Invariance of the generalized oscillator under a linear transformation of the related system of orthogonal polynomials

We consider the families of polynomials P = { P _n (x)} _n=0 ^∞ and Q = { Q _n (x)} _n=0 ^∞ orthogonal on the real line with respect to the respective probability measures μ and ν. We assume that { Q _n (x)} _n=0 ^∞ and {P _n (x)} _n=0 ^∞ are connected by linear relations. In the case k = 2, we describe all pairs (P,Q) for which the algebras A _P and A _Q of generalized oscillators generated by { Qn(x)} _n=0 ^∞ and { Pn(x)} _n=0 ^∞ coincide. We construct generalized oscillators corresponding to pairs (P,Q) for arbitrary k ≥ 1.     

Hermite—Padé Approximants of the Mittag-Leffler Functions

The convergence rate of type II Hermite–Padé approximants for a system of degenerate hypergeometric functions {₁F₁(1, γ; λ_jz)} _j=1 ^k is found in the case when the numbers {λ_j} _j=1 ^k are the roots of the equation λ^k = 1 or real numbers and \(\gamma\in\mathbb{C}\;\backslash\left\{0,-1,-2,...\right\}\). More general statements are obtained for approximants of this type (including nondiagonal ones) in the case of k = 2. The theorems proved in the paper complement and generalize the results obtained earlier by other authors.     

On some properties of Hermite–Padé approximants to an exponential system

Extremal properties and localization of zeros of general (including nondiagonal) type I Hermite–Padé polynomials are studied for the exponential system {e ^λjz } _j=0 ^k with arbitrary different complex numbers λ₀, λ₁,..., λ_k. The theorems proved in the paper complement and generalize the results obtained earlier by other authors.     

On Representation of a Solution to the Cauchy Problem by a Fourier Series in Sobolev-Orthogonal Polynomials Generated by Laguerre Polynomials

We consider the problem of representing a solution to the Cauchy problem for an ordinary differential equation as a Fourier series in polynomials l _r,k ^α (x) (k = 0, 1,...) that are Sobolev-orthonormal with respect to the inner product

$$\left\langle {f,g} \right\rangle = \sum\limits_{v = 0}^{r - 1} {{f^{(v)}}(0){g^{(v)}}} (0) + \int\limits_0^\infty {{f^{(r)}}(t)} {g^{(r)}}(t){t^\alpha }{e^{ - t}}dt$$

, and generated by the classical orthogonal Laguerre polynomials L _k ^α (x) (k = 0, 1,...). The polynomials l _r,k ^α (x) are represented as expressions containing the Laguerre polynomials L _n ^α?r (x). An explicit form of the polynomials l _r,k+r ^α (x) is established as an expansion in the powers x ^r+l , l = 0,..., k. These results can be used to study the asymptotic properties of the polynomials l _r,k ^α (x) as k→∞and the approximation properties of the partial sums of Fourier series in these polynomials.

     

On the approximation of continuous functions of two variables

It is shown that if P _m ^α,β (x) (α, β > ?1, m = 0, 1, 2, …) are the classical Jaboci polynomials, then the system of polynomials of two variables {Ψ _mn ^α,β (x, y)} _m,n=0 ^r = {P _m ^α,β (x)P _n ^α,β (y)} _{m, n=0} ^r (r = m + n ≤ N ? 1) is an orthogonal system on the set Ω _N×N = ?ub;(x _i , y _i ) _i,j=0 ^N , where x _i and y _i are the zeros of the Jacobi polynomial P _n ^α,β (x). Given an arbitrary continuous function f(x, y) on the square [?1, 1]², we construct the discrete partial Fourier-Jacobi sums of the rectangular type S _{m, n, N} ^α,β (f; x, y) by the orthogonal system introduced above. We prove that the order of the Lebesgue constants ∥S _{m, n, N} ^α,β ∥ of the discrete sums S _{m, n, N} ^α,β (f; x, y) for ?1/2 < α, β < 1/2, m + n ≤ N ? 1 is O((mn) ^{q + 1/2}), where q = max?ub;α,β?ub;. As a consequence of this result, several approximate properties of the discrete sums S _{m, n, N} ^α,β (f; x, y) are considered.     

Asymptotic behavior of the maximal zero of a polynomial orthogonal on a segment with a nonclassical weight

Let {p _n (t)} _n=0 ^t8 be a system of algebraic polynomials orthonormal on the segment [?1, 1] with a weight p(t); let {x _n,ν ^(p) } _ν=1 ⁿ be zeros of a polynomial p _n (t) (x _x,ν ^(p) = cosθ _n,ν ^(p) ; 0 < θ _n,1 ^(p) < θ _n,2 ^(p) < ... < θ _n,n ^(p) < π). It is known that, for a wide class of weights p(t) containing the Jacobi weight, the quantities θ _n,1 ^(p) and 1 ? x _n,1 ^(p) coincide in order with n ^?1 and n ^?2, respectively. In the present paper, we prove that, if the weight p(t) has the form p(t) = 4(1 ? t ²)^?1{ln²[(1 + t)/(1 ? t)] + π ²}^?1, then the following asymptotic formulas are valid as n → ∞:

$$\theta _{n,1}^{(p)} = \frac{{\sqrt 2 }}{{n\sqrt {\ln (n + 1)} }}\left[ {1 + {\rm O}\left( {\frac{1}{{\ln (n + 1)}}} \right)} \right],x_{n,1}^{(p)} = 1 - \left( {\frac{1}{{n^2 \ln (n + 1)}}} \right) + O\left( {\frac{1}{{n^2 \ln ^2 (n + 1)}}} \right).$$

     

Series by Haar system with monotone coefficients

Let χ = {χ _n } _n=0 ^∞ be the Haar system normalized in L ²(0, 1) and M = {M _s } _s=1 ^∞ be an arbitrary, increasing sequence of nonnegative integers. For any subsystem of χ of the form {φ _k } = χS = {χ _n } _n∈S , where S = S(M) = {n _k } _k=1 ^∞ = {n ∈ V[p]: p ∈ M}, V[0] = {1, 2} and V[p] = {2 ^p + 1, 2 ^p + 2, …, 2 ^p+1} for p = 1, 2, … a series of the form Σ _i=1 ^∞ a _i φ _i with a _i ↘ 0 is constructed, that is universal with respect to partial series in all classes L ^r (0, 1), r ∈ (0, 1), in the sense of a.e. convergence and in the metric ofL ^r (0, 1). The constructed series is universal in the class of all measurable, finite functions on [0, 1] in the sense of a.e. convergence. It is proved that there exists a series by Haar system with decreasing coefficients, which has the following property: for any ? > 0 there exists a measurable function µ(x), x ∈ [0, 1], such that 0 ≤ µ(x) ≤ 1 and |{x ∈ [0, 1], µ(x) ≠ = 1}| < ?, and the series is universal in the weighted space L _µ[0, 1] with respect to subseries, in the sense of convergence in the norm of L _µ[0, 1].     

The program iteration method in a game problem of guidance

Let a sequence of d-dimensional vectors n _k = (n _k ¹ , n _k ² ,..., n _k ^d ) with positive integer coordinates satisfy the condition n _k ^j = α _j m _k +O(1), k ∈ ?, 1 ≤ j ≤ d, where α ₁ > 0,..., α _d > 0 and {m _k } _k=1 ^∞ is an increasing sequence of positive integers. Under some conditions on a function φ: [0,+∞) → [0,+∞), it is proved that, if the sequence of Fourier sums \({S_{{m_k}}}\) (g, x) converges almost everywhere for any function g ∈ φ(L)([0, 2π)), then, for any d ∈ ? and f ∈ φ(L)(ln⁺ L) ^d?1([0, 2π) ^d ), the sequence \({S_{{n_k}}}\) (f, x) of rectangular partial sums of the multiple trigonometric Fourier series of the function f and the corresponding sequences of partial sums of all conjugate series converge almost everywhere.     

Almost everywhere divergence of lacunary subsequences of partial sums of fourier series

If an increasing sequence {n _m } of positive integers and a modulus of continuity ω satisfy the condition Σ _m=1 ^∞ ω(1/n _m )/m < ∞, then it is known that the subsequence of partial sums \(S_{n_m } \left( {f,x} \right)\) converges almost everywhere to f(x) for any function f ∈ H ₁ ^ω . We show that this sufficient convergence condition is close to a necessary condition for a lacunary sequence {n _m }.     

Regularized traces of singular differential operators with canonical boundary conditions

A self-adjoint differential operator \(\mathbb{L}\) of order 2m is considered in L ₂[0,∞) with the classic boundary conditions \(y^{(k_1 )} (0) = y^{(k_2 )} (0) = y^{(k_3 )} (0) = \ldots = y^{(k_m )} (0) = 0\), where 0 ≤ k ₁ < k ₂ < ... < k _m ≤ 2m ? 1 and {k _s } _s=1 ^m ∪ {2m ? 1 ? k _s } _s=1 ^m = {0, 1, 2, ..., 2m ? 1}. The operator \(\mathbb{L}\) is perturbed by the operator of multiplication by a real measurable bounded function q(x) with a compact support: \(\mathbb{P}\) f(x) = q(x)f(x), f ∈ L ₂[0,∞). The regularized trace of the operator \(\mathbb{L} + \mathbb{P}\) is calculated.     

Exact distribution of MLE of covariance matrix in a GMANOVA-MANOVA model

For a GMANOVA-MANOVA model with normal error: Y = XB1Z1 T B2Z2 T E, E- Nq×n(0, In (?) ∑), the present paper is devoted to the study of distribution of MLE, ∑, of covariance matrix ∑. The main results obtained are stated as follows: (1) When rk(Z) -rk(Z2) ≥ q-rk(X), the exact distribution of ∑ is derived, where z = (Z1,Z2), rk(A) denotes the rank of matrix A. (2) The exact distribution of |∑| is gained. (3) It is proved that ntr{[S-1 - ∑-1XM(MTXT∑-1XM)-1MTXT∑-1]∑}has X2(q_rk(x))(n-rk(z2)) distribution, where M is the matrix whose columns are the standardized orthogonal eigenvectors corresponding to the nonzero eigenvalues of XT∑-1X.     

New schemes with fractal error compensation for PDE eigenvalue computations

With an error compensation term in the fractal Rayleigh quotient of PDE eigen-problems,we propose a new scheme by perturbing the mass matrix Mhto Mh=Mh+Ch2mKh,where Khis the corresponding stif matrix of a 2m 1 degree conforming finite element with mesh size h for a 2m-order self-adjoint PDE,and the constant C exists in the priority error estimationλh jλj~Ch2mλ2j.In particular,for Laplace eigenproblems over regular domains in uniform mesh,e.g.,cube,equilateral triangle and regular hexagon,etc.,we find the constant C=I h 1Mh2 hKh and show that in this case the computation accuracy can raise two orders,i.e.,fromλh jλj=O(h2)to O(h4).Some numerical tests in 2-D and 3-D are given to verify the above arguments.     

Generalizations of Schöbi’s Tetrahedral Dissection

Let v ₁,…,v _n be unit vectors in ? ⁿ such that v _i ?v _j =?w for i≠j, where \(-1. The points ∑ _i=1 ⁿ λ _i v _i (1≥λ ₁≥???≥λ _n ≥0) form a “Hill-simplex of the first type,” denoted by \(\mathcal {Q}_{n}(w)\). It was shown by Hadwiger in 1951 that \(\mathcal {Q}_{n}(w)\) is equidissectable with a cube. In 1985, Schöbi gave a three-piece dissection of \(\mathcal {Q}_{3}(w)\) into a triangular prism \(c\mathcal {Q}_{2}(\frac{1}{2})\times I\), where I denotes an interval and \(c=\sqrt{2(w+1)/3}\). In this paper, we generalize Schöbi’s dissection to an n-piece dissection of \(\mathcal {Q}_{n}(w)\) into a prism \(c\mathcal {Q}_{n-1}(\frac{1}{n-1})\times I\), where \(c=\sqrt{(n-1)(w+1)/n}\). Iterating this process leads to a dissection of \(\mathcal {Q}_{n}(w)\) into an n-dimensional rectangular parallelepiped (or “brick”) using at most n! pieces. The complexity of computing the map from \(\mathcal {Q}_{n}(w)\) to the brick is O(n ²). A second generalization of Schöbi’s dissection is given which applies specifically in ?⁴. The results have applications to source coding and to constant-weight binary codes.     

A numerical algorithm for solving a nonstationary problem of optimal control

Let {φ _n ^(α,β) (z)} _n=0 ^∞ be a system of Jacobi polynomials orthonormal on the circle |z| = 1 with respect to the weight (1 ? cos τ)^α+1/2(1 + cos τ)^β+1/2 (α, β > ?1), and let \(\psi _n^{\left( {\alpha ,\beta } \right)*} \left( z \right): = z^n \overline {\psi _n^{\left( {\alpha ,\beta } \right)} \left( {{1 \mathord{\left/ {\vphantom {1 {\bar z}}} \right. \kern-\nulldelimiterspace} {\bar z}}} \right)}\)). We establish relations between the polynomial φ _n ^(α,?1/2) (z) and the nth (C, α ? 1/2)-mean of the Maclaurin series for the function (1 ? z)^?α?3/2 and also between the polynomial φ _n ^(α,?1/2)* (z) and the nth (C, α + 1/2)-mean of the Maclaurin series for the function (1 ? z)^?α?1/2. We use these relations to derive an asymptotic formula for φ _n ^(α,?1/2) (z); the formula is uniform inside the disk |z| < 1. It follows that φ _n ^(α,?1/2) (z) ≠ 0 in the disk |z| ≤ ρ for fixed φ ∈ (0, 1) and α > ?1 if n is sufficiently large.     

Symmetric quasi-norms of sums of independent random variables in symmetric function spaces with the Kruglov property

Let X be a symmetric Banach function space on [0, 1] and let E be a symmetric (quasi)-Banach sequence space. Let f = {f _k } _k=1 ⁿ , n ≥ 1 be an arbitrary sequence of independent random variables in X and let {e _k } _k=1 ^∞ ? E be the standard unit vector sequence in E. This paper presents a deterministic characterization of the quantity

$||||\sum\limits_{k = 1}^n {{f_k}{e_k}|{|_E}|{|_X}} $

in terms of the sum of disjoint copies of individual terms of f. We acknowledge key contributions by previous authors in detail in the introduction, however our approach is based on the important recent advances in the study of the Kruglov property of symmetric spaces made earlier by the authors. Authors acknowledge support from the ARC.

     

Several extremal approximation problems for the characteristic function of a spherical layer

We discuss three interrelated extremal problems on the set P _n,m of algebraic polynomials of a given degree n on the unit sphere \(\mathbb{S}^{m - 1}\) of the Euclidean space ? ^m of dimension m ≥ 2. (1) Find the norm of the functional \(F\left( \eta \right) = F_h P_n = \int_{\mathbb{G}\left( \eta \right)} {P_n (x)dx}\), which is the integral over the spherical layer \(\mathbb{G}\left( \eta \right) = \left\{ {x = \left( {x_1 , \ldots ,x_m } \right) \in \mathbb{S}^{m - 1} :h' \leqslant x_m \leqslant h''} \right\}\) defined by a pair of real numbers η = (h′, h″), ?1 ≤ h′ < h″ ≤ 1, on the set P _n,m with the norm of the space \(L\left( {\mathbb{S}^{m - 1} } \right)\) of functions summable on the sphere. (2) Find the best approximation in \(L_\infty \left( {\mathbb{S}^{m - 1} } \right)\) of the characteristic function χ _η of the layer \(\mathbb{G}\left( \eta \right)\) by the subspace P _n,m ^⊥ of functions from \(L_\infty \left( {\mathbb{S}^{m - 1} } \right)\) that are orthogonal to the space of polynomials P _n,m . (3) Find the best approximation in the space \(L\left( {\mathbb{S}^{m - 1} } \right)\) of the function χ _η by the space of polynomials P _n,m . We present a solution of all three problems for the values h′ and h″ that are neighboring roots of the polynomial in a single variable of degree n + 1 that deviates least from zero in the space L ₁ ^φ (?1, 1) of functions summable on the interval (?1, 1) with ultraspherical weight φ(t) = (1 ? t ²) ^α , α = (m ? 3)/2. We study the respective one-dimensional problems in the space of functions summable on (?1, 1) with an arbitrary not necessarily ultraspherical weight.     

Anisotropic Hardy-Lorentz spaces and their applications

Let p ∈(0, 1], q ∈(0, ∞] and A be a general expansive matrix on Rn. We introduce the anisotropic Hardy-Lorentz space H~(p,q)_A(R~n) associated with A via the non-tangential grand maximal function and then establish its various real-variable characterizations in terms of the atomic and the molecular decompositions, the radial and the non-tangential maximal functions, and the finite atomic decompositions. All these characterizations except the ∞-atomic characterization are new even for the classical isotropic Hardy-Lorentz spaces on Rn.As applications, we first prove that Hp,q A(Rn) is an intermediate space between H~(p1,q1)_A(Rn) and H~(p2,q2)_A(R~n) with 0 p1 p p2 ∞ and q1, q, q2 ∈(0, ∞], and also between H~(p,q1)_A(Rn) and H~(p,q2)_A(R~n) with p ∈(0, ∞)and 0 q1 q q2 ∞ in the real method of interpolation. We then establish a criterion on the boundedness of sublinear operators from H~(p,q)_A(R~n) into a quasi-Banach space; moreover, we obtain the boundedness of δ-type Calder′on-Zygmund operators from H~(p,∞)_A(R~n) to the weak Lebesgue space L~(p,∞)(R~n)(or to H~p_A(R~n)) in the ln λcritical case, from H~(p,q)_A(R~n) to L~(p,q)(R~n)(or to H~(p,q)_A(R~n)) with δ∈(0,(lnλ)/(ln b)], p ∈(1/(1+,δ),1] and q ∈(0, ∞], as well as the boundedness of some Calderon-Zygmund operators from H~(p,q)_A(R~n) to L~(p,∞)(R~n), where b := | det A|,λ_:= min{|λ| : λ∈σ(A)} and σ(A) denotes the set of all eigenvalues of A.     

Asymptotic Properties of Orthogonal Polynomials with Respect to a Non-discrete Jacobi-Sobolev Inner Product

Let {Q _n ^(α,β) (x)} _n=0 ^∞ denote the sequence of polynomials orthogonal with respect to the non-discrete Sobolev inner product

$\langle f,g\rangle=\int_{-1}^{1}f(x)g(x)d\mu_{\alpha,\beta}(x)+\lambda\int_{-1}^{1}f'(x)g'(x)d\nu_{\alpha,\beta}(x)$

where λ>0 and d μ _α,β(x)=(x?a)(1?x)^α?1(1+x)^β?1 dx, d ν _α,β(x)=(1?x) ^α (1+x) ^β dx with aα,β>0. Their inner strong asymptotics on (?1,1), a Mehler-Heine type formula as well as some estimates of the Sobolev norms of Q _n ^(α,β) are obtained.

     

Rearranged series by Haar system

For the orthonormal Haar system {X _n} the paper proves that for each 0 < ? < 1 there exist a measurable set E ? [0, 1] with measure | E | > 1 ? ? and a series of the form Σ _n=1 ^∞ a _n X _n with a _i ↘ 0, such that for every function f ∈ L ¹(0, 1) one can find a function \(\tilde f\) ∈ L ¹(0, 1) coinciding with f on E, and a series of the form

$\sum\limits_{i = 1}^\infty {\delta _i a_i \chi _i } where \delta _i = 0 or 1$

, that would converge to \(\tilde f\) in L ¹(0, 1).

     

On the Bessel Heat Equation Related to the Bessel Diamond Operator

In this article, we study the equation

$\frac{\partial }{\partial t}u(x,t)=c^{2}\Diamond _{B}^{k}u(x,t)$

with the initial condition u(x,0)=f(x) for x∈? _n ⁺ . The operator ? _B ^k is named to be Bessel diamond operator iterated k-times and is defined by

$\Diamond _{B}^{k}=\bigl[(B_{x_{1}}+B_{x_{2}}+\cdots +B_{x_{p}})^{2}-(B_{x_{p+1}}+\cdots +B_{x_{p+q}})^{2}\bigr]^{k},$

where k is a positive integer, p+q=n, \(B_{x_{i}}=\frac{\partial ^{2}}{\partial x_{i}^{2}}+\frac{2v_{i}}{x_{i}}\frac{\partial }{\partial x_{i}},\) 2v _i =2α _i +1,\(\;\alpha _{i}>-\frac{1}{2}\), x _i >0, i=1,2,…,n, and n is the dimension of the ? _n ⁺ , u(x,t) is an unknown function of the form (x,t)=(x ₁,…,x _n ,t)∈? _n ⁺ ×(0,∞), f(x) is a given generalized function and c is a positive constant (see Levitan, Usp. Mat. 6(2(42)):102–143, 1951; Y?ld?r?m, Ph.D. Thesis, 1995; Y?ld?r?m and Sar?kaya, J. Inst. Math. Comput. Sci. 14(3):217–224, 2001; Y?ld?r?m, et al., Proc. Indian Acad. Sci. (Math. Sci.) 114(4):375–387, 2004; Sar?kaya, Ph.D. Thesis, 2007; Sar?kaya and Y?ld?r?m, Nonlinear Anal. 68:430–442, 2008, and Appl. Math. Comput. 189:910–917, 2007). We obtain the solution of such equation, which is related to the spectrum and the kernel, which is so called Bessel diamond heat kernel. Moreover, such Bessel diamond heat kernel has interesting properties and also related to the kernel of an extension of the heat equation.