The Laguerre functions l(n,tau)(alpha), n = 0,1,. .., are constructed from generalized Laguerre polynomials. The functions l alpha n,tau depend on two parameters: the scale tau>0 and the order of generalization alpha>-1, and form an orthonormal basis in L2[0, infinity). Let the spectrum of a square matrix A lie in the open left half-plane. Then the matrix exponential H(t) = eAt, t > 0, belongs to L2[0, infinity). Hence the matrix exponential H P infinity N can be expanded in a series H = n=0 Sn,tau,alpha l alpha n,tau. An estimate of the norm H-n=0 Sn,tau,alpha l alpha n,tau L2[0,infinity) is proposed. Finding the minimum of this estimate over tau and alpha is discussed. Numerical examples show that the optimal alpha is often almost 0, which essentially simplifies the problem.

Let B and C be square complex matrices. The differential equation x”(t)+Bx'(t)+Cx(t)=f(t) is considered. A solvent is a matrix solution X of the equation X^2+BX+C=0. A pair of solvents X and Z is called complete if the matrix X-Z is invertible. Knowing a complete pair of solvents X and Z allows us to reduce the solution of the initial value problem to the calculation of two matrix exponentials e^Xt and e^Zt. The problem of finding a complete pair X and Z, which leads to small rounding errors in solving the differential equation, is discussed.

Let U,V⊆ℂ be open convex sets, and z_1 , z_2 , …,z_N∈ U and w_1 , w_2 , …,w_M∈ V be (maybe repetitive) points. Let f: U× V→ℂ be an analytic function. Let the interpolating polynomial p be determined by the values of f on the rectangular grid (z_i,w_j) , i=1,2,…,N , j=1,2,…,M . Let A and B be matrices of the sizes n× n and m× m , respectively. The function f of A and B can be defined by the formula f(A,B)=1/(2π i)^2∫_Γ_1∫_Γ_2f(λ,μ)(λ1-A)^-1⊗(μ1-B)^-1 dμ dλ, where Γ_1 and Γ_2 surround the spectra σ(A) and σ(B) , respectively; p(A,B) is defined in the same way. An estimate of ||f(A,B)-p(A,B)|| is given.

Let X be a Banach space and T be a bounded linear operator acting in lp(ℤc,X), 1 ≤ p ≤ ∞. The operator T is called locally nuclear if it can be represented in the form (Tx)_k = ∑_m ∈ℤ^cb_kmx_k - m, k ∈ℤ^c, where bkm: X → X are nuclear, b_km_𝔖_1≤β _m, k,m ∈ℤ^c, ·_𝔖_1 is the nuclear norm, β ∈ l1(ℤc,ℂ) or β ∈ l1,g(ℤc,ℂ), and g is an appropriate weight on ℤc. It is established that if T is locally nuclear and the operator 1 + T is invertible, then the inverse operator (1 + T)−1 has the form 1 + T1, where T1 is also locally nuclear. This result is refined for the case of operators acting in Lp (ℝc,ℂ).

This paper is a survey devoted to the transformations $$\begin{aligned} C&\mapsto \frac{1}{(2\pi i)^2}\int _{\Gamma _1}\int _{\Gamma _2}f(\lambda ,\mu )\,R_{1,\,\lambda }\,C\, R_{2,\,\mu }\,{\mathrm{d}}\mu \,{\mathrm{d}}\lambda ,\\ C&\mapsto \frac{1}{2\pi i}\int _{\Gamma }g(\lambda )R_{1,\,\lambda }\,C\, R_{2,\,\lambda }\,{\mathrm{d}}\lambda , \end{aligned}$$ where $$R_{1,\,(\cdot )}$$ and $$R_{2,\,(\cdot )}$$ are pseudo-resolvents acting in a Banach space, i. e., the resolvents of bounded, unbounded, or multivalued linear operators, and f and g are analytic functions; here $$\Gamma _1$$ , $$\Gamma _2$$ , and $$\Gamma$$ surround the singular sets (spectra) of the pseudo-resolvents $$R_{1,\,(\cdot )}$$ , $$R_{2,\,(\cdot )}$$ , and the both, respectively. Several applications are considered: a representation of the impulse response of a second-order linear differential equation with operator coefficients, a representation of the solution of the Sylvester equation, and properties of the differential of the ordinary functional calculus.

Let $A$ be a square complex matrix; $z_1$, ..., $z_{N}\in\mathbb C$ be arbitrary (possibly repetitive) points of interpolation; $f$ be an analytic function defined on a neighborhood of the convex hull of the union of the spectrum $\sigma(A)$ of the matrix $A$ and the points $z_1$, ..., $z_{N}$; and the rational function $r=\frac uv$ (with the degree of the numerator $u$ less than $N$) interpolates $f$ at these points (counted according to their multiplicities). Under these assumptions estimates of the kind $$ \bigl\Vert f(A)-r(A)\bigr\Vert\le \max_{t\in[0,1];\mu\in\text{convex hull}\{z_1,z_{2},\dots,z_{N}\}}\biggl\Vert\Omega(A)[v(A)]^{-1} \frac{\bigl(vf\bigr)^{{(N)}} \bigl((1-t)\mu\mathbf1+tA\bigr)}{N!}\biggr\Vert, $$ where $\Omega(z)=\prod_{k=1}^N(z-z_k)$, are proposed. As an example illustrating the accuracy of such estimates, an approximation of the impulse response of a dynamic system obtained using the reduced-order Arnoldi method is considered, the actual accuracy of the approximation is compared with the estimate based on this paper.

Let T be a square matrix with a real spectrum, and let f be an analytic function. The problem of the approximate calculation of f(T) is discussed. Applying the Schur triangular decomposition and the reordering, one can assume that T is triangular and its diagonal entries tii are arranged in increasing order. To avoid calculations using the differences tii − tjj with close (including equal) tii and tjj, it is proposed to represent T in a block form and calculate the two main block diagonals using interpolating polynomials. The rest of the f(T) entries can be calculated using the Parlett recurrence algorithm. It is also proposed to perform some scalar operations (such as the building of interpolating polynomials) with an enlarged number of significant decimal digits.

The integral operator of the form $$\begin{aligned} \bigl (Nu\bigr )(x)=\sum _{k=1}^\infty e^{i\langle \omega _k,x\rangle } \int _{\mathbb {R}^c}n_k(x-y)\,u(y)\,dy \end{aligned}$$acting in $$L_p(\mathbb {R}^c)$$, $$1\le p\le \infty $$, is considered. It is assumed that $$\omega _k\in \mathbb {R}^c$$, $$n_k\in L_1(\mathbb {R}^c)$$, and $$\begin{aligned} \sum _{k=1}^\infty \Vert n_k\Vert _{L_1}<\infty . \end{aligned}$$We prove that if the operator $$\mathbf {1}+N$$ is invertible, then $$(\mathbf {1}+N)^{-1}=\mathbf {1}+M$$, where M is an integral operator possessing the analogous representation.

Let A be a square complex matrix; z(1), ..., z(n) is an element of C be (possibly repetitive) points of interpolation; f be a function analytic in a neighborhood of the convex hull of the union of the spectrum of A and the points z(1), ..., z(n); and p be the interpolation polynomial of f constructed by the points z(1), ..., z(n). It is proved that under these assumptions parallel to f(A) - p(A)parallel to <= 1/n! max(t is an element of[0, 1] mu is an element of co{z1, z2, ..., zn})parallel to Omega (A) f((n)) ((1 - t)mu 1 + tA)parallel to, where Omega(z) = Pi(n )(k=1)(z - z(k)) and the symbol co means the convex hull.

Abstract It is well known that the equation x′⁢(t)=A⁢x⁢(t)+f⁢(t){x^{\prime}(t)=Ax(t)+f(t)}, where A is a square matrix, has a unique bounded solution x for any bounded continuous free term f, provided the coefficient A has no eigenvalues on the imaginary axis. This solution can be represented in the form x⁢(t)=∫-∞∞𝒢⁢(t-s)⁢f⁢(s)⁢𝑑s.x(t)=\int_{-\infty}^{\infty}\mathcal{G}(t-s)f(s)\,ds. The kernel 𝒢{\mathcal{G}} is called Green’s function. In this paper, for approximate calculation of 𝒢{\mathcal{G}}, the Newton interpolating polynomial of a special function gt{g_{t}} is used. An estimate of the sensitivity of the problem is given. The results of numerical experiments are presented.

An estimate of Green's function of the bounded solutions problem for the ordinary differential equation $x'(t)-Bx(t)=f(t)$ is proposed. It is assumed that the matrix coefficient $B$ is triangular. This estimate is a generalization of the estimate of the matrix exponential proved by Ch. F. Van Loan.

The paper considers the equation where the operator-valued bounded functions a(j) and b(j) are 2-periodic, and the operator-valued kernels m and n are 2-periodic with respect to the first argument. The connection between the input-output stability of the equation and the invertibility of a family of operators acting on the space of periodic functions is investigated.

An analogue of the Gelfand–Shilov estimate of the matrix exponential is proved for Green’s function of the problem of bounded solutions of the ordinary differential equation \(x'(t)-Ax(t)=f(t)\).

An algorithm for computing an analytic function of a matrix $A$ is described. The algorithm is intended for the case where $A$ has some close eigenvalues, and clusters (subsets) of close eigenvalues are separated from each other. This algorithm is a modification of some well known and widely used algorithms. A novel feature is an approximate calculation of divided differences for the Newton interpolating polynomial in a special way. This modification does not require to reorder the Schur triangular form and to solve Sylvester equations.

Let N be an integral operator of the form(Nu)(x)=∫Rcn(x,x−y)u(y)dy acting in Lp(Rc) with a measurable kernel n satisfying the estimate|n(x,y)|≤β(y), where β∈L1. It is proved that if the function t↦n(t,⋅) is continuous in the norm of L1 and the operator 1+N has an inverse, then (1+N)−1=1+M, where M is an integral operator possessing the same properties.

We introduce the notion of an extended tensor product of Banach spaces and . It is defined as a triple consisting of a Banach space and two full subalgebras and of the algebras and of all bounded linear operators on and respectively. It is assumed that is an extension of the ordinary tensor product , and the functionals on and operators on have a canonical extension from to . Every pseudo- resolvent generates a functional calculus that sends analytic - valued functions in a neighbourhood of the singular set of the pseudo- resolvent to operators on . We prove an analogue of the spectral mapping theorem for such a functional calculus.

Вводится понятие расширенного тензорного произведения банаховых пространств $X$ и $Y$. Оно определяется как набор трех объектов: банахова пространства $X\boxtimes Y$ и двух наполненных подалгебр $\mathbf B_0(X)$ и $\mathbf B_0(Y)$ алгебр $\mathbf B(X)$ и $\mathbf B(Y)$ всех линейных ограниченных операторов, действующих в $X$ и $Y$ соответственно. Предполагается, что $X\boxtimes Y$ - расширение обычного тензорного произведения $X\otimes Y$, а функционалы из $X^*\otimes Y^*$ и операторы из $\mathbf B_0(X)\otimes\mathbf B_0(Y)$ имеют каноническое продолжение с $X\otimes Y$ на $X\boxtimes Y$. Всякая псевдорезольвента в алгебре $\mathbf B_0(Y)$ порождает функциональное исчисление, сопоставляющее аналитическим функциям, определенным в окрестности сингулярного множества псевдорезольвенты и принимающим значения в $\mathbf B_0(X)$, операторы, действующие в $X\boxtimes Y$. Доказано, что для такого функционального исчисления справедлив аналог теоремы об отображении спектра. Библиография: 33 наименования.

The paper deals with projection methods of approximate solving the problemFx' = Gx + bu(t), y = < x, d >which consist in passage to the reduced-order problemFx' = Gx + bu(t), y = < x, d >,whereF = Lambda FV, G = Lambda GV, b = Lambda b, d = V*d.It is shown that, if V and Lambda are constructed on the basis of Krylov's subspaces, a projection method is equivalent to the replacement in the formula expressing the impulse response via the exponential function of the pencil lambda bar right arrow lambda F - G, of the exponential function by its rational interpolation satisfying some interpolation conditions. Special attention is paid to the case when F is not invertible.

It is known that eigenfunctions of many elliptic operators such as Schrodinger operators decrease exponentially. It this paper we suggest a different idea of the proof of this fact. This idea is based on a special transformation psi(E).

Let 𝕊 be a cone in ℝ n . A bounded linear operator T : L p (ℝ n ) → L p (ℝ n ) is said to be causal with respect to 𝕊 if the implication x ( s ) = 0 ( s ε W − 𝕊 ) ⇒ ( Tx ) ( s ) = 0 ( s ε W − 𝕊 ) is valid for any x ε L p (ℝ n ) and any open subset W ⊆ ℝ n . The set of all causal operators is a Banach algebra. We describe the spectrum of the operator (Tx)(t) = ∑_n = 1^∞a_n x(t - t_n ) + ∫𝕊g(s)x(t - s)ds, t ∈ℝ^n , in this algebra. Here x ranges in a Banach space 𝔼 , the a n are bounded linear operators in 𝔼 , and the function g ranges in the set of bounded operators in 𝔼 .