Quantitative Runge type approximation theorems for zero solutions of certain partial differential operators

Analysis on Real and Complex Manifolds

This volume on “Topics in Function Spaces, Differential Operators, Harmonic and Fractal Analysis” is dedicated to our teacher and friend Hans Triebel on the occasion of his seventy-fifth birthday. Hans Triebel was born on February 7, 1936 in Dessau, Germany. He studied mathematics and physics at the Friedrich-Schiller-University of Jena from 1954 to 1959, graduating with a Diploma Degree in mathematics. At the beginning of his academic career he worked in classical complex analysis and received a Ph.D. in Mathematics at the Friedrich-Schiller-University in 1962. Motivated by an interest in both mathematics and physics, he studied Sobolev's famous 1950 book and learned about the theory of distributions as developed by L. Schwartz. This might have been the catalyst for his change of research topic towards partial differential operators and function spaces. As a postdoc he spent one year at the University of Leningrad (St. Petersburg), where he enjoyed the intellectual ferment of the atmosphere created by such great mathematicians as Uraltseva, Ladyzhenskaya, Birman and Solomyak, and attended lectures by Birman on functional analysis, spectral theory and quantum mechanics. Inspired by the Russian School of Mathematics he focused his research on recent developments in linear and nonlinear partial differential operators, spectral theory and functional analysis, rapidly obtaining far-reaching results with a deep impact on further research in this field. In particular, he realized the significance of function spaces and contributed to both theory and applications in a decisive way. He completed his Habilitation Thesis on function spaces and nonlinear analysis in 1966, becoming Full Professor of Analysis at the Friedrich-Schiller-University in 1970. His further studies were also strongly influenced and motivated by personal contacts with S. G. Krein, J. Peetre, as well as by new approaches to the theory of function spaces based on Fourier-analytical techniques due to S. M. Nikol'skij, E. M. Stein and C. Fefferman. The development of the modern theory of function spaces in the last 40 years and its application to various branches in both pure and applied mathematics owes much to his seminal contributions. The bare facts are impressive: he has published more than 200 papers in internationally acknowledged journals, and has written no less than 18 monographs and textbooks; the rate of production of new and interesting results shows no sign of decreasing! Perhaps he is best known by the series of books he has written which present systematic treatments of the theory of function spaces from different points of view, thus revealing its symbiotic relationship with interpolation theory, harmonic analysis, partial differential equations, nonlinear operators, entropy, spectral theory, fractal analysis, wavelet theory, and theoretical numerical analysis. In particular, his books Interpolation Theory, Differential Operators, Function Spaces (finished in 1974 and published in 1978) as well as Theory of Function Spaces (based on earlier lecture notes and published in 1983) are much-quoted standard references and have been translated into Russian. At the textbook level, his Higher Analysis and Analysis and Mathematical Physics are masterpieces, relating mathematical theory to physical applications in an extraordinarily convincing way, and made even more vivid in a series of remarkable lecture courses at Jena. Hans Triebel has supervised nearly 40 Ph.D. students, many of whom have become internationally recognised mathematicians. He is on the editorial boards of various international journals, and in particular was an editor of Mathematische Nachrichten for many years. The reputation of the analysis department at the university of Jena owes much to his pioneering scientific work and the activities of his research group on function spaces. His outstanding scientific achievements were recognised by a National Award of the German Democratic Republic for Science and Technology in 1983 and the award of a Doctor of Science honoris causa by the University of Sussex in 1990. He was elected as (Corresponding) Member of the Academy of Science of the German Democratic Republic in 1978. Since 1993 he has been a Member of the Berlin-Brandenburg Academy of Science (formerly the Prussian Academy of Science). No less remarkable than his mathematical ability are his personal qualities, coupling total integrity, resolution and great warmth with an irreverent sense of humour; stories abound of his preparedness to lecture very early in the morning, sometimes to the surprise of the students! We are glad to have this opportunity to express our deep gratitude to him for sharing with so many of his colleagues his ideas and encyclopaedic knowledge. The present collection of papers is a tribute to his distinguished work and reflects recent developments in the theory of function spaces and related fields by outstanding experts. It is a pleasure to thank all the authors for their contributions.

On contact manifolds we describe a notion of (contact) finite-type for linear partial differential operators satisfying a natural condition on their leading terms. A large class of linear differential operators are of finite-type in this sense but are not well understood by currently available techniques. We resolve this in the following sense. For any such D we construct a partial connection r H on a (finite rank) vector bundle with the property that sections in the null space of D correspond bijectively, and via an explicit map, with sections parallel for the partial connection. It follows that the solution space of D is finite dimensional and bounded by the corank of the holonomy algebra of r H. The treatment is via a uniform procedure, even though in most cases no normal Cartan connection is available. The prolongations of a k th order linear differential operator between vector bundles arise by differentiating the given operator D : E ! F, and forming a new system comprising D along with auxiliary operators that capture some of this derived data. To exploit this effectively it is crucial to determine what part of this information should be retained, and then how best to manage it. With this understood, for many classes of operators the resulting prolonged operator can expose key properties of the original differential operator and its equation. Motivated by questions related to integrability and deformations of structure, a theory of overdetermined equations and prolonged systems was developed during the 1950s and 1960s by Goldschmidt, Spencer, and others (2, 17). Generally, results in these works are derived abstractly using jet bundle theory, and are severely restricted in the sense that they apply most readily to differential operators satisfying involutivity conditions. These features mean the theory can be difficult to apply. In the case that the given partial differential operator D : E ! F, has surjective symbol there is an effective algorithmic approach to this problem. The prolongations are constructed from the leading symbol �(D) : J k � 1 E ! F, where J k � 1 is the bundle of symmetric covariant tensors on M of rank k. At a point of M, denoting by K the kernel of �(D), the spaces K ` = ( J ` � 1 K) ( J k+` � 1 E), `� 0, capture spaces of new variables to be introduced, and the system closes up if K ` = 0 for

We consider the Fuchsian linear differential equation obtained (modulo a prime) for , the five-particle contribution to the susceptibility of the square lattice Ising model. We show that one can understand the factorization of the corresponding linear differential operator from calculations using just a single prime. A particular linear combination of and can be removed from and the resulting series is annihilated by a high order globally nilpotent linear ODE. The corresponding (minimal order) linear differential operator, of order 29, splits into factors of small orders. A fifth-order linear differential operator occurs as the left-most factor of the ‘depleted’ differential operator and it is shown to be equivalent to the symmetric fourth power of LE, the linear differential operator corresponding to the elliptic integral E. This result generalizes what we have found for the lower order terms and . We conjecture that a linear differential operator equivalent to a symmetric (n − 1) th power of LE occurs as a left-most factor in the minimal order linear differential operators for all 's.

The main objective of our paper is to focus on the study of sequences (finite or countable) of groups and hypergroups of linear differential operators of decreasing orders. By using a suitable ordering or preordering of groups linear differential operators we construct hypercompositional structures of linear differential operators. Moreover, we construct actions of groups of differential operators on rings of polynomials of one real variable including diagrams of actions–considered as special automata. Finally, we obtain sequences of hypergroups and automata. The examples, we choose to explain our theoretical results with, fall within the theory of artificial neurons and infinite cyclic groups.

This article is a sequel to the earlier articles, which describe the invertible ordinary differential operators and their generalizations. The generalizations are invertible mappings of filtered modules generated by one differentiation, and are called invertible D-operators. In particular, invertible ordinary linear differential operators, invertible linear difference operators with periodic coefficients, maps defined by unimodular matrices, and C-transformations of control systems are invertible D-operators. C-Transformations are those invertible transformations for which the variables of one system are expressed in terms of the variables of the other system and their derivatives.In the article we consider the invertible D-operators whose inverses are D-operators of the same type. In previous papers, a classification of invertible D-operators was obtained. Namely, a table of integers was associated to each invertible D-operator. These tables were described in a clear elementary-geometric language. Thus, to each invertible D-operator one assigns an elementary-geometric model, which is called a d-scheme of squares. The class of invertible D-operators having the same d-scheme was also described. In this paper, the invertible D-operators whose d-schemes consist of a single square are called unicellular. It is proved that any unicellular operator in some bases is given by an upper triangular matrix that differs from the identity matrix only by the first row. The main result is representation of the arbitrary invertible D-operator as a composition of unicellular operators. The minimum number of unicellular operators in such a composition is equal to the number of squares of the d-scheme of the original D-operator. As in previous papers, the used method is based on the description of d-schemes in the language of spectral sequences of algebraic complexes.The results obtained can be useful in the transformation and classification of control systems, in particular to describe flat systems.

Complexity of factoring and calculating the GCD of linear ordinary differential operators

1.1. In the study of the regularity of generalized solutions of various problems for partial differential equations the notion of the hypoellipticity of linear partial differential operators has a central role. Originally the hypoelliptic operators were introduced by L. Hormander in 1955, and for the results of the development one can today consult his monograph [13], especially chapters LL and 1.3. A linear differential operator P(.,D) defined in some open set O in R through

We study tight bounds and fast algorithms for LCLMs of several linear differential operators with polynomial coefficients. We analyze the arithmetic complexity of existing algorithms for LCLMs, as well as the size of their outputs. We propose a new algorithm that recasts the LCLM computation in a linear algebra problem on a polynomial matrix. This algorithm yields sharp bounds on the coefficient degrees of the LCLM, improving by one order of magnitude the best bounds obtained using previous algorithms. The complexity of the new algorithm is almost optimal, in the sense that it nearly matches the arithmetic size of the output.

An $L_p $ theory $(1 < p < \infty )$ of existence and regularity of solutions of the partial differential equation $(1 - \gamma \mathcal{M}(t)){{\partial u} / {\partial t}}) - \mathcal{L}(t)u = f$ satisfying general boundary conditions is given. For each t, $\mathcal{M}(t)$ is a linear elliptic partial differential operator in the space variables, $\mathcal{L}(t)$ is a linear differential operator whose order does not exceed that of $\mathcal{M}(t)$ and $\gamma $ is a nonzero complex constant.

Partial Differential Equations and Bivariate Orthogonal Polynomials

We study tight bounds and fast algorithms for LCLMs of several linear differential operators with polynomial coef- ficients. We analyse the arithmetic complexity of existing algorithms for LCLMs, as well as the size of their outputs. We propose a new algorithm that recasts the LCLM compu- tation in a linear algebra problem on a polynomial matrix. This algorithm yields sharp bounds on the coefficient degrees of the LCLM, improving by one order of magnitude the best bounds obtained using previous algorithms. The complexity of the new algorithm is almost optimal, in the sense that it nearly matches the arithmetic size of the output.

Given a collection of n curves that are independent realizations of a functional variable, we are interested in finding patterns in the curve data by exploring low-dimensional approximations to the curves. It is assumed that the data curves are noisy samples from the vector space span <texlscub>f 1, …, f m </texlscub>, where f 1, …, f m are unknown functions on the real interval (0, T) with square-integrable derivatives of all orders m or less, and m<n. Ramsay [Principal differential analysis: Data reduction by differential operators, J. R. Statist. Soc. Ser. B 58 (1996), pp. 495–508] first proposed the method of regularized principal differential analysis (PDA) as an alternative to principal component analysis for finding low-dimensional approximations to curves. PDA is based on the following theorem: there exists an annihilating linear differential operator (LDO) ℒ of order m such that ℒf i =0, i=1, …, m [E.A. Coddington and N. Levinson, Theory of Ordinary Differential Equations, McGraw-Hill, New York, 1955, Theorem 6.2]. PDA specifies m, then uses the data to estimate an annihilating LDO. Smooth estimates of the coefficients of the LDO are obtained by minimizing a penalized sum of the squared norm of the residuals. In this context, the residual is that part of the data curve that is not annihilated by the LDO. PDA obtains the smooth low dimensional approximation to the data curves by projecting onto the null space of the estimated annihilating LDO; PDA is thus useful for obtaining low-dimensional approximations to the data curves whether or not the interpretation of the annihilating LDO is intuitive or obvious from the context of the data. This paper extends PDA to allow for the coefficients in the LDO to smoothly depend upon a single continuous covariate. The estimating equations for the coefficients allowing for a continuous covariate are derived; the penalty of Eilers and Marx [Flexible smoothing with B-splines and penalties, Statist. Sci. 11(2) (1996), pp. 89–121] is used to impose smoothness. The results of a small computer simulation study investigating the bias and variance properties of the estimator are reported.

Darboux transformations from the Appell-Lauricella operator

In this paper a generalized version of the classical Hardy-Littlewood-Polya inequality is given. Furthermore, the Stechkin's problem for a linear differential operator is solved in\(L_2 (\mathbb{R})\), and the optimal recovery problem for such differential operator is considered.