Amol Sasane

The London School of Economics and Political Science | LSE · Department of Mathematics

The ring of periodic distributions on ${\mathbb{R}}^{\tt d}$ with usual addition and with convolution is considered. Via Fourier series expansions, this ring is isomorphic to the ring ${\mathcal{S}}'({\mathbb{Z}}^{\tt d})$ of all maps $f:{\mathbb{Z}}^{\tt d}\rightarrow \mathbb{C}$ of at most polynomial growth (i.e., there exist a real $M>0$ and a n...

New algebraic-analytic properties of a previously studied Banach algebra $\mathcal{A}({\bf{p}})$ of entire functions are established. For a given fixed sequence $(\bf{p}(n))_{n\geq 0}$ of positive real numbers, such that $\lim_{n\rightarrow \infty} {\bf{p}}(n)^{\frac{1}{n}}=\infty$, the Banach algebra $\mathcal{A}({\bf{p}})$ is the set of all entir...

For a commutative unital ring R, and n ε N, let SLn(R) denote the special linear group over R, and En(R) the subgroup of elementary matrices. Let M+ be the Banach algebra of all complex Borel measures on [0,+8] with the norm given by the total variation, the usual operations of addition and scalar multiplication, and with convolution. It is first s...

The classical Gleason-Kahane-Żelazko Theorem states that a linear functional on a complex Banach algebra not vanishing on units, and such that $$\Lambda (\textbf{1})=1$$ Λ ( 1 ) = 1 , is multiplicative, that is, $$\Lambda (ab)=\Lambda (a)\Lambda (b)$$ Λ ( a b ) = Λ ( a ) Λ ( b ) for all $$a,b\in A$$ a , b ∈ A . We study the GKŻ property for associa...

A result of Archimedes states that for perpendicular chords passing through a point P in the interior of the unit circle, the sum of the squares of the lengths of the chord segments from P to the circle is equal to 4. A generalization of this result to n≥2 chords is given. This is done in the backdrop of revisiting Problem 1325 from Crux Mathematic...

The paper studies projective freeness and Hermiteness of algebras of complex-valued continuous functions on topological spaces, Stein algebras, and commutative unital Banach algebras. New sufficient cohomology conditions on the maximal ideal spaces of the algebras are given that guarantee the fulfilment of these properties. The results are illustra...

The classical Gleason-Kahane-\.{Z}elazko Theorem states that a linear functional on a complex Banach algebra not vanishing on units, and such that $\Lambda(\mathbf 1)=1$, is multiplicative, that is, $\Lambda(ab)=\Lambda(a)\Lambda(b)$ for all $a,b\in A$. We study the GK\.Z property for associative unital algebras, especially for function algebras. I...

The decay of solutions to the Klein–Gordon equation is studied in two expanding cosmological spacetimes, namely the de Sitter universe in flat Friedmann–Lemaître–Robertson–Walker (FLRW) form and the cosmological region of the Reissner–Nordström–de Sitter (RNdS) model. Using energy methods, for initial data with finite higher-order energies, decay r...

Let ${\mathbb{D}}=\{z\in \mathbb{C}:|z|<1\}$ and for an integer $d\geq 1$, let $S_d$ denote the symmetric group, consisting of of all permutations of the set $\{1,\cdots, d\}$. A function $f:{\mathbb{D}}^d\rightarrow \mathbb{C}$ is symmetric if $f(z_1,\cdots, z_d)=f(z_{\sigma(1)},\cdots, z_{\sigma (d)})$ for all $\sigma \in S_d$ and all $(z_1,\cdot...

The book aims to give a mathematical presentation of the theory of general relativity (that is, spacetime-geometry-based gravitation theory) to advanced undergraduate mathematics students. Mathematicians will find spacetime physics presented in the definition-theorem-proof format familiar to them. The given precise mathematical definitions of physi...

Given a linear, constant coefficient partial differential equation in ℝd+1, where one independent variable plays the role of ‘time’, a distributional solution is called a null solution if its past is zero. Motivated by physical considerations, distributional solutions that are tempered in the spatial directions alone (with no restriction in the tim...

Let $\mathcal{A}$ be a unital complex semisimple Banach algebra, and $M_{\mathcal{A}}$ denote its maximal ideal space. For a matrix $M\in {\mathcal{A}}^{n\times n}$, $\widehat{M}$ denotes the matrix obtained by taking entry-wise Gelfand transforms. For a matrix $M\in {\mathbb{C}}^{n\times n}$, $\sigma(M)\subset \mathbb{C}$ denotes the set of eigenv...

Let A be a unital complex semisimple Banach algebra, and MA denote its maximal ideal space. For a matrix M∈An×n, Mˆ denotes the matrix obtained by taking entry-wise Gelfand transforms. For a matrix M∈Cn×n, σ(M)⊂C denotes the set of eigenvalues of M. It is shown that if A∈An×n and B∈Am×m are such that for all φ∈MA, σ(Aˆ(φ))∩σ(Bˆ(φ))=∅, then for all...

For a commutative unital ring $R$, and $n\in \mathbb{N}$, let $\textrm{SL}_n(R)$ denote the special linear group over $R$, and $\textrm{E}_n(R)$ the subgroup of elementary matrices. Let ${\mathcal{M}}^+$ be the Banach algebra of all complex Borel measures on $[0,+\infty)$ with the norm given by the total variation, the usual operations of addition...

A classical result of Norbert Wiener characterises doubly shift-invariant subspaces for square integrable functions on the unit circle with respect to a finite positive Borel measure $\mu$, as being the ranges of the multiplication maps corresponding to the characteristic functions of $\mu$-measurable subsets of the unit circle. An analogue of this...

The set 𝒜 := 𝔺δ 0 + 𝒟 + ′, obtained by attaching the identity δ 0 to the set 𝒟 + ′ of all distributions on 𝕉 with support contained in (0, ∞), forms an algebra with the operations of addition, convolution, multiplication by complex scalars. It is shown that 𝒜 is a Hermite ring, that is, every finitely generated stably free 𝒜-module is free, or equi...

A generalised summation method is considered based on the Fourier series of periodic distributions. It is shown that eit−2e2it+3e3it−4e4it+−⋯=Pfeit(1+eit)2+iπ∑n∈Zδ(2n+1)π′, where Pfeit(1+eit)2∈D′(R) is the 2π-periodic distribution given by Pfeit(1+eit)2,φ=∑n∈Z∫02π(t−π)2eit(1+eit)2∫01(1−θ)φ′′((2n+1)π+θ(t−π))dθdt, for φ∈D(R). Applying the generalised...

An algebraic characterization of the property of approximate controllability is given, for behaviours of spatially invariant dynamical systems, consisting of distributional solutions w, that are periodic in the spatial variables, to a system of partial differential equations M∂∂x1,…,∂∂xd,∂∂tw=0, corresponding to a polynomial matrix M∈(ℂ[ξ1,…,ξd,τ])...

Given a linear, constant coefficient partial differential equation in ${\mathbb{R}}^{d+1}$, where one independent variable plays the role of `time', a distributional solution is called a null solution if its past is zero. Motivated by physical considerations, we consider distributional solutions that are tempered in the spatial directions alone (an...

The set E′(R) of all compactly supported distributions, with the operations of addition, convolution, multiplication by complex scalars, and with the strong dual topology is a topological algebra. In this article, it is shown that the topological stable rank of E′(R) is 2.

The decay of solutions to the Klein-Gordon equation is studied in two expanding cosmological spacetimes, namely the de Sitter universe in flat Friedmann-Lema\^{i}tre-Robertson-Walker (FLRW) form, and the cosmological region of the Reissner-Nordstr\"om-de Sitter (RNdS) model. Using energy methods, for initial data with finite higher order energies,...

Let $A=[a_{ij}]\in O_3(\mathbb{R})$. We give several different proofs of the fact that the vector $$ V:=\left[\begin{array}{ccc} \displaystyle \frac{1}{a_{23}+a_{32}} & \displaystyle \frac{1}{a_{13}+a_{31}} & \displaystyle \frac{1}{a_{12}+a_{21}} \end{array}\right]^T, $$ if it exists, is an eigenvector of $A$ corresponding to the eigenvalue $1$.

A generalised summation method is considered based on the Fourier series of periodic distributions. It is shown that $$ e^{it}-2e^{2it}+3e^{3it}-4e^{4it}+-\cdots = {\mathrm P\mathrm f} {\displaystyle \frac{e^{it}}{(1+e^{it})^2}} +i\pi \displaystyle \sum_{n\in \mathbb{Z}} \delta'_{(2n+1)\pi}, $$ where ${\mathrm P\mathrm f} {\displaystyle \frac{e^{it...

Let Λ be either a subgroup of the integers ℤ, a semigroup in ℕ, or Λ = ℚ (resp., Q +). We determine the Bass and topological stable ranks of the algebras AP Λ = {f ∈ AP : σ(f) ⊆ Λ} of almost periodic functions on the real line and with Bohr spectrum in Λ. This answers a question in the first part of this series of articles under the same heading, w...

The set $\mathcal{E}'(\mathbb{R})$ of all compactly distributions, with the operations of addition, convolution, multiplication by complex scalars, and with the strong dual topology is a topological algebra. In this article, it is shown that the topological stable rank of $\mathcal{E}'(\mathbb{R})$ is 2.

Problem 1325 from the journal {\em Crux Mathematicorum} is revisited, and two new solutions are presented.

The set of periodic distributions, with usual addition and convolution, forms a ring, which is isomorphic, via taking a Fourier series expansion, to the ring ${\mathcal{S}}'({\mathbb{Z}}^d)$ of sequences of at most polynomial growth with termwise operations. In this article we make the following observations: ${\mathcal{S}}'({\mathbb{Z}}^d)$ is not...

It is shown that the ring of periodic distributions is a coherent ring (with the operations of pointwise addition and convolution) by showing that the isomorphic ring $s'$ of the Fourier coefficients (of sequences of at most polynomial growth) with termwise operations is coherent. Moreover, it is shown that the subring $\ell^\infty$ of $s'$ of all...

A version of the Krull intersection theorem states that for Noetherian integral domains the Krull intersection ki(I) of every proper ideal I is trivial; that is We investigate the validity of this result for various function algebras R, present ideals I of R for which ki(I) ≠ {0}, and give conditions on I so that ki(I) = {0}.

A corona type theorem is given for the ring R of periodic distributions in R^d in terms of the sequence of Fourier coefficients of these distributions, which have at most polynomial growth. It is also shown that the Bass stable rank and the topological stable rank of R are both equal to 1.

A geometric construction for the Poincare formula for relativistic addition of velocities in one dimension was given by Jerzy Kocik in "Geometric Diagram for Relativistic Addition of Velocities", American Journal of Physics, volume 80, page 737, 2012. While the proof given there used Cartesian coordinate geometry, three alternative approaches are g...

It is shown that the algebra of bounded Dirichlet series is not a coherent ring, and has infinite Bass stable rank. As corollaries of the latter result, it is derived that the algebra of bounded Dirichlet series has infinite topological stable rank and infinite Krull dimension.

This chapter describes the frequency domain approach to the robust stabilization problem in linear control theory. The exposition is restricted to single-input single-output systems. After introducing the preliminaries on linear control systems, their transfer functions, stable and nonstable systems, the stabilization problem and its solution are d...

This chapter describes the frequency domain approach to the robust stabilization problem in linear control theory. The exposition is restricted to single-input single-output systems. After introducing the preliminaries on linear control systems, their transfer functions, stable and nonstable systems, the stabilization problem and its solution are d...

The Bohl algebra $\textrm{B}$ is the ring of linear combinations of functions $t^k e^{\lambda t}$, where $k$ is any nonnegative integer, and $\lambda$ is any complex number, with pointwise operations. We show that the Bass stable rank and the topological stable rank of $\textrm{B}$ (where we use the topology of uniform convergence) are infinite.

Let H_n^2 denote the Drury-Arveson Hilbert space on the unit ball B_n in C^n, and let M(H_n^2) be its multiplier algebra. We show that for n>=3, the ring M(H_n^2) is not coherent.

Let $R$ be the polydisc algebra or the Wiener algebra. It is shown that the group $SL_n(R)$ is generated by the subgroup of elementary matrices with all diagonal entries $1$ and at most one nonzero off-diagonal entry. The result an easy consequence of the deep result due to Ivarsson and Kutzschebauch (Ann. of Math. 2012).

Using the facts that the disk algebra and the Wiener algebra are not coherent, we prove that the polydisc algebra, the ball algebra and the Wiener algebra of the polydisc are not coherent.

We give a sufficient condition for the surjectivity of partial differential operators with constant coefficients on a class of distributions on R^{n+1} (here we think of there being n space directions and one time direction), that are periodic in the spatial directions and tempered in the time direction.

We denote by A_0+AP_+ the Banach algebra of all complex-valued functions f defined in the closed right half plane, such that f is the sum of a holomorphic function vanishing at infinity and a ``causal'' almost periodic function. We give a complete description of the maximum ideal space M(A_0+AP_+) of A_0+AP_+. Using this description, we also establ...

The classical -metric introduced by Vinnicombe in robust control theory for rational plants was extended to classes of nonrational transfer functions in Ball (2012) [1]. In Sasane (2012) [11], an extension of the classical -metric was given when the underlying ring of stable transfer functions is the Hardy algebra, . However, this particular extens...

For single input single output systems, we give a refinement of the generalized chordal metric. Our metric is given in terms of coprime factorizations, but it coincides with the extension of Vinnicombe's nu-metric given in earlier work by Ball and Sasane if the coprime factorizations happens to be normalized. The advantage of the metric introduced...

In this note we establish a vector-valued version of Beurling's Theorem (the Lax-Halmos Theorem) for the polydisc. As an application of the main result, we provide necessary and sufficient conditions for the completion problem in $H^\infty(\mathbb{D}^n)$.

The book constitutes a basic, concise, yet rigorous course in complex analysis, for students who have studied calculus in one and several variables, but have not previously been exposed to complex analysis. The textbook should be particularly useful and relevant for undergraduate students in joint programmes with mathematics, as well as engineering...

The classical nu-metric introduced by Vinnicombe in robust control theory for rational plants was extended to classes of nonrational transfer functions in Ball and Sasane [Complex Analysis and Operator Theory; 2012]. In Sasane [Mathematics of Control and Related Fields; 2012], an extension of the classical nu-metric was given when the underlying ri...

The classical Shannon sampling theorem states that a signal f with Fourier transform F in L^2(R) having its support contained in (-\pi,\pi) can be recovered from the sequence of samples (f(n))_{n in Z} via f(t)=\sum_{n in Z} f(n) (sin(\pi (t -n)))/(\pi (t-n)) (t in R). In this article we prove a generalization of this result under the assumption th...

We give algebraic characterizations of the properties of autonomy and of controllability of behaviours of spatially invariant dynamical systems, consisting of distributional solutions, that are periodic in the spatial variables, to a system pf partial differential equations M (partial derivative/partial derivative x(1), . . . , partial derivative/p...

AS(L (E, E∗)) denotes the space of all functions f: D ∪ S → L (E, E∗) that are holomorphic in D, and bounded and continuous on D ∪ S. In this article we prove the following main results: 1. A theorem concerning the approximation of f ∈ AS(L (E, E∗)) by a function F that is holomorphic in a neighbourhood of D∪S and such that the error F −f is unifor...

An abstract chordal metric is defined on linear control systems described by their transfer functions. Analogous to a previous result due to Jonathan Partington ("Robust control and approximation in the chordal metric", in Robust Control, LNCIS 183, Springer, 1992) for H^infty, it is shown that strong stabilizability is a robust property in this me...

An abstract ν-metric was introduced by J.A. Ball and A.J. Sasane [”Extension of the ν-metric”, Complex Anal. Oper. Theory, to appear] with a view towards extending the classical ν-metric of Vinnicombe from the case of rational transfer functions to more general non-rational transfer function classes of infinite-dimensional linear control systems. I...

Let A_+ be the ring of Laplace transforms of complex Borel measures on R with support in [0,+\infty) which do not have a singular nonatomic part. We compare the nu-metric d_{A_+} for stabilizable plants over A_+ given in the article by Ball and Sasane [2010], with yet another metric d_{H^\infty}|_{A_+}, namely the one induced by the metric d_{H^\in...

An abstract $\nu$-metric was introduced by Ball and Sasane, with a view towards extending the classical $\nu$-metric of Vinnicombe from the case of rational transfer functions to more general nonrational transfer function classes of infinite-dimensional linear control systems. In this short note, we give an important concrete special instance of th...

Let $R$ be a commutative complex unital semisimple Banach algebra with the involution $\cdot ^\star$. Sufficient conditions are given for the existence of a stabilizing solution to the $H^\infty$ Riccati equation when the matricial data has entries from $R$. Applications to spatially distributed systems are discussed.

Let D denote the open unit disk in C centered at 0. Let H∞ R denote the set of all bounded and holomorphic functions defined in D that also satisfy f (z) = f (z) for all z ∈ D. It is shown that H∞ R is a coherent ring.

Let \({\mathbb {D}^n:=\{z=(z_1,\ldots, z_n)\in \mathbb {C}^n:|z_j| < 1, \;j=1,\ldots, n\}}\), and let \({\overline{\mathbb{D}}^n}\) denote its closure in \({\mathbb {C}^n}\). Consider the ring$$C_{\rm r}(\overline{\mathbb{D}}^n;\mathbb {C}) =\left\{f: \overline{\mathbb{D}}^n\rightarrow \mathbb {C}:f \,\, {\rm is \,\, continuous \,\, and}\,\, f(z)=\...

Let M +denote the Banach algebra of all complex Borel measures with support contained in [0,+8), with the usual addition and scalar multiplication, and with convolution *, and the norm being the total variation of μ. We show that the maximal ideal space X(M +) of M +, equipped with the Gelfand topology, is contractible as a topological space. In pa...

Let D, T denote the unit disc and unit circle, respectively, in C, with center 0. If S T, then let AS denote the set of complex-valued functions dened on D( S that are analytic in D, and continuous and bounded on D ( S. Then AS is a ring with pointwise addition and multiplication. We prove that if the intersection of S with the set of limit points...

Let $\calA_+$ denote the set of Laplace transforms of complex Borel measures $\mu$ on $[0,+\infty)$ such that $\mu$ does not have a singular non-atomic part. In \cite{BalSas}, an extension of the classical $\nu$-metric of Vinnicombe was given, which allowed one to address robust stabilization problems for unstable plants over $\calA_+$. In this art...

Let Ω be a circular domain, that is, an open disk with finitely many closed disjoint disks removed. Denote by H ∞(Ω) the Banach algebra of all bounded holomorphic functions on Ω, with pointwise operations and the supremum norm. We show that the topological stable rank of H ∞(Ω) is equal to 2. The proof is based on Suárez’s theorem that the topologi...

Let I be any index set. We consider the Banach algebra ${\mathbb {C} e+ \ell^2(I)}$ with the Hadamard product, and prove that its Bass and topological stable ranks are both equal to 1. We also characterize divisors, maximal ideals, closed ideals and closed principal ideals. For ${I=\mathbb {N}}$ we also characterize all prime z-ideals in this B...

An abtract $\nu$-metric was introduced by Ball and Sasane, with a view towards extending the classical $\nu$-metric of Vinnicombe from the case of rational transfer functions to more general nonrational transfer function classes of infinite-dimensional linear control systems. In this short note, we give an additional concrete special instance of th...

Let N and D be two matrices over the algebra H ∞ of bounded analytic functions in the disk, or its real counterpart . Suppose that N and D have the same number n of columns. In a generalization of the notion of topological stable rank 2, it is shown that N and D can be approximated (in the operator norm) by two matrices Ñ and , so that the Aryabhat...

Let $R$ be a commutative complex Banach algebra with the involution $\cdot ^\star$ and suppose that $A\in R^{n\times n}$, $B\in R^{n\times m}$, $C\in R^{p\times n}$. The question of when the Riccati equation $$ PBB^\star P-PA-A^\star P-C^\star C=0 $$ has a solution $P\in R^{n\times n}$ is investigated. A counterexample to a previous result in the l...

Let Aℝ() denote the set of functions belonging to the disc algebra having real Fourier coefficients. We show that Aℝ() has Bass and topological stable ranks equal to 2, which settles the conjecture made by Brett Wick in [18]. We also give a necessary and sufficient condition for reducibility in some real algebras of functions on symmetric domains w...

We denote by W +(ℂ+) the set of all complex-valued functions defined in the closed right half plane ℂ+ ≔ {s ∈ ℂ | Re(s) ≥ 0} that differ from the Laplace transform of functions from L 1(0, ∞) by a constant. Equipped with pointwise operations, W +(ℂ+) forms a ring. It is known that W +(ℂ+) is a pre-Bézout ring. The following properties are shown for...

We compute the Bass stable rank and the topological stable rank of several convolution Banach algebras of complex measures on (-∞,∞) or on [0,∞) consisting of a discrete measure and/or of an absolutely continuous measure. We also compute the stable ranks of the convolution algebras , , ℓ1(S) and , where S is an arbitrary subgroup of , of the almost...

Let $C$ denote a closed convex cone $C$ in $\mathbb{R}^d$ with apex at 0. We denote by $\mathcal{E}'(C)$ the set of distributions having compact support which is contained in $C$. Then $\mathcal{E}'(C)$ is a ring with the usual addition and with convolution. We give a necessary and sufficient analytic condition on $\hat{f}_1,..., \hat{f}_n$ for $f_...

Let \({{\mathbb{C}}}_{+} := \{s \in {{\mathbb{C}}}\quad | \quad {\rm Re}(s) \geq 0\}\) and let \(\mathcal{A}\) denote the Banach algebra $${{{\mathcal{A}}}} = \left\{ s( \in {{{\mathbb{C}}}}_ + ) \mapsto \hat{f}_a (s) + \sum\limits_{k = 0}^\infty {f_k e^{ - st_k }}\bigg | \bigg.{\begin{array}{l}{f_a \in L^1 (0,\infty ),(f_k )_{k \geq 0} \in \ell^{1...

Let H(n) be the canonical half space in Rn, that is, H(n) = {(t1,...,tn) ∈ R n \ {0} | ∀j, (tj 6= 0 and t1 = t2 = � � � = tj 1 = 0) ⇒ tj > 0} ∪ {0}. Let M(H(n)) denote the Banach algebra of all complex Borel measures with support contained in H(n), with the usual addition and scalar multiplication, and with convolution ∗, and the norm being the tot...

We extend the $\nu$-metric introduced by Vinnicombe in robust control theory for rational plants to the case of infinite-dimensional systems/classes of nonrational transfer functions. Comment: 23 pages

We prove an abstract Nyquist criterion in a general set up. As applications, we recover various versions of the Nyquist criterion, some of which are new.

We extend the ν-metric introduced by Vinnicombe in robust control theory for rational plants to the case of infinite-dimensional systems/classes of nonrational transfer functions. Keywords ν-metric–Robust control–Banach algebras

Let B be a Blaschke product. We prove in several different ways the corona theorem for the algebra HB∞ := C + BH ∞. That is, we show the equivalence of the classical corona condition on data f1,...,fn ε HB∞ ∀ z ε D, Σ k=1n |fk(z)| ≥ δ > 0, and the solvability of the Bezout equation for 91,..., 9nB∞ ∀ z ε D, Σ k=1n 9k(z)fk(z) = 1. Estimates on solut...

Let R be a unital semi-simple commutative complex Banach algebra, and let M(R) denote its maximal ideal space, equipped with the Gelfand topology. Sufficient topological conditions are given on M(R) for R to be a projective free ring, that is, a ring in which every finitely generated projective R-module is free. Several examples are included, notab...

Let AR denote the set of functions from the disk algebra having real Fourier coefficients. Generalizing a result of A.Quadrat we show that every unstabilizable multi-input multi-output plant is as close as we want to a stabilizable multi-input multi-output plant in the product topology.

This accessible, undergraduate-level text illustrates the role of algebras of holomorphic functions in the solution of an important engineering problem: the stabilization of a linear control system. Its concise and self-contained treatment avoids the use of higher mathematics and forms a bridge to more advanced treatments.

We show that there do not exist finitely generated, non-principal ideals of denominators in the disk-algebra A (&Dopf;). Our proof involves a new factorization theorem for A (&Dopf;) that is based on Treil's determination of the Bass stable rank for H ∞.

Let E be a separable infinite-dimensional Hilbert space, and let H(\mathbbD; L(E))H({\mathbb{D}}; {\mathcal{L}}(E)) denote the algebra of all functions f : \mathbbD ® L(E)f : {\mathbb{D}} \rightarrow {\mathcal{L}}(E) that are holomorphic. If A{\mathcal{A}} is a subalgebra of H(\mathbbD; L(E))H({\mathbb{D}}; {\mathcal{L}}(E)) , then using an a...

Let K denote a compact real symmetric subset of C and let AR(K) denote the real Banach algebra of all real symmetric continuous functions on K which are analytic in the interior K ◦ of K, endowed with the supremum norm. We characterize all unimodular pairs (f, g) in AR(K)2 which are reducible. In addition, for an arbitrary compact K in C, we give a...

Format: HTML LaTeX MathJax Abstract Let $K$ denote a compact real symmetric subset of $\mathbb{C}$ and let $A_{\mathbb R}(K)$ denote the real Banach algebra of all real symmetric continuous functions on $K$ that are analytic in the interior $K^\circ$ of $K$, endowed with the supremum norm. We characterize all unimodular pairs $(f,g)$ in $A_{\mathb...

We consider a quasilinear PDE system which models nonlinear vibrations of a thermoelastic plate defined on a bounded domain in \mathbbRn\mathbb{R}^n, n≤3. Existence of finite energy solutions describing the dynamics of a nonlinear thermoelastic plate is established. In addition asymptotic long time behavior of weak solutions is discussed. It is s...

In this article, we prove that the Krull dimension of several commonly used classes of transfer functions of infinite dimensional linear control systems is infinite. On the other hand, we also show that the weak Krull dimension of the Hardy algebra H¥(\mathbbD)H^{\infty}(\mathbb{D}) , the disk algebra A(\mathbbD)A(\mathbb{D}) and the Wiener al...

Let $B$ be a Blaschke product. We prove in several different ways the corona theorem for the algebra $H^\infty_B:=\mC+BH^\infty$. That is, we show the equivalence of the classical {\em corona condition} on data $f_1, ..., f_n \in H^\infty_B$: \[ \forall z \in \mD, \sum_{k=1}^{n} |f_k(z)| \geq \delta >0, \] and the {\em solvability of the Bezout equ...

Let 𝔻 denote the open unit disc in ℂ. Let 𝕋 denote the unit circle and let S ⊂ T. We denote by AS(𝔻) the set of all functions f : 𝔻 ∪ S → ℂ that are holomorphic in 𝔻 and are bounded and continuous in 𝔻 ∪ S. Equipped with the supremum norm, AS(𝔻) is a Banach algebra, and it lies between the extreme cases of the disc algebra A(𝔻) and the Hardy space...

Let D denote the open unit diskfz2 Cjjzj < 1g, and C+ denote the closed right half-planefs2 Cj Re(s) 0g. (1) Let W +(D) be the Wiener algebra of the disc, that is the set of all absolutely convergent Taylor series in the open unit disk D, with pointwise operations. (2) Let W +(C+) be the set of all functions dened in the right half- plane C+ that d...

Let ℂ ≥ 0 : = { s ∈ ℂ ∣ Re ( s ) ≥ 0 } , and let 𝒲 + denote the ring of all functions f : ℂ ≥ 0 → ℂ such that f ( s ) = f a ( s ) + ∑ k = 0 ∞ f k e − s t k ( s ∈ ℂ ≥ 0 ) , where f a ∈ L 1 ( 0 , ∞ ) , ( f k ) k ≥ 0 ∈ ℓ 1 , and 0 = t 0 < t 1 < t 2 < ⋯ equipped with pointwise operations. (Here ⋅ ^ denotes the Laplace transform.) It is shown that the r...

In this article we prove a representation theorem for \(H^{\infty}({\mathbb{D}})\) functions, such that the realization formula is spectrally minimal in the following sense: the spectrum of the main operator in the realization intersects the unit circle precisely at those points where the given function has no holomorphic extension. We also extend...

Let E, E * be separable Hilbert spaces. If S is an open subset of \({\mathbb{T}}\), then \(A_S({\mathcal{L}}(E, E_{*}))\) denotes the space of all functions \(f : {\mathbb{D}} \cup S \rightarrow {\mathcal{L}}(E, E_{*})\) that are holomorphic in \(\mathbb{D}\), and bounded and continuous on \(\mathbb{D} \cup S\). In this article we prove the followi...

This paper presents a new method for approximate dynamic inversion of nonaffine-in-control systems via time-scale separation. The control signal is sought as a solution of the “fast” dynamics and is shown to asymptotically stabilize the original nonaffine system. Sufficient conditions are formulated, which satisfy the assumptions of the Tikhonov th...