This article investigates the extended best polynomial approximation operator, defined on the space of measurable functions, within the setting of Lorentz Gamma spaces \(\Gamma_{ w,q}\) for \(0 < q \le 1\). We derive quantitative estimates for extended best approximations and the outer operator quasi-norm. As a consequence, we show that extended best approximations in \(\Gamma_{ w,1}\) are near-best approximations in \(\Gamma_{ w,q}\).
This paper is devoted to the investigation of several notions associated with the shadowing behavior of ${\mathscr{G}}$-processes. We introduce and explore various types of shadowing, including standard shadowing, $h$-shadowing, limit shadowing, $s$-limit shadowing, and exponential limit shadowing for ${\mathscr{G}}$-processes. It is shown that if two ${\mathscr{G}}$-processes are uniformly conjugate, then the presence of any of these shadowing properties in one implies the same property holds for the other. Moreover, we establish that a finite direct product of ${\mathscr{G}}$-processes possesses the $h$-shadowing property if and only if each constituent ${\mathscr{G}}$-process in the product exhibits this property. The same equivalence also holds for the other types of shadowing considered.
In this paper, we investigate the weighted product Hardy spaces $H_w^{p}(\mathbb{R}^{d_1}\times \mathbb{R}^{d_2})$. Under some conditions on the weight, we prove that the Riesz potential operator $I_\alpha$ is bounded from $L_w^{p}(\mathbb{R}^{d_1}\times \mathbb{R}^{d_2})$ to $L_{w^{q/p}}^{q}(\mathbb{R}^{d_1}\times \mathbb{R}^{d_2})$ when $\alpha=(\alpha_1, \alpha_2)$ and $\frac{1}{p}- \frac{1}{q} = \frac{\alpha_1}{d_1} =\frac{\alpha_2}{d_2}$. We also verify the boundedness of $I_\alpha$ from $H_w^{p}(\mathbb{R}^{d_1}\times \mathbb{R}^{d_2})$ to $H_{w^{q/p}}^{q}(\mathbb{R}^{d_1}\times \mathbb{R}^{d_2})$ and from $H_w^{p}(\mathbb{R}^{d_1}\times \mathbb{R}^{d_2})$ to $L_{w^{q/p}}^{q}(\mathbb{R}^{d_1}\times \mathbb{R}^{d_2})$. We consider similar questions for the maximal fractional operator, too.
We consider when local finite-rank positive definite kernels comefrom a single global finite-rank kernel. The first part treats thecase where the global kernel is fixed and shows that the missing mixedblocks control when a global rank-$r$ realization exists and howrank grows when it does not. The second part treats the completionproblem, where only local kernels are given. In that setting, rank-preservingcompletion becomes a unitary patching problem on overlap generatedsubspaces. Forests always patch, full overlaps are governed by cycleconditions for the induced unitaries, and partial overlaps lead tofinite-dimensional subspace compatibility conditions. We also givea rank bound for finite completions obtained by joining two completedpieces along their common feature subspace.
We establish key results and properties of the Banach space of vector-valued weakly Lorentz sequence, denoted wp pound,q(& centerdot;). These results extend a classical result due to Grothendieck for p-spaces pound to the Lorentz setting. Additionally, we introduce and study a new concept, namely p pound,q-factorable operators. We establish fundamental properties of this class and provide a characterization of p pound,q-factorable operators. Specifically, we show that an operator T between Banach spaces is p pound,q-factorable if and only if it admits a continuous linear factorization through the Lorentz space p pound,q.
We prove a Voronovskaya-type inequality for the Kantorovich-type modification of Meyer-K & ouml;nig and Zeller operator Mfn(f, x) = where X infinity k =0 Z k+1 mn,k(x) (n + k + 1)(n + k + 2) n+k+2 n + 1 k n+k+1 (n + kl mn,k(x) = xk(1-x)n+1. k f(u)du,
The main result of this paper is the boundedness of integral operators on Riesz-Morrey spaces. As applications of the main result, we extend Hardy's inequalities to Riesz-Morrey spaces and establish the boundedness of the Hadamard fractional integrals on Riesz-Morrey spaces.
In the paper, the authors introduce the concept of strongly (alpha, m)-convex functions with modulus, establish several new integral identities, and, by virtue of these newly-established identities and H & ouml;lder's inequality, present some new integral inequalities of the Hermite-Hadamard type for strongly (alpha, m)-convex functions with modulus.
In this paper, we identify a large class of hyponormal block Toeplitz operators whose self-commutators are of finite rank. Recall that an operator T-phi is hyponormal and [T-phi & lowast;,T-phi]is a finite rank operator if and only if there exists a finite Blaschke product bin E(phi), where E(phi) := {k E H infinity(T) : IIkII infinity < 1 and phi - k & centerdot;phi & strns;E H infinity(T)} . An analogous set E(Phi) can be defined for a matrix-valued symbol Phi. In the block Toeplitz operator case, we first establish that if a symbol Phi is in L infinity(T, Mn) and if epsilon(Phi) contains a constant unitary matrix U, then T-Phi is normal. We then obtain a suitable converse, under a mild assumption on the symbol. Next, we provide a partial answer to a conjecture recently posed by R.E. Curto, I.S. Hwang and W.Y. Lee [10, Conjecture 6.1]. Concretely, assume that Phi is an element of H infinity(T, Mn)is such that Phi & lowast; is of bounded type and T Phi is hyponormal. Then [T-Phi & lowast;, T-Phi] is a finite rank operator if and only if there exists a finite Blaschke-Potapov product in epsilon(Phi), where Phi e := Phi & lowast; and Phi(e(-i theta)) := Phi(e(-i theta)).
We study the quantum invariants of projective varieties over the number fields. Namely, explicit formulas for a functor Q on such varieties are proved. The case of abelian varieties with complex multiplication is treated in detail.
We study uniqueness of best approximation in Orlicz spaces L Phi, for different types of not necessarily strictly convex functions Phi and for some finite dimensional approximation classes of functions, where Tchebycheff spaces, and more general approximation ones, are involved.
Let B be a Banach algebra, and let Ccb(B) denote the set of all closed convex bounded subsets of B. Assume that >= is a partial order defined on Ccb(B), and define D := {A is an element of Ccb(B) : A > 0}, where 0 denotes the zero element of Ccb(B). Furthermore, suppose that for every A, B is an element of D, the set A (R) B also belongs to D, where A (R) B means the closure of the product set AB. In this paper, general solutions F : D -> D of the multiplicative set-valued functional equation F(X (R) Y ) = F(X) (R) F(Y) for all X, Y is an element of D are determined. These solutions are closely involved with some set-valued mappings. Moreover, its stability is also proved on Banach algebras. The results not only generalize classical findings in functional equations but also open avenues for further exploration in nonlinear analysis and set-valued operator theory.
In this paper, we develop new upper bounds for the numerical radii of the tensor products of two operators. These inequalities improve and generalize some earlier related inequalities.
A sequence of positive linear operators acting on suitable function spaces on convex Borel cones is introduced and studied. Such operators generalize the well-known Bernstein-Chlodovsky operators on [0, +infinity[ and, in addition, they unify many of their more recent extensions to other settings. The study is mainly addressed to highlight their pointwise convergence as well as their uniform convergence on compact subsets for particular classes of bounded Borel measurable functions. In some particular cases, by means of such operators, a Weierstrass-type density result is obtained which concerns the approximation of such class of functions in terms of polynomials or, more generally, of elements of some function algebras. In order to achieve the main results, some new Korovkin-type theorems are also discussed in the framework of completely regular spaces. In a final section, some examples and applications are discussed as well.
In this paper, we investigate fixed point results for self-mappings defined on perturbed metric spaces endowed with a graph structure. By combining the notions of a perturbed metric and graph-preserving mappings, we establish a Banach-type fixed point theorem that ensures the existence and uniqueness of fixed points under certain contractive conditions. The presented result generalizes several known fixed point theorems in both standard and generalized metric settings. Illustrative examples are provided to demonstrate the validity and applicability of the main theorem.
In this work, we investigate the second BVP (boundary value problem) associated with the linear equilibrium theory of thermoelasticity with microtemperatures. We obtain a solution of the second BVP in terms of a double-layer thermoelastic potential, unlike the results reported in the literature, where a solution is represented by a single-layer thermoelastic potential.
Applying the martingale transform and $K$-method of interpolation spaces, we investigate the interchanging relations between Hardy-Lorentz-Karamata spaces of predictable martingales. More precisely, let $0
In this paper, we consider a sequence of operators as a wavelet type extension of univariate generalized Kantorovich operators depending on a positive real parameter given in [3]. We establish quantitative estimates for the rate of convergence of these operators in the continuous functions space and $L^{p}$-spaces in terms of modulus of continuity and $K$-functionals, respectively. Furthermore, some inequalities such as Bernstein-Markov type for continuous functions and variation preservation type property of the operators when the involved function is of bounded variation are provided.
We study sequences of bounded operators \((T_n)_{n \ge 0}\) on a complex separable Hilbert space \(\mathcal{H}\) that satisfy a linear recurrence relation of the form $$ T_{n+r} = A_0 T_n + A_1 T_{n+1} + \cdots + A_{r-1} T_{n+r-1} \quad(\textrm{for all } n\ge 0), $$ where the coefficients \(A_0, A_1, \dots, A_{r-1}\) are pairwise commuting bounded operators on \(\mathcal{H}\). \ Such relations naturally arise in the context of the operator-valued moment problem, particularly in the study of flat extensions of block Hankel operators. \ Our first goal is to derive an explicit combinatorial formula for \(T_n\). As a concrete application, we provide an explicit expression for the powers of an operator-valued companion matrix. \ In the special case of scalar coefficients $A_k=a_kI_\mathcal{H}$, with $a_k\in\mathbb{R}$, we recover a Binet-type formula that allows the explicit computation of the powers and the exponential of algebraic operators in terms of Bell polynomials.
