The best approximation to a class of functions of several variables by another class and related extremum problems

We study the relationship between several extremum problems for unbounded linear operators of convolution type in the spaces , m ≥ 1, 1 ≤ γ ≤ ∞. For the problem of calculating the modulus of continuity of the convolution operatorA on the function classQ defined by a similar operator and for the Stechkin problem on the best approximation of the operatorA on the classQ by bounded linear operators, we construct dual problems in dual spaces, which are the problems on, respectively, the best and the worst approximation to a class of functions by another class. Translated fromMatematicheskie Zametki, Vol. 64, No. 3, pp. 323–340, September, 1998. This research was supported by INTAS under grant No. 94-4070.     

Normal Matrices and an Extension of Malyshev's Formula

Let A be a complex matrix of order n with n ≥ 3. We associate with A the 3n × 3n matrix $Q\left( {\gamma } \right) = \left( \begin{gathered} A \gamma _1 I_n \gamma _3 I_n \\ 0 A \gamma _2 I_n \\ 0 0 A \\ \end{gathered} \right)$ where $\gamma _1 ,\gamma _2 ,\gamma _3 $ are scalar parameters and γ=(γ₁,γ₂,γ₃). Let σ_i, 1 ≤ i ≤ 3n, be the singular values of Q(γ) in the decreasing order. We prove that, for a normal matrix A, its 2-norm distance from the set $\mathcal{M}$ of matrices with a zero eigenvalue of multiplicity at least 3 is equal to $\mathop {max}\limits_{\gamma _1 ,\gamma _2 \geqslant 0,\gamma _3 \in \mathbb{C}} \sigma _{3n - 2} (Q\left( \gamma \right)).$ This fact is a refinement (for normal matrices) of Malyshev's formula for the 2-norm distance from an arbitrary n × n matrix A to the set of n × n matrices with a multiple zero eigenvalue.     

Γ-inverses of Bounded Linear Operators

Let B(H) be the algebra of all the bounded linear operators on a Hilbert space H.For A,P and Q in B(H),if there exists an operator X∈ B(H) such thatAP X QA=A,X QAP X=X,(QAP X)*=QAP X and(X QAP)*=X QAP,then X is said to be the Γ-inverse of A associated with P and Q,and denoted by AP,Q+.In this note,we present some necessary and su?cient conditions for which A+P,Qexists,and give an explicit representation of AP,Q+(if AP,Q+exists).     

Justification of a Malyshev-Type Formula in the Nonnormal Case

Let A be a complex matrix of order n, n ≥ 3 . We associate with A the 3n $$Q(\gamma ) = \left( {\begin{array}{*{20}c} A & {_{\gamma 1} I_n } & {_{\gamma 3} I_n } \\ 0 & A & {_{\gamma 2} I_n } \\ 0 & 0 & A \\ \end{array} } \right),$$ where γ₁, γ₂, γ₃ are scalar parameters and γ = (γ₁, γ₂, γ₃). Let σ_i, 1 ≤ i ≤ 3n, be the singular values of Q(γ), in decreasing order. Under certain assumptions on A, the authors have proved earlier that the 2-norm distance from A to the set $\mathcal{M}$ of matrices with a zero eigenvalue of multiplicity at least 3 is equal to max $$\begin{array}{*{20}c} {\max } \\ {\gamma 1,\gamma 2,\gamma 3 \in \mathbb{C}} \\ \end{array} \;^\sigma 3n - 2(Q(\gamma )).$$ Now, the justification of this formula for the distance is given for an arbitrary matrix A.     

Linear deviations of the classes\tilde W_p^\alpha and approximations in spaces of multipliers

We consider the problem of the asymptotically best linear method of approximation in the metric of L_s[?π, π] of the set \(\tilde W_p^\alpha (1)\) of periodic functions with a bounded in L_p[?π, π] fractional derivative, by functions from \(\tilde W_p^\beta (M)\) ,β >α, for sufficiently large M, and the problem about the best approximation in L_s[?π, π] of the operator of differentiation on \(\tilde W_p^\alpha (1)\) by continuous linear operators whose norm (as operators from L_r[?π, π] into L_q[?π, π])does not exceed M. These problems are reduced to the approximation of an individual element in the space of multipliers, and this allows us to obtain estimates that are exact in the sense of the order.     

Unitary Equivalence of Proper Extensions of a Symmetric Operator and the Weyl Function

Let A be a densely defined simple symmetric operator in ${\mathfrak{H}}$ , let ${\Pi=\{\mathcal{H},\Gamma_0, \Gamma_1}\}$ be a boundary triplet for A ^* and let M(·) be the corresponding Weyl function. It is known that the Weyl function M(·) determines the boundary triplet Π, in particular, the pair {A, A ₀}, uniquely up to the unitary similarity. Here ${A_0 := A^* \upharpoonright \text{ker}\, \Gamma_0 ( = A^*_0)}$ . At the same time the Weyl function corresponding to a boundary triplet for a dual pair of operators defines it uniquely only up to the weak similarity. We consider a symmetric dual pair {A, A} with symmetric ${A \subset A^*}$ and a special boundary triplet ${\widetilde{\Pi}}$ for{A, A} such that the corresponding Weyl function is ${\widetilde{M}(z) = K^*(B-M(z))^{-1} K}$ , where B is a non-self-adjoint bounded operator in ${\mathcal{H}}$ . We are interested in the problem whether the result on the unitary similarity remains valid for ${\widetilde{M}(\cdot)}$ in place of M(·). We indicate some sufficient conditions in terms of the operators A ₀ and ${A_B= A^* \upharpoonright \text{ker}\, (\Gamma_1-B \Gamma_0)}$ , which guaranty an affirmative answer to this problem. Applying the abstract results to the minimal symmetric 2nth order ordinary differential operator A in ${L^2(\mathbb{R}_+)}$ , we show that ${\widetilde{M}(\cdot)}$ defined in ${\Omega_+ \subset \mathbb{C}_+}$ determines the Dirichlet and Neumann realizations uniquely up to the unitary equivalence. At the same time similar result for realizations of Dirac operator fails. We obtain also some negative abstract results demonstrating that in general the Weyl function ${\widetilde{M}(\cdot)}$ does not determine A _B even up to the similarity.     

Polaroid Operators with SVEP and Perturbations of Property (gw)

Let ${\mathcal{L}(X)}$ be the algebra of all bounded linear operators on X and ${\mathcal{P}S(X)}$ be the class of polaroid operators with the single-valued extension property. The property (gw) holds for ${T \in \mathcal{L}(X)}$ if the complement in the approximate point spectrum of the semi-B-essential approximate point spectrum coincides with the set of all isolated points of the spectrum which are eigenvalues of the spectrum. In this note we focus on the stability of the property (gw) under perturbations: we prove that, if ${T \in \mathcal{P}S(X)}$ and A (resp. Q) is an algebraic (resp. quasinilpotent) operator, then the property (gw) holds for f(T ^* + A ^*) (resp. f(T ^* + Q*)) for every analytic function f in σ(T + A) (resp. σ(T + Q)). Some applications are also given.     

The characterization of a class of quantum Markov semigroups and the associated operator-valued Dirichlet forms based on Hilbert W*-module

In this paper, we introduce the concept of operator-valued quadratic form based on Hilbert W*-module l2 A, and give a one to one correspondence between the set of positive self-adjoint regular module operators on l2A and the set of regular quadratic forms, where A is a finite and σ-finite von Neumann algebra. Furthermore, we obtain that a strict continuous symmetric regular module operator semigroup {Tt }t∈R + L(l2A) is Markovian if and only if the associated A-valued quadratic form is a Dirichlet form, where L(l2A) is the von Neumann algebra of all adjointable module maps on l2A.     

On Marcus-Wang conjecture

LetA _m ,B _m ,m=1, ...,p, be linear operators on ann-dimensional unitary space \(V.L = \sum\limits_{m = 1}^p {A_m \otimes B_m } \) is a linear operator on ?² V, the tensor product space with the customarily induced inner product. The numerical range ofL is defined as $$W\tfrac{1}{2}(L) = \left\{ {(L)x \otimes y,x \otimes y):x,y o.n.} \right\}$$ where “o.n.” means “orthonormal”. In [1], M.Marcus and B.Y. Wang conjecture: There exists no non-zero operatorL of minimum length less thann for whichW ₂ ¹ (L)=0. In this paper, we prove that this conjecture is true.     

Abstract wave equations and associated Dirac-type operators

We discuss the unitary equivalence of generators G _A,R associated with abstract damped wave equations of the type ${\ddot{u} + R \dot{u} + A^*A u = 0}$ in some Hilbert space ${\mathcal{H}_1}$ and certain non-self-adjoint Dirac-type operators Q _A,R (away from the nullspace of the latter) in ${\mathcal{H}_1 \oplus \mathcal{H}_2}$ . The operator Q _A,R represents a non-self-adjoint perturbation of a supersymmetric self-adjoint Dirac-type operator. Special emphasis is devoted to the case where 0 belongs to the continuous spectrum of A*A. In addition to the unitary equivalence results concerning G _A,R and Q _A,R , we provide a detailed study of the domain of the generator G _A,R , consider spectral properties of the underlying quadratic operator pencil ${M(z) = |A|^2 - iz R - z^2 I_{\mathcal{H}_1}, z\in\mathbb{C}}$ , derive a family of conserved quantities for abstract wave equations in the absence of damping, and prove equipartition of energy for supersymmetric self-adjoint Dirac-type operators. The special example where R represents an appropriate function of |A| is treated in depth, and the semigroup growth bound for this example is explicitly computed and shown to coincide with the corresponding spectral bound for the underlying generator and also with that of the corresponding Dirac-type operator. The cases of undamped (R?=?0) and damped (R ≠ 0) abstract wave equations as well as the cases ${A^* A \geq \varepsilon I_{\mathcal{H}_1}}$ for some ${\varepsilon > 0}$ and ${0 \in \sigma (A^* A)}$ (but 0 not an eigenvalue of A*A) are separately studied in detail.     

On Riesz type kernels over local fields

Based on a representation lemma, Riesz type kernels on the local fieldK and on the integer ringO inK are constructed. Furthermore, we discuss approximation theorems for the Lipschitz classLip(L^r;a) and the L^p boundedness of such operators motivated by the open problem: Does $\sigma _n f\xrightarrow{{a,e}}f$ for f ∈ L¹(0) (see M. H. Taibleson [6] and [5])?     

A Note on Quasi-paranormal Operators

Let T be a bounded linear operator on an infinite dimensional complex Hilbert space. In this paper, we introduce the new class, denoted ${{\mathcal{QP}}}$ , of operators satisfying ${{\|T^{2}x\|^{2}\leq \|T^{3}x\|\|Tx\|}}$ for all ${{x \in \mathcal{H}}}$ . This class includes the classes of paranormal operators and quasi-class A operators. We prove basic structural properties of these operators. Using these results, we also prove that if E is the Riesz idempotent for a nonzero isolated point λ₀ of the spectrum of ${{T \in \mathcal{QP}}}$ , then E is self-adjoint if and only if ${{N(T-\lambda_{0}) \subseteq N(T^{*}-\overline{\lambda}_{0})}}$ .     

The explicit solutions ofC_{p,q}^{k + \alpha } -form for the\bar \partial -equations on pseudoconvex open sets

In this paper, we study the integral solution operators for the $\bar \partial $ -equations on pseudoconvex domains. As a generalization of [1] for the $\bar \partial $ -equations on pseudoconvex domains with boundary of classC ^∞, we obtain the explicit integral operator solutions of $C_{p,q}^{k + \alpha } $ -form for the $\bar \partial $ -equations on pseudoconvex open sets with boundary ofC ^k (k≥0) and the sup-norm estimates of which solutions have similar as that [1] in form.     

Uniform regularization of the problem of calculating the values of an operator

Let X and Y be linear normed spaces, W a set in X, A an operator from W into Y, and \(\mathfrak{W}\) the set of all operators or the set ? of linear operators from X into Y. With δ>0 we put $$v\left( {\delta ,\mathfrak{M}} \right) = \mathop {\inf }\limits_{T \in \mathfrak{M}} \mathop {\sup }\limits_{x \in W} \mathop {\sup }\limits_{\left\| {\eta - x} \right\|_X \leqslant \delta } \left\| {Ax - T\eta } \right\|_Y $$ . We discuss the connection of \(v\left( {\delta , \mathfrak{M}} \right)\) with the Stechkin problem on best approximation of the operator A in W by linear bounded operators. Estimates are obtained for \(v\left( {\delta , \mathfrak{M}} \right)\) e.g., we write the inequality , where H(Y) is Jung's constant of the space Y, and Ω(t) is the modulus of continuity of A in W.     

A generalization of the katona theorem for crosst-intersecting families

LetX be ann-element set and letA and? be families of subsets ofX. We say thatA and? are crosst-intersecting if |A ∩ B| ≥ t holds for all A ∈A and for allB ∈ ?. Suppose thatA and ? are crosst-intersecting. This paper first proves a crosst-intersecting version of Harper's Theorem:

1. There are two crosst-intersecting Hamming spheresA ₀,? ₀ with centerX such that |A| ≤ |A ₀| and|?| ≤ |? ₀| hold.
2. Suppose thatt ≥ 2 and that the pair of integers (|A) is maximal with respect to direct product ordering among pairs of crosst-intersecting families. Then,A and? are Hamming spheres with centerX.

Using these claims, the following conjecture of Frankl is proven:

1. Ifn + t = 2k ? 1 then |A| |?| ≤ max \(\left\{ {\left( {K_k^n + \left( {_{k - 1}^{n - 1} } \right)} \right)^2 ,K_k^n K_{k - 1}^n } \right\}\) holds, whereK _l ⁿ is defined as \(\left( {_n^n } \right)\left( {_{n - 1}^n } \right) + \cdots + \left( {_l^n } \right).\)
2. Ifn + t = 2k then |A| |? ≤ (K _k ⁿ )² holds.

The extremal configurations are also determined.     

A multivariate analog of Marsden's identity and a quasi-interpolation scheme

Let ? be a linear combination of certain box splines and \(\hat \phi \) its Fourier transform, such that \(\hat \phi \left( 0 \right) \ne 0\) and \(D^\beta \hat \phi \left( {2\pi k} \right) = 0\) for all κ∈Z^N{0} and β≤α. In this paper we construct an expression of the multivariate polynomial (·-y)^α in terms of a linear combination of the integer translates of ?(·), where the coefficients can be computed recursively using only the information on \(D^\beta \hat \phi \left( 0 \right)\) , β ≤ α. As an application, a quasi-interpolation scheme based only on function values on (scaled) integers κ∈Z^N is constructed that gives a “multivariate order” of approximation that includes both coordinate and total orders.     

Weak Order for the Discretization of the Stochastic Heat Equation Driven by Impulsive Noise

We study the approximation of the distribution of X _T , where (X _t ) _t?∈?[0,?T] is a Hilbert space valued stochastic process that solves a linear parabolic stochastic partial differential equation driven by an impulsive space time noise, $$ dX_t+AX_t\,dt= Q^{1/2}\,dZ_t,\quad X_0=x_0\in H,\quad t\in [0,T]. $$ Here (Z _t ) _t?∈?[0,?T] is an impulsive cylindrical process and the operator Q describes the spatial covariance structure of the noise; we assume that A ^??α has finite trace for some α?>?0 and that A ^β Q is bounded for some β?∈?(α???1, α]. A discretized solution $(X_h^n)_{n\in\{0,1,\ldots,N\}}$ is defined via the finite element method in space (parameter h?>?0) and a θ-method in time (parameter Δt?=?T/N). For $\varphi \in C^2_b(H;{\mathbb R})$ we show an integral representation for the error $|{\mathbb E}\varphi(X^N_h)-{\mathbb E}\varphi(X_T)|$ and prove that $$ \left|{\mathbb E}\varphi\left(X^N_h\right)-{\mathbb E}\varphi(X_T)\right|=O\left(h^{2\gamma}+\left(\Delta t\right)^{\gamma}\right) $$ where γ?<?1???α?+?β. This is the same order of convergence as in the case of a Gaussian space time noise, which has been obtained in a paper by Debussche and Printems (Math Comput 78:845–863, 2009). Our result also holds for a combination of impulsive and Gaussian space time noise.     

A class of singular integrals on then-complex unit sphere

The operaton on the n-complex unit sphere under study have three forms: the singular integrals with holomorphic kernels, the bounded and holomorphic Fourier multipliers, and the Cauchy-Dunford bounded and holomorphic functional calculus of the radial Dirac operator $D = \sum\nolimits_{k = 1}^n {z_k \frac{\partial }{{\partial _{z_k } }}} $ . The equivalence between the three fom and the strong-type (p,p), 1 <p < ∞, and weak-type (1,1)-boundedness of the operators is proved. The results generalise the work of L. K. Hua, A. Korányli and S. Vagi, W. Rudin and S. Gong on the Cauchy-Szegö, kemel and the Cauchy singular integral operator.     

L p -theory for second-order elliptic operators with unbounded coefficients in an endpoint class

The m-accretivity and m-sectoriality of the minimal and maximal realizations of second-order elliptic operators of the form ${Au=-{\rm div}(a \nabla u)+F\cdot \nabla u +Vu}$ in ${L^p(\mathbb{R}^N)}$ are shown, where the coefficients a, F and V are unbounded. The result may be regarded as an endpoint assertion of the previous result in Sobajima (J Evol Equ 12:957–971, 2012) and an improvement of that in Metafune et al. (Forum Math 22:583–601, 2010). Moreover, an L ^p -generalization of Kato’s self-adjoint problem in Kato (1981, Appendix 2) is discussed. The proof is based on Sobajima (J Evol Equ 12:957–971, 2012). As examples, the operators ${-\Delta \pm |x|^{\beta-1}x \cdot \nabla +c|x|^{\gamma}}$ are also dealt with, which are mentioned in Metafune et al. (Forum Math 22:583–601, 2010).     

Number theoretic applications of a class of Cantor series fractal functions. I

Suppose that \({{(P, Q) \in {\mathbb{N}_{2}^\mathbb{N}} \times {\mathbb{N}_{2}^\mathbb{N}}}}\) and x = E ₀.E ₁ E ₂ · · · is the P-Cantor series expansion of \({x \in \mathbb{R}}\) . We define $$\psi_{P,Q}(x) := {\sum_{n=1}^{\infty}} \frac{{\rm min}(E_n, q_{n}-1)}{q_1 \cdots q_n}.$$ The functions \({\psi_{P,Q}}\) are used to construct many pathological examples of normal numbers. These constructions are used to give the complete containment relation between the sets of Q-normal, Q-ratio normal, and Q-distribution normal numbers and their pairwise intersections for fully divergent Q that are infinite in limit. We analyze the Hölder continuity of \({\psi_{P,Q}}\) restricted to some judiciously chosen fractals. This allows us to compute the Hausdorff dimension of some sets of numbers defined through restrictions on their Cantor series expansions. In particular, the main theorem of a paper by Y. Wang et al. [29] is improved. Properties of the functions \({\psi_{P,Q}}\) are also analyzed. Multifractal analysis is given for a large class of these functions and continuity is fully characterized. We also study the behavior of \({\psi_{P,Q}}\) on both rational and irrational points, monotonicity, and bounded variation. For different classes of ergodic shift invariant Borel probability measures \({\mu_1}\) and \({\mu_2}\) on \({{\mathbb{N}_2^\mathbb{N}}}\) , we study which of these properties \({\psi_{P,Q}}\) satisfies for \({\mu_1 \times \mu_2}\) -almost every (P,Q) \({{\in {\mathbb{N}_{2}^{\mathbb{N}}} \times {\mathbb{N}_{2}^{\mathbb{N}}}}}\) . Related classes of random fractals are also studied.