagin couple.

Smooth Deep Image Generator from Noises

Generative Adversarial Networks (GANs) have demonstrated a strong ability to fit complex distributions since they were presented, especially in the field of generating natural images. Linear interpolation in the noise space produces a continuously changing in the image space, which is an impressive property of GANs. However, there is no special consideration on this property in the objective function of GANs or its derived models. This paper analyzes the perturbation on the input of the generator and its influence on the generated images. A smooth generator is then developed by investigating the tolerable input perturbation. We further integrate this smooth generator with a gradient penalized discriminator, and design smooth GAN that generates stable and high-quality images. Experiments on real-world image datasets demonstrate the necessity of studying smooth generator and the effectiveness of the proposed algorithm.

ClusterGAN: Latent Space Clustering in Generative Adversarial Networks

Generative Adversarial networks (GANs) have obtained remarkable success in many unsupervised learning tasks and unarguably, clustering is an important unsupervised learning problem. While one can potentially exploit the latent-space back-projection in GANs to cluster, we demonstrate that the cluster structure is not retained in the GAN latent space. In this paper, we propose ClusterGAN as a new mechanism for clustering using GANs. By sampling latent variables from a mixture of one-hot encoded variables and continuous latent variables, coupled with an inverse network (which projects the data to the latent space) trained jointly with a clustering specific loss, we are able to achieve clustering in the latent space. Our results show a remarkable phenomenon that GANs can preserve latent space interpolation across categories, even though the discriminator is never exposed to such vectors. We compare our results with various clustering baselines and demonstrate superior performance on both synthetic and real datasets.