On Generalized Friedrichs and Krein‐von Neumann Extensions and Canonical Systems

Let S be a densely defined and closed symmetric relation in a Hilbert space ℋ︁ with defect numbers (1,1), and let A be some of its canonical selfadjoint extensions. According to Krein's formula, to S and A corresponds a so‐called Q‐function from the Nevanlinna class N . In this note we show to which subclasses N _γ of N the Q‐functions corresponding to S and its canonical selfadjoint extensions belong and specify the Q‐functions of the generalized Friedrichs and Krein‐von Neumann extensions. A result of L. de Branges implies that to each function Q ∈ N there corresponds a unique Hamiltonian H such that Q is the Titchmarsh‐Weyl coefficient of the two‐dimensional canonical system J_y′ = —zHy on [0, ∞) where Weyl's limit point case prevails at ∞. Then the boundary condition y(0) = 0 corresponds to a symmetric relation T_min with defect numbers (1,1) in the Hilbert space L²_H, and Q is equal to the Q‐function with respect to the extension corresponding to the boundary condition y₁(0) = 0. If H satisfies some growth conditions at 0 or ∞, wepresent results on the corresponding Q‐functions and show under which conditions the generalized Friedrichs or Krein‐von Neumann extension exists.     

Particular formulae for the Moore-Penrose inverse of a columnwise partitioned matrix

An essential part of Cegielski’s [Obtuse cones and Gram matrices with non-negative inverse, Linear Algebra Appl. 335 (2001) 167-181] considerations of some properties of Gram matrices with nonnegative inverses, which are pointed out to be crucial in constructing obtuse cones, consists in developing some particular formulae for the Moore-Penrose inverse of a columnwise partitioned matrix A = (A₁ : A₂) under the assumption that it is of full column rank. In the present paper, these results are generalized and extended. The generalization consists in weakening the assumption mentioned above to the requirement that the ranges of A₁ and A₂ are disjoint, while the extension consists in introducing the conditions referring to the class of all generalized inverses of A.     

Strong linear preservers of rank reverse permutability on triangular matrices

Let F be a field with ∣F∣ > 2 and T_n(F) be the set of all n × n upper triangular matrices, where n ? 2. Let k ? 2 be a given integer. A k-tuple of matrices A₁, …, A_k ∈ T_n(F) is called rank reverse permutable if rank(A₁ A₂ ? A_k) = rank(A_k A_k−1 ? A₁). We characterize the linear maps on T_n(F) that strongly preserve the set of rank reverse permutable matrix k-tuples.     

Zeros of nonpositive type of generalized Nevanlinna functions with one negative square

A generalized Nevanlinna function Q(z) with one negative square has precisely one generalized zero of nonpositive type in the closed extended upper halfplane. The fractional linear transformation defined by Qτ(z)=(Q(z)−τ)/(1+τQ(z)), τ∈R∪{∞}, is a generalized Nevanlinna function with one negative square. Its generalized zero of nonpositive type α(τ) as a function of τ defines a path in the closed upper halfplane. Various properties of this path are studied in detail.     

The Kernel of the Eisenstein Ideal

Let N be a prime number, and let J₀(N) be the Jacobian of the modular curve X₀(N). Let T denote the endomorphism ring of J₀(N). In a seminal 1977 article, B. Mazur introduced and studied an important ideal I⊆T, the Eisenstein ideal. In this paper we give an explicit construction of the kernel J₀(N)[I] of this ideal (the set of points in J₀(N) that are annihilated by all elements of I). We use this construction to determine the action of the group Gal(Q/Q) on J₀(N)[I]. Our results were previously known in the special case where N−1 is not divisible by 16.     

The characterization of a class of multivariate MRA and semi-orthogonal parseval frame wavelets

Let A be a d × d expansive matrix with ∣detA∣ = 2. This paper addresses Parseval frame wavelets (PFWs) in the setting of reducing subspaces of L²(R^d). We prove that all semi-orthogonal PFWs (semi-orthogonal MRA PFWs) are precisely the ones with their dimension functions being non-negative integer-valued (0 or 1). We also characterize all MRA PFWs. Some examples are provided.     

Nevanlinna Functions,Perturbation Formulas,and Triplets of Hilbert Spaces

Let S be a closed symmetric operator with defect numbers (1,1) in a Hilbert space ?? and let A be a selfadjoint operator extension of S in ??. Then S is necessarily a graph restriction of A and the selfadjoint extensions of S can be considered as graph perturbations of A, cf. [8]. Only when S is not densely defined and, in particular, when S is bounded, 5 is given by a domain restriction of A and the graph perturbations reduce to rank one perturbations in the sense of [23]. This happens precisely when the Q - function of S and A belongs to the subclass No of Nevanlinna functions. In this paper we show that by going beyond the Hilbert space ?? the graph perturbations can be interpreted as compressions of rank one perturbations. We present two points of view: either the Hilbert space ?? is given a one-dimensional extension, or the use of Hilbert space triplets associated with A is invoked. If the Q - function of S and A belongs to the subclass N₁ of Nevanlinna functions, then it is convenient to describe the selfadjoint extensions of S including its generalized Friedrichs extension (see [6]) by interpolating the original triplet, cf. [5]. For the case when A is semibounded, see also [4]. We prove some invariance properties, which imply that such an interpolation is independent of the (nonexceptional) extension.     

Max-min of polynomials and exponential diophantine equations

In this paper we study the maximum-minimum value of polynomials over the integer ring Z. In particular, we prove the following: Let F(x,y) be a polynomial over Z. Then, maxx∈Z(T)miny∈Z|F(x,y)|=o(T1/2) as T→∞ if and only if there is a positive integer B such that maxx∈Zminy∈Z|F(x,y)|?B. We then apply these results to exponential diophantine equations and obtain that: Let f(x,y), g(x,y) and G(x,y) be polynomials over Q, G(x,y)∈(Q[x,y]−Q[x])∪Q, and b a positive integer. For every α in Z, there is a y in Z such that f(α,y)+g(α,y)bG(α,y)=0 if and only if for every integer α there exists an h(x)∈Q[x] such that f(x,h(x))+g(x,h(x))bG(x,h(x))≡0, and h(α)∈Z.     

On the uniqueness of overcomplete dictionaries, and a practical way to retrieve them

A full-rank under-determined linear system of equations Ax = b has in general infinitely many possible solutions. In recent years there is a growing interest in the sparsest solution of this equation—the one with the fewest non-zero entries, measured by ∥x∥₀. Such solutions find applications in signal and image processing, where the topic is typically referred to as “sparse representation”. Considering the columns of A as atoms of a dictionary, it is assumed that a given signal b is a linear composition of few such atoms. Recent work established that if the desired solution x is sparse enough, uniqueness of such a result is guaranteed. Also, pursuit algorithms, approximation solvers for the above problem, are guaranteed to succeed in finding this solution.Armed with these recent results, the problem can be reversed, and formed as an implied matrix factorization problem: Given a set of vectors {b_i}, known to emerge from such sparse constructions, Ax_i = b_i, with sufficiently sparse representations x_i, we seek the matrix A. In this paper we present both theoretical and algorithmic studies of this problem. We establish the uniqueness of the dictionary A, depending on the quantity and nature of the set {b_i}, and the sparsity of {x_i}. We also describe a recently developed algorithm, the K-SVD, that practically find the matrix A, in a manner similar to the K-Means algorithm. Finally, we demonstrate this algorithm on several stylized applications in image processing.     

Friedrichs extensions of Schrödinger operators with singular potentials

The Friedrichs extension for the generalized spiked harmonic oscillator given by the singular differential operator −d²/dx²+Bx²+Ax⁻²+λx^−α (B>0, A?0) in L₂(0,∞) is studied. We look at two different domains of definition for each of these differential operators in L₂(0,∞), namely C₀^∞(0,∞) and D(T_2,F)∩D(M_λ,α), where the latter is a subspace of the Sobolev space W_2,2(0,∞). Adjoints of these differential operators on C₀^∞(0,∞) exist as result of the null-space properties of functionals. For the other domain, convolutions and Jensen and Minkowski integral inequalities, density of C₀^∞(0,∞) in D(T_2,F)∩D(M_λ,α) in L₂(0,∞) lead to the other adjoints. Further density properties C₀^∞(0,∞) in D(T_2,F)∩D(M_λ,α) yield the Friedrichs extension of these differential operators with domains of definition D(T_2,F)∩D(M_λ,α).     

Power series solutions to Volterra integral equations

Power series type solutions are given for a wide class of linear and q-dimensional nonlinear Volterra equations on R^p. The basic assumption on the kernel K(x, y) is that K(x, xt) has a power series in x. For example, this holds for any analytic kernel.The kernel may be strongly singular, provided certain constants are finite. One and only one such power series solution exists. Its coefficients are given by a simple iterative formula. In many cases this may be solved explicitly. In particular an explicit formula for the resolvent is given.     

Reduced Weyl asymptotics for pseudodifferential operators on bounded domains II. The compact group case

Let G⊂O(n) be a compact group of isometries acting on n-dimensional Euclidean space Rn, and X a bounded domain in Rn which is transformed into itself under the action of G. Consider a symmetric, classical pseudodifferential operator A₀ in L²(Rn) that commutes with the regular representation of G, and assume that it is elliptic on X. We show that the spectrum of the Friedrichs extension A of the operator is discrete, and using the method of the stationary phase, we derive asymptotics for the number Nχ(λ) of eigenvalues of A equal or less than λ and with eigenfunctions in the χ-isotypic component of L²(X) as λ→∞, giving also an estimate for the remainder term for singular group actions. Since the considered critical set is a singular variety, we recur to partial desingularization in order to apply the stationary phase theorem.     

Mixed finite element methods for general quadrilateral grids

We study a new mixed finite element of lowest order for general quadrilateral grids which gives optimal order error in the H(div)-norm. This new element is designed so that the H(div)-projection Π_h satisfies ∇ · Π_h = P_hdiv. A rigorous optimal order error estimate is carried out by proving a modified version of the Bramble-Hilbert lemma for vector variables. We show that a local H(div)-projection reproducing certain polynomials suffices to yield an optimal L²-error estimate for the velocity and hence our approach also provides an improved error estimate for original Raviart-Thomas element of lowest order. Numerical experiments are presented to verify our theory.     

Alexandrov’s inequality and conjectures on some Toeplitz matrices

We study determinant inequalities for certain Toeplitz-like matrices over C. For fixed n and N ? 1, let Q be the n × (n + N − 1) zero-one Toeplitz matrix with Q_ij = 1 for 0 ? j − i ? N − 1 and Q_ij = 0 otherwise. We prove that det(QQ^∗) is the minimum of det(RR^∗) over all complex matrices R with the same dimensions as Q satisfying ∣R_ij∣ ? 1 whenever Q_ij = 1 and R_ij = 0 otherwise. Although R has a Toeplitz-like band structure, it is not required to be actually Toeplitz. Our proof involves Alexandrov’s inequality for polarized determinants and its generalizations. This problem is motivated by Littlewood’s conjecture on the minimum 1-norm of N-term exponential sums on the unit circle. We also discuss polarized Bazin-Reiss-Picquet identities, some connections with k-tree enumeration, and analogous conjectured inequalities for the elementary symmetric functions of QQ^∗.     

Two local operators and the BLUE

For a given matrix A, a matrix P such that PA = A is said to be a local identity, and such that P²A = PA is said to be a local idempotent. In the paper, some simple properties of such operators are presented. Their relation to the best linear unbiased estimation in the general Gauss-Markov model is also demonstrated.     

Strong Hypercontractivity and Relative Subharmonicity

A Hermitian metric, g, on a complex manifold, M, together with a smooth probability measure, μ, on M determine minimal and maximal Dirichlet forms, Q_D and Q_max, given by Q(f)=∫_M g(grad f(z), grad f(z)) dμ(z). Q_D is the form closure of Q on C^∞_c(M) and Q_max is the form closure of Q on C¹_b(M). The corresponding operators, A_D and A_max, generate semigroups having standard hypercontractivity properties in the scale of L^p spaces, p>1, when the corresponding form, Q, satisfies a logarithmic Sobolev inequality. It was shown by the author (1999, Acta Math.182, 159-206) that the semigroup e^−tA_D has even stronger hypercontractivity properties when restricted to certain holomorphic subspaces of L^p. These results are extended here to A_max. When (M, g) is not complete it is necessary that the elliptic differential operator A_max degenerate on the boundary of M. A second proof of these strong hypercontractive inequalities for both A_D and A_max is given, which depends on an extension of the submean value property of subharmonic functions. The Riemann surface for z^1/n and the weighted Bergman spaces in the unit disc are given as examples.     

Twostep-by-twostep continuous PIRKN-type PC methods for nonstiff IVPs

In this paper, we start with the consideration of direct collocation-based Runge-Kutta-Nyström (RKN) methods with continuous output formulas for solving nonstiff initial-value problems (IVPs) for systems of special second-order differential equations y″(t) = f(t, y(t)). At nth step, the continuous output formulas can be used for calculating the step values at (n + 2)th step and the integration processes can be proceeded twostep-by-twostep. In this case, we obtain twostep-by-twostep RKN methods with continuous output formulas (continuous TBTRKN methods). Furthermore, we consider a parallel predictor-corrector (PC) iteration scheme using the continuous TBTRKN methods as corrector methods with predictor methods defined by the continuous output formulas. The resulting twostep-by-twostep parallel-iterated RKN-type PC methods with continuous output formulas (twostep-by-twostep continuous PIRKN-type PC methods or TBTCPIRKN methods) give us a faster integration processes. Numerical comparisons based on the solution of a few widely-used test problems show that the new TBTCPIRKN methods are much more efficient than the well-known PIRKN methods, the famous nonstiff sequential ODEX2, DOP853 codes and comparable with the CPIRKN methods.