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In this paper, the maximal‐weighted Hölder regularity for the fractional abstract Cauchy problem is presented. First, two classes of novel interpolation spaces constructed by resolvent families are portrayed. Afterward, by applying the resolvent method and the properties of interpolation spaces, the maximal‐weighted Hölder regularity of the mild solution is established in weighted Hölder space C0β,γ(X)$C_0^{\beta ,\gamma }(X)$ (0≤γ≤β$0\le \gamma \le \beta$, 0<β<1$0<\beta <1$). Eventually, an instance for the fractional differential equation of 2m‐order multidimensional parabolic type is provided to verify the reasonability of main conclusions. [ABSTRACT FROM AUTHOR]

This paper is devoted to the existence of random attractors for rough partial differential equations driven by nonlinear multiplicative Hölder rough paths with exponents in (1 / 3 , 1 / 2 ]. Our approach relies upon rough paths theory and stopping times analysis in a suitable scale of interpolation spaces. The core step is to derive the adequate algebraic and analytical properties of a sequence of stopping times, which allows us to establish the required compact tempered absorbing set. The existence of a pullback attractor for the generated random dynamical system is straightforward. An illustrative example is presented by reaction-diffusion equations subjected to fractional Brownian rough paths. [ABSTRACT FROM AUTHOR]

In this paper, we studied the Hölder regularities of solutions to an abstract fractional differential equation, which is regarded as an abstract version of fractional Rayleigh–Stokes problems, rising up to describing a non-Newtonian fluid with a Riemann–Liouville fractional derivative. The purpose of this article was to establish the Hölder regularities of mild solutions, classical solutions, and strict solutions. We introduced an interpolation space in terms of an analytic resolvent to lower the spatial regularity of initial value data. By virtue of the properties of analytic resolvent and the interpolation space, the Hölder regularities were obtained. As applications, the main conclusions were applied to the regularities of fractional Rayleigh–Stokes problems. [ABSTRACT FROM AUTHOR]

This paper extends the work of Fasshauer and Ye [Reproducing kernels of Sobolev spaces via a Green kernel approach with differential operators and boundary operators, Adv. Comput. Math. 38(4) (2011) 891921] in two different ways, namely, new kernels and associated native spaces are identified as crucial Hilbert spaces in applied mathematics. These spaces include the following spaces defined in bounded domains Ω ⊂ ℝ d with smooth boundary: homogeneous Sobolev–Slobodeckij̆ spaces, denoted by H 0 s (Ω ¯) , and Sobolev–Slobodeckij̆ spaces, denoted by H s (Ω) , where s > d 2 . Our goal is accomplished by obtaining the Green's solutions of equations involving the fractional Laplacian and fractional differential operators defined through interpolation theory. We provide a proof that the Green's kernels satisfying these problems are symmetric and positive definite reproducing kernels of H 0 s (Ω ¯) and H s (Ω) , respectively. Constructing kernels in these two ways enables the characterization of functions in native spaces based on their regularity. The Galerkin/collocation method, based on these kernels, is employed to solve various fractional problems, offering explicit or simplified calculations and efficient solutions. This method yields improved results with reduced computational costs, making it suitable for complex domains. [ABSTRACT FROM AUTHOR]

In this paper, we first characterize the behaviour of fractional resolvent families on the real interpolation spaces (X , D (A)) θ , p , θ ∈ (0 , 1) , p ∈ [ 1 , ∞ ]. Second, we establish strict Hölder regularity in space and time to an abstract degenerate fractional order differential equations. We also provide an application to illustrate our results. [ABSTRACT FROM AUTHOR]

Little seems to be known about the invariant manifolds for stochastic partial differential equations (SPDEs) driven by nonlinear multiplicative noise. Here we contribute to this aspect and analyze the Lu-Schmalfuß conjecture [Garrido-Atienza, et al., (2010) [14] ] on the existence of stable manifolds for a class of parabolic SPDEs driven by nonlinear multiplicative fractional noise. We emphasize that stable manifolds for SPDEs are infinite-dimensional objects, and the classical Lyapunov-Perron method cannot be applied, since the Lyapunov-Perron operator does not give any information about the backward orbit. However, by means of interpolation theory, we construct a suitable function space in which the discretized Lyapunov-Perron-type operator has a unique fixed point. Based on this we further prove the existence and smoothness of local stable manifolds for such SPDEs. [ABSTRACT FROM AUTHOR]

We study interpolation spaces obtained via a general discrete interpolation method based on a Banach space with an unconditional basis. We find conditions which guarantee that such interpolation spaces have the Opial property and the uniform Opial property. [ABSTRACT FROM AUTHOR]

We know that interpolation spaces in terms of analytic semigroup have a significant role into the study of strict Hölder regularity of solutions of classical abstract Cauchy problem (ACP). In this paper, we first construct interpolation spaces in terms of solution operators in fractional calculus and characterize these spaces. Then we establish strict Hölder regularity of mild solutions of fractional order ACP. [ABSTRACT FROM AUTHOR]

We know that interpolation spaces in terms of analytic semigroup have a significant role into the study of strict Hölder regularity of solutions of classical abstract Cauchy problem (ACP). In this paper, we first construct interpolation spaces in terms of solution operators in fractional calculus and characterize these spaces. Then we establish strict Hölder regularity of mild solutions of fractional order ACP. [ABSTRACT FROM AUTHOR]

We study kernel least-squares estimation under a norm constraint. This form of regularisation is known as Ivanov regularisation and it provides better control of the norm of the estimator than the well-established Tikhonov regularisation. Ivanov regularisation can be studied under minimal assumptions. In particular, we assume only that the RKHS is separable with a bounded and measurable kernel. We provide rates of convergence for the expected squared L² error of our estimator under the weak assumption that the variance of the response variables is bounded and the unknown regression function lies in an interpolation space between L² and the RKHS. We then obtain faster rates of convergence when the regression function is bounded by clipping the estimator. In fact, we attain the optimal rate of convergence. Furthermore, we provide a high-probability bound under the stronger assumption that the response variables have subgaussian errors and that the regression function lies in an interpolation space between L∞ and the RKHS. Finally, we derive adaptive results for the settings in which the regression function is bounded. [ABSTRACT FROM AUTHOR]

Abstract Let 1 ≤ p < ∞. A Banach lattice E is said to be disjointly homogeneous (resp. p -disjointly homogeneous) if two arbitrary normalized disjoint sequences from E contain equivalent in E subsequences (resp. every normalized disjoint sequence contains a subsequence equivalent in E to the unit vector basis of l p). Answering a question raised in the paper [11] , for each 1 < p < ∞ , we construct a reflexive p -disjointly homogeneous rearrangement invariant space on [ 0 , 1 ] whose dual is not disjointly homogeneous. Employing methods from interpolation theory, we provide new examples of disjointly homogeneous rearrangement invariant spaces; in particular, we show that there is a Tsirelson type disjointly homogeneous rearrangement invariant space, which contains no subspace isomorphic to l p , 1 ≤ p < ∞ , or c 0. [ABSTRACT FROM AUTHOR]

Abstract We give a complete characterization of limiting interpolation spaces for the real method of interpolation using extrapolation theory. For this purpose the usual tools (e.g., Boyd indices or the boundedness of Hardy type operators) are not appropriate. Instead, our characterization hinges upon the boundedness of some simple operators (e.g. f ↦ f (t 2) ∕ t , or f ↦ f (t 1 ∕ 2)) acting on the underlying lattices that are used to control the K - and J -functionals. Reiteration formulae, extending Holmstedt's classical reiteration theorem to limiting spaces, are also proved and characterized in this fashion. The resulting theory gives a unified roof to a large body of literature that, using ad-hoc methods, had covered only special cases of the results obtained here. Applications to Matsaev ideals, Grand Lebesgue spaces, Bourgain–Brezis–Mironescu–Maz'ya–Shaposhnikova limits, as well as a new vector valued extrapolation theorems, are provided. [ABSTRACT FROM AUTHOR]

A new method is presented for simulation of nonlinear transmission line circuits based on proper orthogonal decomposition reduction techniques coupled with an efficient interpolatory algorithm. Evaluation of the nonlinear function and corresponding Jacobian is performed in the reduced domain. A key criterion is developed for a priori determination of the dimension of the interpolation space leading to a substantial reduction in the computational cost. The proposed algorithm is applicable to general nonlinear circuits and does not impose any constraints on the topology of the pertinent circuit or type of the nonlinear components. [ABSTRACT FROM AUTHOR]

It is a widely used method, for instance in perturbation theory, to associate with a given C 0 -semigroup its so-called interpolation and extrapolation spaces. In the model case of the shift semigroup acting on L 2 ( R ) , the resulting chain of spaces recovers the classical Sobolev scale. In 2014, the second named author defined the universal interpolation space as the projective limit of the interpolation spaces and the universal extrapolation space as the completion of the inductive limit of the extrapolation spaces, provided that the latter is Hausdorff. In this note we use the notion of the dual with respect to a pivot space in order to show that the aforementioned inductive limit is Hausdorff and already complete if we consider a C 0 -semigroup acting on a reflexive Banach space. If the space is Hilbert, then the inductive limit can be represented as the dual of the projective limit whenever a power of the generator of the initial semigroup is a self-adjoint operator. In the case of the classical Sobolev scale we show that the latter duality holds, and that the two universal spaces were already studied by Laurent Schwartz in the 1950s. Our results and examples complement the approach of Haase, who in 2006 gave a different definition of universal extrapolation spaces in the context of functional calculi. Haase avoids the inductive limit topology precisely for the reason that it a priori cannot be guaranteed that the latter is always Hausdorff. We show that this is indeed the case provided that we start with a semigroup defined on a reflexive Banach space. [ABSTRACT FROM AUTHOR]

For a certain special class of symmetric sequence spaces, we give an explicit relation between the interpolation and extrapolation representations. This relation is carried over to symmetrically normed ideals of compact operators. [ABSTRACT FROM AUTHOR]

In this paper, we consider an infinite dimensional exponential family P of probability densities, which are parametrized by functions in a reproducing kernel Hilbert space H, and show it to be quite rich in the sense that a broad class of densities on Rd can be approximated arbitrarily well in Kullback-Leibler (KL) divergence by elements in P. Motivated by this approximation property, the paper addresses the question of estimating an unknown density p0 through an element in P. Standard techniques like maximum likelihood estimation (MLE) or pseudo MLE (based on the method of sieves), which are based on minimizing the KL divergence between p0 and P, do not yield practically useful estimators because of their inability to efficiently handle the log-partition function. We propose an estimator pn based on minimizing the Fisher divergence, J(po||p) between p0 and p G P, which involves solving a simple finite-dimensional linear system. When p0 G P, we show that the pro- ...in Fisher divergence under the smoothness assumption that logp0 G R(Cß) for some ß > 0, where C is a certain Hilbert-Schmidt operator on H and R(Cß) denotes the image of Cß. We also investigate the misspecified case of p0 G P and show that J(p0||pn) ...infpep J(p0||p) as n...oe, and provide a rate for this convergence under a similar smoothness condition as above. Through numerical simulations we demonstrate that the proposed estimator outperforms the non-parametric kernel den. [ABSTRACT FROM AUTHOR]

We study two variants of measures of non-compactness of operators associated to a Banach operator ideal in the sense of Pietsch. These measures are motivated by the notions of surjective-ideal-compactness and injective-ideal-compactness, defined respectively by Carl and Stephani and by Stephani. Interpolation results on these measures in the cases of Banach couples generated by a single Banach space are given. As an application, we obtain interpolation theorems on p -compact operators and quasi p -nuclear operators. [ABSTRACT FROM AUTHOR]

Consider the abstract algebraic-delay differential system, xi (t) = Ax(t) + F(x(t), a(t)), a(t) = H(xt, at). Here A is a linear operator on D(A) ⊆ X satisfying the Hille-Yosida conditions, x(t) ∊ D(A) ⊆ X, a(t) ∊ Rn, and X is a real Banach space. Let C0 ⊆ D(A) be closed and convex, and K ⊆ Rn be a compact set contained in the ball of radius h > 0 centered at 0. Under suitable Lipschitz conditions on the nonlinearities F and H and a subtangential condition, the system generates a continuous semiflow on a subset of the space of continuous functions C([-h, 0], C0 ×Rn), which is induced by the algebraic constraint. The object of this paper is to find conditions under which this semiflow is also differentiable with respect to initial data. In the motivating example coming from modelling the dynamics of an age structured population, the nonlinearities F and H are not Fré-chet differentiable on the sets C0 × K and C([-h, 0], C0 × K), respectively. The main challenge of obtaining the differentiability of the semiflow is to determine the right type of differentiability and the right phase space. We develop a novel approach to address this problem which also shows how the spaces on which the derivatives of solution operators act reflect the model structure [ABSTRACT FROM AUTHOR]

Let Lp(M) be the space of bounded Lp(M)-quasimartingales. We prove that, with equivalent norms, (Lp0(M), Lp1(M))θ, p = Lp(M), where 1 < p0, p1 ≤ ∞ 1, and .... We also prove that, for 1 < p < q < ∞, (BMOc(M), Hpc(M))... = Hqc(M) and (BMOr(M),Hpr(M))... = Hqr(M), where Hp(M) and BMOˆ(M) are, respectively, the Hardy space and the bounded mean oscillation space of noncommutative quasimartingales. [ABSTRACT FROM AUTHOR]

We consider abstract initial boundary value problems for parabolic differential-operator equations on the rectangle [0,T] x [0,1] with singular data. We use our previous results on normestimates of solutions and R-boundedness of some sets of boundary value problems for abstract elliptic equations with a parameter on [0,1] in a UMD Banach space. Unique solvability of these problems is proved in the Sobolev spaces of vector-valued functions with values in some interpolation spaces. The corresponding estimates for the solutions are also established. We also show completeness of elementary solutions of abstract parabolic boundary value problems. Abstract results are provided by a relevant application to parabolic PDEs. In some cases, the boundary conditions may contain the intermediate points of the interval [0,1] or may be integro-dilfcrcntial. [ABSTRACT FROM AUTHOR]

We study functional calculus properties of C0-groups on real interpolation spaces using transference principles. We obtain interpolation versions of the classical transference principle for bounded groups and of a recent transference principle for unbounded groups. Then we show that each group generator on a Banach space has a bounded -calculus on real interpolation spaces. Additional results are derived from this. [ABSTRACT FROM AUTHOR]

We apply interpolation techniques to study behaviour of the canonical inclusion maps of quasi-Banach spaces of analytic functions on the open unit disk of the plane into (quasi)-Banach function lattices on the closed or open unit disk equipped with a Borel measure. These results are applied to abstract Hardy spaces generated by symmetric spaces. We investigate relationships between boundedness or compactness of the canonical inclusion maps and generalized variants of Carleson measures and show applications to composition operators on abstract Hardy spaces. We specialize our results to Hardy–Lorentz spaces. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we study the singularity of multivariate Hermite interpolation of type total degree. We present two methods to judge the singularity of the interpolation schemes considered and by methods to be developed, we show that all Hermite interpolation of type total degree on points in is singular if . And then we solve the Hermite interpolation problem on nodes completely. Precisely, all Hermite interpolations of type total degree on points with are singular; only three cases for and one case for can produce regular Hermite interpolation schemes, respectively. Besides, we also present a method to compute the interpolation space for Hermite interpolation of type total degree. [Copyright &y& Elsevier]

The article discusses the space of the initial data of the three-dimensional (3D) boundary-value problem for a parabolic differential-difference equation. It says that linear bounded operators such as extension of a function by zero and the projectrion operator of a function into the interval were introduced. It mentions that the Sobolev space of complex-value was denoted.

One of the OECD principles for model validation requires defining the Applicability Domain (AD) for the QSAR models. This is important since the reliable predictions are generally limited to query chemicals structurally similar to the training compounds used to build the model. Therefore, characterization of interpolation space is significant in defining the AD and in this study some existing descriptor-based approaches performing this task are discussed and compared by implementing them on existing validated datasets from the literature. Algorithms adopted by different approaches allow defining the interpolation space in several ways, while defined thresholds contribute significantly to the extrapolations. For each dataset and approach implemented for this study, the comparison analysis was carried out by considering the model statistics and relative position of test set with respect to the training space. [ABSTRACT FROM AUTHOR]

Abstract: Estimates for the James constant for various norms in real interpolation spaces for finite families of Banach spaces are given. As a corollary it is shown that if a family contains at least one space which is uniformly nonsquare, then the interpolation space is uniformly nonsquare. [Copyright &y& Elsevier]

New measures of noncompactness for bounded sets and linear operators, in the setting of abstract measures and generalized limits, are constructed. A quantitative version of a classical criterion for compactness of bounded sets in Banach spaces by R. S. Phillips is provided. Properties of those measures are established and it is shown that they are equivalent to the classical measures of noncompactness. Applications to summable families of Banach spaces, interpolations of operators and some consequences are also given. [ABSTRACT FROM AUTHOR]

The SCOLE model is a coupled system consisting of a flexible beam (modelled as an Euler–Bernoulli equation) with one end clamped and the other end linked to a rigid body. Its inputs are the force and the torque acting on the rigid body. It is well-known that the SCOLE model is not exactly controllable with L input signals in the natural energy state space H, because the control operator is bounded from the input space $${\mathbb{C}^2}$$ to H, and hence compact. We regard the velocity and the angular velocity of the rigid body as the output signals of this system. Using the theory of coupled linear systems (one infinite-dimensional and one finite-dimensional) developed by us recently in another paper, we show that the SCOLE model is well-posed, regular and exactly controllable in arbitrarily short time when using a certain smoother state space $${\mathcal{X}\subset H^c}$$. [ABSTRACT FROM AUTHOR]

Abstract: This article is concerned with the question of when the Banach lattices generated by the interpolation method of constants contain no isomorphic copies of . For this we study the Köthe dual spaces of Banach lattices obtained by the methods of constants and means. We find an explicit formula for the Köthe dual of Banach lattices generated by the means method with a mild restriction on the parameters. We apply these results to describe a large class of Banach lattices, generated by the method of constants, containing no subspaces isomorphic to . Moreover, we present applications to the strict singularity of certain inclusion maps between Banach sequence lattices. [Copyright &y& Elsevier]

An embedding criterion for interpolation spaces is formulated and applied to the study of the Riesz basis property in the L2,|g| space of eigenfunctions of an indefinite Sturm–Liouville problem u″=λgu on the interval (-1,1) with the Dirichlet boundary conditions, provided that the function g(x) changes sign at the origin. In particular, the basis property criterion is established for an odd g(x). Some connections with stability in interpolation scales are discussed. [ABSTRACT FROM AUTHOR]