Narrow Results

We initiate a detailed study of two-parameter Besov spaces on the unit ball of ${ℝ}^n$ consisting of harmonic functions whose sufficiently high-order radial derivatives lie in harmonic Bergman spaces. We compute the reproducing kernels of those Besov spaces that are Hilbert spaces. The kernels are weighted infinite sums of zonal harmonics and natural radial fractional derivatives of the Poisson kernel. Estimates of the growth of kernels lead to characterization of integral transformations on Lebesgue classes. The transformations allow us to conclude that the order of the radial derivative is not a characteristic of a Besov space as long as it is above a certain threshold. Using kernels, we define generalized Bergman projections and characterize those that are bounded from Lebesgue classes onto Besov spaces. The projections provide integral representations for the functions in these spaces and also lead to characterizations of the functions in the spaces using partial derivatives. Several other applications follow from the integral representations such as atomic decomposition, growth at the boundary and of Fourier coefficients, inclusions among them, duality and interpolation relations, and a solution to the Gleason problem.

This paper discusses the uniformly strong convergence of multivariate density estimation with moderately ill-posed noise over a bounded set. We provide a convergence rate over Besov spaces by using a compactly supported wavelet. When the model degenerates to one-dimensional noise-free case, the convergence rate coincides with that of Giné and Nickl’s (Ann. Probab., 2009 or Bernoulli, 2010). Our result can also be considered as an extension of Masry’s theorem (Stoch. Process. Appl., 1997) to some extent.

This paper considers wavelet estimation for density derivatives based on negatively associated and size-biased data. We provide upper bounds of nonlinear wavelet estimator on $L^p(1\le p<\infty )$ risk. It turns out that the convergence rate of the nonlinear estimator is better than that of the linear one.

We establish a well-posedness theory and a blow-up criterion for the Chaplygin gas equations in ${ℝ}^N$ for any dimension $N\ge 1$. First, given $\omega =\frac{1}{\rho }$, ${\mathscr{H}}\hookrightarrow {{𝒞}}^{0,1}$, we prove the well-posedness property for solutions $v=(\omega ,u)$ in the space ${\mathscr{H}}$ for the Cauchy problem associated with the Chaplygin gas equations, provided the initial density ${\rho }_0$ is bounded below. We also prove that the solution of the Chaplygin gas equations depends continuously upon its initial data $v_0$ in ${𝒞}([0,T[;B_{2,r}^{\sigma })$ if $B_{2,r}^{\sigma }\hookrightarrow {{𝒞}}^{0,1}$, and we state a blow-up criterion for the solutions in the classical BMO space. Finally, using Osgood’s modulus of continuity, we establish a refined blow-up criterion of the solutions.

Mittal, Rhoades (1999–2001), Mittal et al. (2005, 2006, 2011) have initiated a study of error estimates through trigonometric Fourier approximation (tfa) for the situation in which the summability matrix $T\equiv (a_{n,k})$ does not have monotone rows. Recently Mohanty et al. (2011) have obtained a theorem on the degree of approximation of functions in Besov space $B_q^{\alpha }(L_p)$ by choosing $T$ to be a Nörlund ($N_p$)-matrix with non-increasing weights $\{p_n\}$. In this paper, we continue the work of Mittal et al. and extend the result of Mohanty et al. (2011) to the general matrix $T$.

We describe a recent progress in finding optimal bounds of Hausdorff dimension of singular sets of functions in Lebesgue spaces, Sobolev spaces, Bessel potential spaces, Besov spaces, Lizorkin-Triebel spaces, and Hardy spaces. We are also interested in the question of existence of maximally singular functions in a given space of functions, that is, functions such that the Hausdorff dimension of their singular sets is maximal possible.

We define and study generalized homogeneous Besov spaces connected with the Riemann–Liouville operator. We establish some results of density of subspaces, completeness and continuous embedding. Also, a discrete equivalent norm is examined.

In this paper, we consider the embeddings of weighted Besov spaces into Besov-type spaces with w being a (local) Muckenhoupt weight, and give sufficient and necessary conditions on the continuity and the compactness of these embeddings. As special cases, we characterize the continuity and the compactness of embeddings in case of some polynomial or exponential weights. The proofs of these conclusions strongly depend on the geometric properties of dyadic cubes.

We construct multidimensional interpolating tensor product multiresolution analyses (MRA's) of the function spaces C₀(ℝⁿ, K), K = ℝ or K = ℂ, consisting of real or complex valued functions on ℝⁿ vanishing at infinity and the function spaces C_u(ℝⁿ, K) consisting of bounded and uniformly continuous functions on ℝⁿ. We also construct an interpolating dual MRA for both of these spaces. The theory of the tensor products of Banach spaces is used. We generalize the Besov space norm equivalence from the one-dimensional case to our n-dimensional construction.

In the present paper, we estimate the rate of convergence of Fourier series of functions by Deferred Nörlund mean in Besov space which is a generalization of $H(\alpha ,p)$ space.