Mathematics

We give an extension of Pizzetti's formula associated with the Dunkl operators. It gives an explicit formula for the Dunkl inner product of an arbitrary function and a homogeneous Dunkl harmonic polynomial on the unit sphere.

In this note, we present an extension of the celebrated Abel-Liouville identity in terms of noncommutative complete Bell polynomials for generalized Wronskians. We also characterize the range equivalence of n -dimensional vector-valued functions in the subclass of n -times differentiable functions with a nonvanishing Wronskian.

A set S of points in R n is called a rationally parameterisable hypersurface if S={σ(t):t∈D} , where σ: R n−1 → R n is a vector function with domain D and rational functions as components. A generalized n -dimensional polytope in R n is a union of a finite number of convex n -dimensional polytopes in R n . The Fourier transform of such a generalized polytope P in R n is defined by F P (s)= ∫ P e −is⋅x dx . We prove that F P 1 (σ(t))= F P 2 (σ(t)) ∀t∈O implies P 1 = P 2 if O is an open subset of D satisfying some well-defined conditions. Moreover we show that this theorem can be applied to quadric hypersurfaces that do not contain a line, but at least two points, i.e., in particular to spheres.

We show that if compact set E⊂ R d has Hausdorff dimension larger than d 2 + 1 4 , where d≥4 is an even integer, then the distance set of E has positive Lebesgue measure. This improves the previously best known result towards Falconer's distance set conjecture in even dimensions.

A method for approximate solution of initial value and spectral problems for one dimensional Dirac equation based on an analytic approximation of the transmutation operator is presented. In fact the problem of numerical approximation of solutions is reduced to approximation of the potential matrix by a finite linear combination of matrix valued functions related to generalized formal powers introduced in arXiv:1904.03361. Convergence rate estimates in terms of smoothness of the potential are proved. The method allows one to compute both lower and higher eigendata with an extreme accuracy.

By extending the approximate weak factorization method to the bilinear setting we identify testing conditions on the function b that characterize the L p ? L q ??L r boundedness of the commutator [b,T ] 1 (f,g)=bT(f,g)?�T(bf,g), for many exponents in the range 1<p,q<??and r> 1 2 , where T is a bilinear Calderón-Zygmund operator.

Multivariate quasi-projection operators Q j (f,φ, φ ˜ ) , associated with a function φ and a distribution/function φ ˜ , are considered. The function φ is supposed to satisfy the Strang-Fix conditions and a compatibility condition with φ ˜ . Using technique based on the Fourier multipliers, we studied approximation properties of such operators for functions f from anisotropic Besov spaces and L p spaces with 1≤p≤∞ . In particular, upper and lower estimates of the L p -error of approximation in terms of moduli of smoothness and best approximations are obtained.

We determine the distance (up to a multiplicative constant) in the Zygmund class Λ ∗ ( R n ) to the subspace J(bmo)( R n ). The latter space is the image under the Bessel potential J:=(1−Δ ) −1/2 of the space bmo( R n ), which is a non-homogeneous version of the classical BMO. Locally, J(bmo)( R n ) consists of functions that together with their first derivatives are in bmo( R n ). More generally, we consider the same question when the Zygmund class is replaced by the Hölder space Λ s ( R n ), with 0<s≤1 and the corresponding subspace is J s (bmo)( R n ), the image under (1−Δ ) −s/2 of bmo( R n ). One should note here that Λ 1 ( R n )= Λ ∗ ( R n ). Such results were known earlier only for n=s=1 with a proof that does not extend to the general case. Our results are expressed in terms of second differences. As a byproduct of our wavelet based proof, we also obtain the distance from f∈ Λ s ( R n ) to J s (bmo)( R n ) in terms of the wavelet coefficients of f. We additionally establish a third way to express this distance in terms of the size of the hyperbolic gradient of the harmonic extension of f on the upper half-space R n+1 + .

We consider a wide class of linear boundary-value problems for systems of r -th order ordinary differential equations whose solutions range over the normed complex space ( C (n) ) m of n≥r times continuously differentiable functions y:[a,b]→ C m . The boundary conditions for these problems are of the most general form By=q , where B is an arbitrary continuous linear operator from ( C (n) ) m to C rm . We prove that the solutions to the considered problems can be approximated in ( C (n) ) m by solutions to some multipoint boundary-value problems. The latter problems do not depend on the right-hand sides of the considered problem and are built explicitly.

We consider an over-determined Falconer type problem on (k+1) -point configurations in the plane using the group action framework introduced in \cite{GroupAction}. We define the area type of a (k+1) -point configuration in the plane to be the vector in $\R^{\binom{k+1}{2}}$ with entries given by the areas of parallelograms spanned by each pair of points in the configuration. We show that the space of all area types is 2k−1 dimensional, and prove that a compact set $E\subset\R^d$ of sufficiently large Hausdorff dimension determines a positve measure set of area types.