Elliptic operators with non-local Wentzell–Robin boundary conditions

This chapter is about metric spaces, an abstract generalization of the real line that allows discussion of open and closed sets, limits, convergence, continuity, and similar properties. The usual distance function for the real line becomes an example of a metric. The other notions are defined in terms of the metric. The advantage of the generalization is that proofs of certain properties of the real line immediately go over to all other examples. Section 1 gives the definition of metric space and open set, and it lists a number of important examples, including Euclidean spaces and certain spaces of functions. Sections 2 through 4 develop properties of open and closed sets, continuity, and convergence of sequences that are simple generalizations of known facts about $\mathbb{R}$. Section 5 shows how a subset of a metric space can be made into a metric space so that the restriction of a continuous function from the whole space to the subset remains continuous. It also shows that three natural metrics for the product of two metric spaces lead to the same open sets, continuous functions, and convergent sequences. Section 6 shows that any metric space is “Hausdorff,” “regular,” and “normal,” and it goes on to exhibit three different countability hypotheses about a metric space as equivalent. A metric space with these properties is called “separable.” Section 7 concerns compactness and completeness. A metric space is defined to be “compact” if every open cover has a finite subcover. This property is equivalent to the condition that every sequence has a convergent subsequence. The Heine–Borel Theorem says that the compact sets of $\mathbb{R}^n$ are exactly the closed bounded sets. A number of the results early in Chapter I that were proved by the Bolzano–Weierstrass Theorem in the context of the real line are seen to extend to any compact metric space. A metric space is “complete” if every Cauchy sequence is convergent. A metric space is compact if and only if it is complete and “totally bounded.” Section 8 concerns connectedness, which is an abstraction of the property of an interval of the line that accounts for the Intermediate Value Theorem. Section 9 proves a fundamental result known as the Baire Category Theorem. A sample consequence of the theorem is that the pointwise limit of a sequence of continuous complex-valued functions on a complete metric space must have points where it is continuous. Section 10 studies the spaces of real-valued and complex-valued continuous functions on a compact metric space. A generalization of Ascoli's Theorem from the setting of Chapter I provides a characterization of compact sets in either of these spaces of continuous functions. A generalization of the Weierstrass Approximation Theorem, known as the Stone–Weierstrass Theorem, gives sufficient conditions for a subalgebra of either of these spaces of continuous functions to be dense. One consequence is that these spaces of continuous functions are separable. Section 11 constructs the “completion” of a metric space out of Cauchy sequences in the given space. The result is a complete metric space and a distance-preserving map of the given metric space into the completion such that the image is dense.

A well known result due to Dixmier and Grothendieck for spaces of continuous scalar-valued functions C(X), X compact Hausdorff, is that C(X) is a Banach dual if, and only if, Xis hyperstonean. Moreover, for hyperstonean X, the predual of C(X) is strongly unique. Here we obtain a formulation of this result for spaces of continuous vector-valued functions. It is shown that if E is a Hilbert space and C(X, (E, σ *) ) denotes the space of continuous functions on X to E when E is provided with its weak * ( = weak) topology, then C(X, (E, σ *) ) is a Banach dual if, and only if, X is hyperstonean. Moreover, for hyperstonean X, the predual of C(X, (E, σ *) ) is strongly unique.

Spaces of continuous functions over a Ψ-space

The main problem considered in this paper is the approximation of a trigonometric polynomial by a trigonometric polynomial with a prescribed number of harmonics. The method proposed here gives an opportunity to consider approximation in different spaces, among them the space of continuous functions, the space of functions with uniformly convergent Fourier series, and the space of continuous analytic functions. Applications are given to approximation of the Sobolev classes by trigonometric polynomials with prescribed number of harmonics, and to the widths of the Sobolev classes. This work supplements investigations by Maiorov, Makovoz and the author where similar results were given in the integral metric.

We study the sectoriality of a class of degenerate second-order elliptic differential operators in the space of continuous functions on the canonical simplex of \({\mathbb{R}^{d}}\) . This kind of operators arises from the theory of Fleming–Viot processes in population genetics.

This chapter is about metric spaces, an abstract generalization of the real line that allows discussion of open and closed sets, limits, convergence, continuity, and similar properties. The usual distance function for the real line becomes an example of a metric. The other notions are defined in terms of the metric. The advantage of the generalization is that proofs of certain properties of the real line immediately go over to all other examples.Section 1 gives the definition of metric space and open set, and it lists a number of important examples, including Euclidean spaces and certain spaces of functions.Sections 2 through 4 develop properties of open and closed sets, continuity, and convergence of sequences that are simple generalizations of known facts about ℝ.Section 5 shows how a subset of a metric space can be made into a metric space so that the restriction of a continuous function from the whole space to the subset remains continuous. It also shows that three natural metrics for the product of two metric spaces lead to the same open sets, continuous functions, and convergent sequences.Section 6 shows that any metric space is “Hausdorff,” “regular,” and “normal,” and it goes on to exhibit three different countability hypotheses about a metric space as equivalent. A metric space with these properties is called “separable.”Section 7 concerns compactness and completeness. A metric space is defined to be “compact” if every open cover has a finite subcover. This property is equivalent to the condition that every sequence has a convergent subsequence. The Heine-Borel Theorem says that the compact sets of ℝn are exactly the closed bounded sets. A number of the results early in Chapter I that were proved by the Bolzano-Weierstrass Theorem in the context of the real line are seen to extend to any compact metric space. A metric space is “complete” if every Cauchy sequence is convergent. A metric space is compact if and only if it is complete and “totally bounded.”Section 8 concerns connectedness, which is an abstraction of the property of an interval of the line that accounts for the Intermediate Value Theorem.Section 9 proves a fundamental result known as the Baire Category Theorem. A sample consequence of the theorem is that the pointwise limit of a sequence of continuous complex-valued functions on a complete metric space must have points where it is continuous.Section 10 studies the spaces of real-valued and complex-valued continuous functions on a compact metric space. A generalization of Ascoli’s Theorem from the setting of Chapter I provides a characterization of compact sets in either of these spaces of continuous functions. A generalization of the Weierstrass Approximation Theorem, known as the Stone-Weierstrass Theorem, gives sufficient conditions for a subalgebra of either of these spaces of continuous functions to be dense. One consequence is that these spaces of continuous functions are separable.Section 11 constructs the “completion” of a metric space out of Cauchy sequences in the given space. The result is a complete metric space and a distance-preserving map of the given metric space into the completion such that the image is dense.

This work deals with the $H^1 $ condition numbers and the distribution of the $\tilde \beta _{N,M} $-singular values of the preconditioned operators $\{ \tilde \beta _{N,M}^{ - 1} W_{N,M} \hat A_{N,M} \} $. $\hat A_{N,M} $ is the matrix representation of the Legendre spectral collocation discretization of the elliptic operator A defined by $Au: = - \Delta u + a_1 u_x + a_2 u_y + a_0 u$ in $\Omega $ (the unit square) with boundary conditions $u = 0$ on $\Gamma _0 $, $\frac{{\partial u}}{{\partial \nu _A }} = \alpha u$ on $\Gamma _1 $. $\tilde \beta _{N,M} $ is the stiffness matrix associated with the finite element discretization of the positive definite elliptic operator B defined by $Bv: = - \Delta v + b_0 v$ in $\Omega $ with boundary conditions $v = 0$ on $\Gamma _0 $, $\frac{{\partial v}}{{\partial \nu _B }} = \beta v$ on $\Gamma _1 $. The finite element space is either the space of continuous functions which are bilinear on the rectangles determined by the Legendre–Gauss–Lobatto (LGL) points or the space of continuous functions which are linear on a triangulation of $\Omega $ determined by the LGL points. $W_{N,M} $ is the matrix of quadrature weights. When $A = B$ we obtain results on the eigenvalues of $\tilde \beta _{N,M}^{ - 1} W_{N,M} \hat B_{N,M} $. In the general case we show that there is an integer $N_0 $ and constants $\alpha $, $\beta $ with $0 < \alpha < \beta $, such that if $\min (N,M) \geq N_0 $, then all the $\tilde \beta _{N,M} $-singular values of $\tilde \beta _{N,M}^{ - 1} W_{N,M} \hat A_{N,M} $ lie in the interval $[\alpha ,\beta ]$. Moreover, there is a smaller interval, $[\alpha _0 ,\beta _0 ]$, independent of the operator A, such that if $\min (N,M) \geq N_0 $, then all but a fixed finite number of the $\tilde \beta _{N,M} $-singular value lie in $[\alpha _0 ,\beta _0 ]$. These results are related to results of Manteuffel and Parter [SIAM J. Numer. Anal., 27 (1990), pp. 656–694], Parter and Wong [J. Sci. Comput., 6 (1991), pp. 129–157] and Wong [Numer. Math., 62 (1992), pp. 391–411, 413–437] for finite element discretizations.

In this note we characterize the compact multiplication operators M π on the weighted locally convex spaces CV 0 (X, E) of vector-valued continuous functions induced by the operator-valued mappings π : X → B(E).

Let be a space of homogeneous type with and a variable exponent function satisfying , where . Assume that is a nonnegative self‐adjoint operator on whose heat kernels satisfy the Gaussian upper bound estimates. In this paper, the authors first establish the atomic, nontangential, and radial maximal function characterizations for the variable Hardy space associated with . Second, as applications, when , is a bounded Lipschitz domain, is a second‐order divergence form elliptic operator with the Robin boundary condition, the authors establish a new atomic characterization of the associated with , and then the authors further show that, for , where and , respectively, denote the variable Hardy spaces on associated with the corresponding elliptic operators having the Dirichlet and the Neumann boundary conditions, and ( and ), respectively, denote “supported type” and “restricted type” Hardy spaces on (its local version), and is a critical index depending on the operators , , and . Third, the authors prove the boundedness of the Riesz transform on the variable Lebesgue space , and from to , or to . Finally, the authors further establish the global regularity estimates in both and for the inhomogeneous Robin problem of on .

Hyperdifferential Operators and Continuous Functions on Function Fields

The Denjoy integral is an integral that extends the Lebesgue integral and can integrate any derivative. In this paper, it is shown that the graph of the indefinite Denjoy integral $f\mapsto \int_a^x f$ is a coanalytic non-Borel relation on the product space $M[a,b]\times C[a,b]$, where $M[a,b]$ is the Polish space of real-valued measurable functions on $[a,b]$ and where $C[a,b]$ is the Polish space of real-valued continuous functions on $[a,b]$. Using the same methods, it is also shown that the class of indefinite Denjoy integrals, called $ACG_{\ast}[a,b]$, is a coanalytic but not Borel subclass of the space $C[a,b]$, thus answering a question posed by Dougherty and Kechris. Some basic model theory of the associated spaces of integrable functions is also studied. Here the main result is that, when viewed as an $\mathbb{R}[X]$-module with the indeterminate $X$ being interpreted as the indefinite integral, the space of continuous functions on the interval $[a,b]$ is elementarily equivalent to the Lebesgue-integrable and Denjoy-integrable functions on this interval, and each is stable but not superstable, and that they all have a common decidable theory when viewed as $\mathbb{Q}[X]$-modules.

Let F be a finite field (or discrete) and let A and B be vector spaces of F-valued continuous functions defined on locally compact spaces X and Y, respectively. We look at the representation of linear bijections H:A⟶B by continuous functions h:Y⟶X as weighted composition operators. In order to do it, we extend the notion of Hamming metric to infinite spaces. Our main result establishes that under some mild conditions, every Hamming isometry can be represented as a weighted composition operator. Connections to coding theory are also highlighted.

A continuous function on the complex plane is harmonic if and only if the span of its compositions with entire functions is not dense in the space of continuous functions in the topology of uniform convergence on compact sets.Let C be the complex plane and let E be the space of all entire functions.We remark that these functions <p (and their conjugates) are precisely those functions for which/ q> is harmonic whenever/is.Let C(C) be the space of all complex-valued continuous functions on C in the topology of uniform convergence on compact sets.The next definition is motivated by the theory of mean-periodic functions (see [2]).Definition.A variety is a closed linear subspace V of C(C) that is invariant under composition with all entire functions.

For realcompact spaces $X$ and $Y$ we give a complete description of the linear biseparating maps between spaces of vector-valued continuous functions on $X$ and $Y$ in two cases: the spaces of all continuous functions and the spaces of {\em bounded} continuous functions. With similar techniques we also describe the linear biseparating maps defined between some other families of spaces, in particular spaces of vector-valued uniformly continuous bounded functions.

Let X and Y be metric spaces and C(X, Y) be the space of all continuous functions from X to Y. If X is a locally connected space, the compact-open topology on C(X, Y) is weaker than the Attouch-Wets topology on C(X, Y). The result is applied on the space of continuous linear functions. Let X be a locally convex topological linear space metrisable with an invariant metric and X* be a continuous dual. X is normable if and only if the strong topology on X* and the Attouch-Wets topology coincide.