The Experts below are selected from a list of 273 Experts worldwide ranked by ideXlab platform
Nathan S. Feldman - One of the best experts on this subject based on the ideXlab platform.
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A Convex Stone-Weierstrass Theorem & Applications
arXiv: Functional Analysis, 2015Co-Authors: Nathan S. Feldman, Paul J. McguireAbstract:A convex-polynomial is a convex combination of the monomials $\{1, x, x^2, \ldots\}$. This paper establishes that the convex-polynomials on $\mathbb R$ are dense in $L^p(\mu)$ and weak$^*$ dense in $L^\infty(\mu)$, precisely when $\mu([-1,\infty)) = 0$. It is shown that the convex-polynomials are dense in $C(K)$ precisely when $K \cap [-1, \infty) = \emptyset$, where $K$ is a compact subset of the real line. Moreover, the closure of the convex-polynomials on $[-1,b]$ are shown to be the functions that have a convex-power series representation. A Continuous Linear Operator $T$ on a locally convex space $X$ is convex-cyclic if there is a vector $x \in X$ such that the convex hull of the orbit of $x$ is dense in $X$. The above results characterize which multiplication Operators on various real Banach spaces are convex-cyclic. It is shown for certain multiplication Operators that every closed invariant convex set is a closed invariant subspace.
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a convex stone weierstrass theorem applications
arXiv: Functional Analysis, 2015Co-Authors: Nathan S. Feldman, Paul J. McguireAbstract:A convex-polynomial is a convex combination of the monomials $\{1, x, x^2, \ldots\}$. This paper establishes that the convex-polynomials on $\mathbb R$ are dense in $L^p(\mu)$ and weak$^*$ dense in $L^\infty(\mu)$, precisely when $\mu([-1,\infty)) = 0$. It is shown that the convex-polynomials are dense in $C(K)$ precisely when $K \cap [-1, \infty) = \emptyset$, where $K$ is a compact subset of the real line. Moreover, the closure of the convex-polynomials on $[-1,b]$ are shown to be the functions that have a convex-power series representation. A Continuous Linear Operator $T$ on a locally convex space $X$ is convex-cyclic if there is a vector $x \in X$ such that the convex hull of the orbit of $x$ is dense in $X$. The above results characterize which multiplication Operators on various real Banach spaces are convex-cyclic. It is shown for certain multiplication Operators that every closed invariant convex set is a closed invariant subspace.
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n-Weakly hypercyclic and n-weakly supercyclic Operators
Journal of Functional Analysis, 2012Co-Authors: Nathan S. FeldmanAbstract:If X is a locally convex topological vector space over a scalar field F=R or C and if E is a subset of X, then we define E to be n-weakly dense in X if for every onto Continuous Linear Operator F:X→Fn we have that F(E) is dense in Fn. If X is a Hilbert space, this is equivalent to requiring that E have a dense orthogonal projection onto every subspace of dimension n. We then consider Continuous Linear Operators on X that have orbits or scaled orbits that are n-weakly dense in X. We show that on a separable Hilbert space there are non-trivial examples of such Operators and establish many of their basic properties. A fundamental tool is Ballʼs solution of the complex plank problem which implies that certain sets are 1-weakly closed.
Nobuaki Obata - One of the best experts on this subject based on the ideXlab platform.
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Transformations for White Noise Functionals
Journal of Functional Analysis, 1993Co-Authors: Takeyuki Hida, Hui-hsiung Kuo, Nobuaki ObataAbstract:Several results concerning the spaces (E) and (E)* of test and generalized white noise functionals, respectively, are obtained. The irreducibility of the canonical commutation relation for Operators on (E) and on (E)* is proved. It is shown that the Fourier-Mehler transform F0 on (E)* is the adjoint of a Continuous Linear Operator G0 on (E). Moreover, a characterization theorem for the Fourier-Mehler transform is proved. In particular, the Fourier transform is the unique (up to a constant) Continuous Linear Operator F on (E)* such that FDξ = qξF and Fqξ = − DξF. Here Dξ and qξ are differentiation and multiplication Operators, respectively. Several one-parameter transformation groups acting on (E) and the Lie algebra generated by their infinitesimal generators are also discussed.
Michał Goliński - One of the best experts on this subject based on the ideXlab platform.
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the invariant subspace problem for the space of smooth functions on the real line
Journal of Mathematical Analysis and Applications, 2020Co-Authors: Michał Goliński, Adam PrzestackiAbstract:Abstract We construct a Continuous Linear Operator acting on the space of smooth functions on the real line without non-trivial invariant subspaces. This is a first example of such an Operator acting on a Frechet space without a Continuous norm. The construction is based on the ideas due to C. Read who constructed a Continuous Operator without non-trivial invariant subspaces on the Banach space l 1 .
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Operator on the space of rapidly decreasing functions with all non-zero vectors hypercyclic
Advances in Mathematics, 2013Co-Authors: Michał GolińskiAbstract:Abstract We construct a Continuous Linear Operator T on the Schwartz space S ( R d ) of rapidly decreasing functions such that each non-zero orbit of T is dense. The construction is inspired by the work of C. Read on similar Operators on the space l 1 . The construction, due to the structure of a Frechet space, can be made significantly simpler than the original construction of Read.
Takeyuki Hida - One of the best experts on this subject based on the ideXlab platform.
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Transformations for White Noise Functionals
Journal of Functional Analysis, 1993Co-Authors: Takeyuki Hida, Hui-hsiung Kuo, Nobuaki ObataAbstract:Several results concerning the spaces (E) and (E)* of test and generalized white noise functionals, respectively, are obtained. The irreducibility of the canonical commutation relation for Operators on (E) and on (E)* is proved. It is shown that the Fourier-Mehler transform F0 on (E)* is the adjoint of a Continuous Linear Operator G0 on (E). Moreover, a characterization theorem for the Fourier-Mehler transform is proved. In particular, the Fourier transform is the unique (up to a constant) Continuous Linear Operator F on (E)* such that FDξ = qξF and Fqξ = − DξF. Here Dξ and qξ are differentiation and multiplication Operators, respectively. Several one-parameter transformation groups acting on (E) and the Lie algebra generated by their infinitesimal generators are also discussed.
Michael Langenbruch - One of the best experts on this subject based on the ideXlab platform.
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Analytic Extension of Smooth Functions
Results in Mathematics, 1999Co-Authors: Michael LangenbruchAbstract:Let F be a closed proper subset of ℝn and let ℰ* be a class of ultradifferentiable functions. We give a new proof for the following result of Schmets and Valdivia on analytic modification of smooth functions: for every function ƒ ∈ ℰ* (ℝn) there is \({\widetilde f} \in {\cal E}_{*}(\rm R ^{n})\)which is real analytic on ℝnF and such that ∂a ƒ ¦F = ∂a ƒ ¦ F for any a ∈ ℕ0n. For bounded ultradifferentiable functions ƒ we can obtain \({\widetilde f}\)by means of a Continuous Linear Operator.