Tensor products of Sobolev–Besov spaces and applications to approximation from the hyperbolic cross

Besov as well as Sobolev spaces of dominating mixed smoothness are shown to be tensor products of Besov and Sobolev spaces defined on R. Using this we derive several useful characterizations from the one-dimensional case to the d-dimensional situation. Finally, consequences for hyperbolic cross approximations, in particular for tensor product splines, are discussed.     

A hyperbolic modulus of smoothness for multivariate functions of bounded variation

In this paper we will introduce a hyperbolic kind of modulus on the space of multivariate functions of bounded variation and discuss the fundamental properties of the smoothness spaces induced by it. The results obtained, here, can be used to analyze the approximation properties of so-called hyperbolic Lebesgue-Stieltjes convolution operators.     

Tractability through increasing smoothness

We prove that some multivariate linear tensor product problems are tractable in the worst case setting if they are defined as tensor products of univariate problems with logarithmically increasing smoothness. This is demonstrated for the approximation problem defined over Korobov spaces and for the approximation problem of certain diagonal operators. For these two problems we show necessary and sufficient conditions on the smoothness parameters of the univariate problems to obtain strong polynomial tractability. We prove that polynomial tractability is equivalent to strong polynomial tractability, and that weak tractability always holds for these problems. Under a mild assumption, the Korobov space consists of periodic functions. Periodicity is crucial since the approximation problem defined over Sobolev spaces of non-periodic functions with a special choice of the norm is not polynomially tractable for all smoothness parameters no matter how fast they go to infinity. Furthermore, depending on the choice of the norm we can even lose weak tractability.     

N–Term Approximation in Anisotropic Function Spaces

In L₂(0, 1)²) infinitely many different biorthogonal wavelet bases may be introduced by taking tensor products of one–dimensional biorthogonal wavelet bases on the interval (0, 1). Most well–known are the standard tensor product bases and the hyperbolic bases. In [23, 24] further biorthogonal wavelet bases are introduced, which provide wavelet characterizations for functions in anisotropic Besov spaces. Here we address the following question: Which of those biorthogonal tensor product wavelet bases is the most appropriate one for approximating nonlinearly functions from anisotropic Besov spaces? It turns out, that the hyperbolic bases lead to nonlinear algorithms which converge as fast as the corresponding schemes with respect to specific anisotropy adapted bases.     

On smoothness of a fundamental solution to a second order hyperbolic equation

We consider the problem of constructing a fundamental solution to a second order hyperbolic linear equation with variable coefficients depending on the space variable x ∈ ? ⁿ . Under the assumption of high but finite smoothness of the coefficients, we write out the structure of a fundamental solution, establish smoothness of the coefficients of the expansion of its singular part, and characterize smoothness of the regular part.     

Function theory for Laplace and Dirac-Hodge Operators in hyperbolic space

We develop basic properties of solutions to the Dirac-Hodge and Laplace equations in upper half space endowed with the hyperbolic metric. Solutions to the Dirac-Hodge equation are called hypermonogenic functions, while solutions to this version of Laplace's equation are called hyperbolic harmonic functions. We introduce a Borel-Pompeiu formula forC ¹ functions and a Green's formula for hyperbolic harmonic functions. Using a Cauchy integral formula, we introduce Hardy spaces of solutions to the Dirac-Hodge equation. We also provide new arguments describing the conformal covariance of hypermonogenic functions and invariance of hyperbolic harmonic functions and introduce intertwining operators for the Dirac-Hodge operator and hyperbolic Laplacian. Research supported by the National Science Foundation of China (Mathematics Tianyuan Foundation, No A324610) and Hebei Province (105129) Research supported by Academy of Finland     

Approximation by Fourier means and generalized moduli of smoothness

The quality of approximation by Fourier means generated by an arbitrary generator with compact support in the spaces L_p, 1 ≤ p ≤ +∞, of 2π-periodic pth integrable functions and in the space C of continuous 2π-periodic functions in terms of the generalized modulus of smoothness constructed froma 2π-periodic generator is studied. Natural sufficient conditions on the generator of the approximation method and values of smoothness ensuring the equivalence of the corresponding approximation error and modulus are obtained. As applications, Fourier means generated by classical kernels as well as the classical moduli of smoothness are considered.     

The differential form spectrum of quaternionic hyperbolic spaces

By using harmonic analysis and representation theory, we determine explicitly the L² spectrum of the Hodge-de Rham Laplacian acting on quaternionic hyperbolic spaces and we show that the unique possible discrete eigenvalue and the lowest continuous eigenvalue can both be realized by some subspace of hypereffective differential forms. Similar results are obtained also for the Bochner Laplacian.     

Smooth center manifolds for nonuniformly partially hyperbolic trajectories

We establish the existence of unique smooth center manifolds for ordinary differential equations v^′=A(t)v+f(t,v) in Banach spaces, assuming that v^′=A(t)v admits a nonuniform exponential trichotomy. This allows us to show the existence of unique smooth center manifolds for the nonuniformly partially hyperbolic trajectories. In addition, we prove that the center manifolds are as regular as the vector field. Our proof of the Ck smoothness of the manifolds uses a single fixed point problem in an appropriate complete metric space. To the best of our knowledge we establish in this paper the first smooth center manifold theorem in the nonuniform setting.     

Hyperbolic Wavelet Approximation

We study the multivariate approximation by certain partial sums (hyperbolic wavelet sums) of wavelet bases formed by tensor products of univariate wavelets. We characterize spaces of functions which have a prescribed approximation error by hyperbolic wavelet sums in terms of a K -functional and interpolation spaces. The results parallel those for hyperbolic trigonometric cross approximation of periodic functions [DPT]. October 16, 1995. Date revised: August 28, 1996.     

Mixed initial boundary value problem for hyperbolic geometric flow on Riemann surfaces

The hyperbolic geometric flow equations is introduced recently by Kong and Liu motivated by Einstein equation and Hamilton Ricci flow. In this paper, we consider the mixed initial boundary value problem for hyperbolic geometric flow, and prove the global existence of classical solutions. The results show that, for any given initial metric on R² in certain class of metric, one can always choose suitable initial velocity symmetric tensor such that the solutions exist, and the scalar curvature corresponding to the solution metric gij keeps bounded. If the initial velocity tensor does not satisfy the certain conditions, the solutions will blow up at a finite time. Some special explicit solutions to the reduced equation are given.     

Ёквивалентные нормировки пространствфункций, ассоциированных с обобшенным сдвигом ГегенбауЭра

Taylor-Delsart formula is elaborated in the paper for functions of the generalized Gegenbauer shift. This formula is utilized to construct a version of the Gegenbauer shift modulus of smoothness of order k which for k = 1 reduces to the modulus of smoothness of the first order. By means of this modulus and Peetre’s K-functional, an interpolation theorem is obtained. Equivalent normalizations are obtained for functional spaces associated with the generalized Gegenbauer shift.     

Mixed modulus of smoothness with Muckenhoupt weights and approximation by angle

Mixed modulus of smoothness in weighted Lebesgue spaces with Muckenhoupt weights are investigated. Using mixed modulus of smoothness we obtain Potapov type direct and inverse estimates of angular trigonometric approximation of functions in these spaces. Also we obtain equivalences between mixed modulus of smoothness and K-functional and realization functional. Fractional order modulus of smoothness is considered as well.     

Some problems of approximation theory in the spacesL p on the line with power weight

In the spaces L _p on the line with power weight, we study approximation of functions by entire functions of exponential type. Using the Dunkl difference-differential operator and the Dunkl transform, we define the generalized shift operator, the modulus of smoothness, and the K-functional. We prove a direct and an inverse theorem of Jackson-Stechkin type and of Bernstein type. We establish the equivalence between the modulus of smoothness and the K-functional.     

On Lipschitz-type maximal functions and their smoothness spaces

In a recent monograph (cf. No. 293 of the Memoirs of the Amer. Math. Soc. 47 (1984)) DeVore and Sharpley study maximal functions of integral type and their related smoothness spaces. One of their central results gives an embedding theorem for the smoothness spaces in terms of Besov spaces. In this paper we consider the related problem when the Besov spaces are substituted by the so-called A-spaces introduced by Popov (take the τ-modulus instead of the ω-modulus). We will define Lipschitz-type maximal functions whose smoothness spaces satisfy a corresponding embedding theorem in terms of A-spaces. By well-known results new insights can only be expected for functions satisfying low order smoothness conditions and, therefore, only function spaces generated by first order differences are considered.     

Weighted Tensor Product Algorithms for Linear Multivariate Problems

We study the ε-approximation of linear multivariate problems defined over weighted tensor product Hilbert spaces of functions f of d variables. A class of weighted tensor product (WTP) algorithms is defined which depends on a number of parameters. Two classes of permissible information are studied. Λ^all consists of all linear functionals while Λ^std consists of evaluations of f or its derivatives. We show that these multivariate problems are sometimes tractable even with a worst-case assurance. We study problem tractability by investigating when a WTP algorithm is a polynomial-time algorithm, that is, when the minimal number of information evaluations is a polynomial in 1/ε and d. For Λ^all we construct an optimal WTP algorithm and provide a necessary and sufficient condition for tractability in terms of the sequence of weights and the sequence of singular values for d=1. ForΛ^std we obtain a weaker result by constructing a WTP algorithm which is optimal only for some weight sequences.     

Energy estimate and fundamental solution for degenerate hyperbolic Cauchy problems

The aim of this paper is to give an uniform approach to different kinds of degenerate hyperbolic Cauchy problems. We prove that a weakly hyperbolic equation, satisfying an intermediate condition between effective hyperbolicity and the C^∞ Levi condition, and a strictly hyperbolic equation with non-regular coefficients with respect to the time variable can be reduced to first-order systems of the same type. For such a kind of systems, we prove an energy estimate in Sobolev spaces (with a loss of derivatives) which gives the well-posedness of the Cauchy problem in C^∞. In the strictly hyperbolic case, we also construct the fundamental solution and we describe the propagation of the space singularities of the solution which is influenced by the non-regularity of the coefficients with respect to the time variable.     

Monotonicity preserving representations of non-polynomial surfaces

We prove that, in contrast to the case for rational surfaces, some tensor product representations through spaces containing algebraic, trigonometric and hyperbolic polynomials are monotonicity preserving. The surface representations provided in this paper are the only known monotonicity preserving surfaces in addition to the tensor product Bézier and tensor product B-spline surfaces.