Interpolation Spaces Research Articles

Interpolation of Protoquantum Spaces

We consider the complex interpolation method in the context of protoquantum spaces. We apply it to construct an ℒp-protoquantization of an arbitrary normed space for 1 < p < ∞.

The periodic version of the Da Prato\u2013Grisvard theorem and applications to the bidomain equations with FitzHugh\u2013Nagumo transport

In this article, the periodic version of the classical Da Prato–Grisvard theorem on maximal {{L}}^p-regularity in real interpolation spaces is developed, as well as its extension to semilinear evolution equations. Applying this technique to the bidomain equations subject to ionic transport described by the models of FitzHugh–Nagumo,Aliev–Panfilov, or Rogers–McCulloch, it is proved that this set of equations admits a unique, strongT-periodic solution in a neighborhood of stable equilibrium points provided it is innervated by T-periodic forces.

A low cost and highly accurate technique for big data spatial-temporal interpolation

The high velocity, variety and volume of data generation by today's systems have necessitated Big Data (BD) analytic techniques. This has penetrated a wide range of industries; BD as a notion has various types and characteristics, and therefore a variety of analytic techniques would be required. The traditional analysis methods are typically unable to analyse spatial-temporal BD. Interpolation is required to approximate the values between the already existing data points, yet since there exist both location and time dimensions, only a multivariate interpolation would be appropriate. Nevertheless, existing software are unable to perform such complex interpolations. To overcome this challenge, this paper presents a layer by layer interpolation approach for spatial-temporal BD. Developing this layered structure provides the opportunity for working with much smaller linear system of equations. Consequently, this structure increases the accuracy and stability of numerical structure of the considered BD interpolation. To construct this layer by layer interpolation, we have used the good properties of Radial Basis Functions (RBFs). The proposed new approach is applied to numerical examples in spatial-temporal big data and the obtained results confirm the high accuracy and low computational cost. Finally, our approach is applied to explore one of the air pollution indices, i.e. daily PM2.5 concentration, based on different stations in the contiguous United States, and it is evaluated by leave-one-out cross validation.

Complex interpolation of various subspaces of Morrey spaces

In this article, we discuss the complex interpolation of various closed subspaces of Morrey spaces.We have been considering some closed subspaces of Morrey spacesin our earlier works.The main property that we need is the lattice property but in connectionwith the diamond spaces defined by Yuan et al. (2015),it seems to be natural to consider the convolution property as well.Our result will extend the resultsby Hakim and Sawano (2017) and Hakim et al. (2017).

Optimal local embeddings of Besov spaces involving only slowly varying smoothness

The aim of the paper is to establish (local) optimal embeddings of Besov spaces Bp,r0,b involving only a slowly varying smoothness b. In general, our target spaces are outside of the scale of Lorentz–Karamata spaces. In particular, we improve results from Caetano et al. (2011), where the targets are (local) Lorentz–Karamata spaces. To derive such results, we apply limiting real interpolation techniques and weighted Hardy-type inequalities.

Calderon's Complex Interpolation of Morrey Spaces

In this note we will discuss some results related to complex interpolation of Morreyspaces. We first recall the Riesz-Thorin interpolation theorem in Section 1.After that, we discuss a partial generalization of this theorem in Morrey spaces proved in \cite{St}.We also discuss non-interpolation property of Morrey spaces given in \cite{BRV99, RV}.In Section 3, we recall the definition of Calder\'on's complex interpolation method andthe description of complex interpolation of Lebesgue spaces.In Section 4, we discuss the description of complex interpolation of Morrey spaces given in\cite{CPP98, HS2, Lemarie, LYY}. Finally, we discuss the description of complex interpolationof subspaces of Morrey spaces in the last section.This note is a summary of the current research about interpolation of Morrey spaces,generalized Morrey spaces, and their subspaces in\cite{CPP98, HS, HS2, H, H4, Lemarie, LYY}.

Improvement of Magnetic-Abrasive Finishing of Nonuniform Products Made of High-Speed Steel in Digital Conditions

The paper considers the implementation of magnetic-abrasive finishing in the conditions of digital technologies, in particular within CNC systems. The process was implemented through the development of a special device for MAF based on vertical processing center with CNC – Emco Concept Mill 250 and the definition of the combination of working motions of a billet in the interpole space of a magnetic system and their speeds. The result of the study are optimal combinations of working motions and their speeds taking into account formed surface roughness and processing efficiency.

Well‐posedness and stability for a mixed order system arising in thin film equations with surfactant

AbstractThe objective of the present work is to provide a well‐posedness result for a capillary driven thin film equation with insoluble surfactant. The resulting parabolic system of evolution equations is not only strongly coupled and degenerated, but also of mixed orders. To the best of our knowledge the only well‐posedness result for a capillary driven thin film with surfactant is provided in [4] by the same author, where a severe smallness condition on the surfactant concentration is assumed to prove the result. Thus, in spite of an intensive analytical study of thin film equations with surfactant during the last decade, a proper well‐posedness result is still missing in the literature. It is the aim of the present paper to fill this gap. Furthermore, we apply a recently established result on asymptotic stability in interpolation spaces [15] to prove that the flat equilibrium of our system is asymptotically stable.

Interpolation of compact bilinear operators

This paper is devoted to the study of the stability of the compactness property of bilinear operators acting on the products of interpolated Banach spaces. We prove one-sided compactness results for bilinear operators on products of Banach spaces generated by abstract methods of interpolation, in the sense of Aronszajn and Gagliardo. To get these results, we prove a key one-sided bilinear interpolation theorem on compactness for bilinear operators on couples satisfying an extra approximation property. We give applications to general cases, including Peetre’s method and the general real interpolation methods.

A complex variable boundary point interpolation method for the nonlinear Signorini problem

A complex variable boundary point interpolation method (CVBPIM) is presented for numerical analysis of the nonlinear Signorini problem. To reduce the computational cost and to improve the stability of the point interpolation method, a complex variable point interpolation method (CVPIM) is developed to construct meshless shape functions with interpolation properties. By using a projection technique to deal with the nonlinear inequality boundary conditions, and combining boundary integral equations with the CVPIM to establish discrete linear algebraic systems, the CVBPIM is an easy-to-implement and boundary-only meshless method for nonlinear Signorini problems. Numerical results are presented to show the accuracy and efficiency of the method.

Shilnikov-type dynamics in three-dimensional piecewise smooth maps

We show the existence of Shilnikov-type dynamics and bifurcation behaviour in general discrete three-dimensional piecewise smooth maps and give analytical results for the occurence of such dynamical behaviour. Our main example in fact shows a `two-sided' Shilnikov dynamics, i.e. simultaneous looping and homoclinic intersection of the one-dimensional eigenmanfolds of fixed points on both sides of the border. We also present two complementary methods to analyse the return time of an orbit to the border: one based on recursion and another based on complex interpolation.

Complex Interpolation and the Adams Theorem

Recently, many researchers investigated the description of the complex interpolation of Morrey spaces. Among others, the second complex interpolation of Morrey spaces turned out to be the Calderon product of Morrey spaces. In this paper as an application of this fact, we propose an improvement of the Adams theorem asserting that the fractional integral operator maps Morrey spaces to other Morrey spaces.

Parabolic evolution equations in interpolation spaces: boundedness, stability, and applications

We develop a general approach for the study of the boundedness and stability of solution to general parabolic semi-linear evolution equations in real-interpolation spaces. Our approach yields a unified method to handle problems in Newtonian viscous incompressible fluid flows. Moreover, our method will be applied to derive new results for the existence and polynomial stability of bounded solutions to the Ornstein–Uhlenbeck semi-linear equations and various diffusion equations with rough coefficients. The method is based on a combination of interpolation functors, duality estimates, smoothing properties of the linearized equation, and fixed point arguments.

Rhodonea Curves as Sampling Trajectories for Spectral Interpolation on the Unit Disk

Rhodonea curves are classical planar curves in the unit disk with the characteristic shape of a rose. In this work, we use these rose curves as sampling trajectories to create novel nodes for spectral interpolation on the disk. By generating the interpolation spaces with a parity-modified Chebyshev–Fourier basis, we will prove the unisolvence of the interpolation on the rhodonea nodes. Properties such as continuity, convergence, and numerical condition of the interpolation scheme depend on the spectral structure of the interpolation space. For rectangular spectral index sets, we show that the interpolant is continuous at the center, that the Lebesgue constant grows only logarithmically, and that the scheme converges fast if the interpolated function is smooth. Finally, we show that the scheme can be implemented efficiently using a fast Fourier method and that it can be applied to define a Clenshaw–Curtis quadrature on the disk.

On the Limit Regularity in Sobolev and Besov Scales Related to Approximation Theory

We study the interrelation between the limit $$L_p(\Omega )$$-Sobolev regularity $$\overline{s}_p$$ of (classes of) functions on bounded Lipschitz domains $$\Omega \subseteq \mathbb {R}^d$$, $$d\ge 2$$, and the limit regularity $$\overline{\alpha }_p$$ within the corresponding adaptivity scale of Besov spaces $$B^\alpha _{\tau ,\tau }(\Omega )$$, where $$1/\tau =\alpha /d+1/p$$ and $$\alpha >0$$ ($$p>1$$ fixed). The former determines the convergence rate of uniform numerical methods, whereas the latter corresponds to the convergence rate of best N-term approximation. We show how additional information on the Besov or Triebel–Lizorkin regularity may be used to deduce upper bounds for $$\overline{\alpha }_p$$ in terms of $$\overline{s}_p$$ simply by means of classical embeddings and the extension of complex interpolation to suitable classes of quasi-Banach spaces due to Kalton et al. (in: De Carli and Milman (ed) Interpolation theory and applications, American Mathematical Society, Providence, 2007). The results are applied to the Poisson equation, to the p-Poisson problem, and to the inhomogeneous stationary Stokes problem. In particular, we show that already established results on the Besov regularity for the Poisson equation are sharp.

Bridging the hybrid high-order and virtual element methods

Abstract We present a unifying viewpoint on hybrid high-order and virtual element methods on general polytopal meshes in dimension $2$ or $3$, in terms of both formulation and analysis. We focus on a model Poisson problem. To build our bridge (i) we transcribe the (conforming) virtual element method into the hybrid high-order framework and (ii) we prove $H^m$ approximation properties for the local polynomial projector in terms of which the local virtual element discrete bilinear form is defined. This allows us to perform a unified analysis of virtual element/hybrid high-order methods, that differs from standard virtual element analyses by the fact that the approximation properties of the underlying virtual space are not explicitly used. As a complement to our unified analysis we also study interpolation in local virtual spaces, shedding light on the differences between the conforming and nonconforming cases.

Ventcel’ boundary value problems for elliptic Waldenfels operators

In this paper we study a class of first-order Ventcel’ boundary value problems for second-order, elliptic Waldenfels integro-differential operators. More precisely, by using real analysis techniques such as Strichartz norms and the complex interpolation method we prove existence and uniqueness theorems in the framework of Sobolev and Besov spaces of \(L^{p}\) type which extend earlier theorems due to Bony–Courrège–Priouret and Runst–Youssfi to the general degenerate case. Our proof is based on various maximum principles for second-order, elliptic Waldenfels operators with discontinuous coefficients in the framework of \(L^{p}\) Sobolev spaces.

Real-Variable Characterizations of Hardy–Lorentz Spaces on Spaces of Homogeneous Type with Applications to Real Interpolation and Boundedness of Calderón–Zygmund Operators

AbstractLet (𝒳,d,μ) be a space of homogeneous type, in the sense of Coifman and Weiss, with the upper dimensionω. Assume that η ∈(0, 1) is the smoothness index of the wavelets on 𝒳 constructed by Auscher and Hytönen. In this article, via grand maximal functions, the authors introduce the Hardy–Lorentz spacesH*p,q(𝒳)H_*^{p,q}\left( \mathcal{X} \right)with the optimal rangep∈(ωω+η,∞)p \in \left( {{\omega \over {\omega + \eta }},\infty } \right)andq ∈(0, ∞]. When andp∈(ωω+η,1]p \in ({\omega \over {\omega + \eta }},1]q ∈(0, ∞], the authors establish its real-variable characterizations, respectively, in terms of radial maximal functions, non-tangential maximal functions, atoms, molecules, and various Littlewood–Paley functions. The authors also obtain its finite atomic characterization. As applications, the authors establish a real interpolation theorem on Hardy–Lorentz spaces, and also obtain the boundedness of Calderón–Zygmund operators on them including the critical cases. The novelty of this article lies in getting rid of the reverse doubling assumption ofμby fully using the geometrical properties of 𝒳 expressed via its dyadic reference points and dyadic cubes and, moreover, the results in the caseq ∈(0, 1) of this article are also new even when 𝒳 satisfies the reverse doubling condition.

Garsia–Rodemich spaces: Local maximal functions and interpolation

We characterize the Garsia-Rodemich spaces associated with a rearrangement invariant space via local maximal operators. Let $Q_{0}$ be a cube in $R^{n}$. We show that there exists $s_{0}\in(0,1),$ such that for all $0<s<s_{0},$ and for all r.i. spaces $X(Q_{0}),$ we have% \[ GaRo_{X}(Q_{0})=\{f\in L^{1}(Q_{0}):\Vert f\Vert_{GaRo_{X}}\simeq\Vert M_{0,s,Q_{0}}^{\#}f\Vert_{X}<\infty\}, \] where $M_{0,s,Q_{0}}^{\#}$ is the Stromberg-Jawerth-Torchinsky local maximal operator. Combined with a formula for the $K-$functional of the pair $(L^{1},BMO)$ obtained by Jawerth-Torchinsky, our result shows that the $GaRo_{X}$ spaces are interpolation spaces between $L^{1}$ and $BMO.$ Among the applications, we prove, using real interpolation, the monotonicity under rearrangements of Garsia-Rodemich type functionals. We also give an approach to Sobolev-Morrey inequalities via Garsia-Rodemich norms, and prove necessary and sufficient conditions for $GaRo_{X}(Q_{0})=X(Q_{0}).$ Using packings, we obtain a new expression for the $K-$functional of the pair $(L^{1},BMO)$.

Study of the Blow Up of the Maximal Solution to the Three-Dimensional Magnetohydrodynamic System in Lei-Lin-Gevrey Spaces

In this paper, we investigate existence, uniqueness and blowup in finite time of the local solution to the three dimensional magnetohydrodynamic system, in Gevrey-Lei-Lin spaces. To prove the blowup results and give the blow profile as a function of time, two key points are used. The first is a frequency decomposition of the spectrum of the initial data. This allows to use Leray theory. The second is a technical lemma we proved to state that the Lei-Lin space is an interpolation space between the Gevrey-Lei-Lin and the Lebesgue square integrable functions spaces. To prove uniqueness, we use a penalization procedure and energy methods. About existence, we split the initial condition into low frequencies part and high frequencies part. The former are considered as initial data to the linear part of the system. The latter will be taken as small as needed, so that smallness theory applies and allows to run a fixed point argument.