Construction of optimal quadrature formulas for Fourier coefficients in Sobolev space $L_{2}^{(m)}(0,1)$

This paper studies the problem of construction of optimal quadrature formulas in the sense of Sard in the \(L_{2}^{(m)}(0,1)\) space for numerical calculation of Fourier coefficients. Using the S.L.Sobolev’s method, we obtain new optimal quadrature formulas of such type for N+1≥m, where N+1 is the number of nodes. Moreover, explicit formulas for the optimal coefficients are obtained. We study the order of convergence of the optimal formula for the case m=1. The obtained optimal quadrature formulas in the \(L_{2}^{(m)}(0,1)\) space are exact for P _m?1(x), where P _m?1(x) is a polynomial of degree m?1. Furthermore, we present some numerical results, which confirm the obtained theoretical results.     

Optimal quadrature formulas for Fourier coefficients in $W_2^{(m,m-1)}$ space

This paper studies the problem of construction of optimal quadrature formulas in the sense of Sard in the $W_2^{(m,m-1)}[0,1]$ space for calculating Fourier coefficients. Using S.~L.\ Sobolev''s method we obtain new optimal quadrature formulas of such type for $N 1\geq m$, where $N 1$ is the number of the nodes. Moreover, explicit formulas for the optimal coefficients are obtained. We investigate the order of convergence of the optimal formula for $m=1$. The obtained optimal quadrature formula in the $W_2^{(m,m-1)}[0,1]$ space is exact for $\exp(-x)$ and $P_{m-2}(x)$, where $P_{m-2}(x)$ is a polynomial of degree $m-2$. Furthermore, we present some numerical results, which confirm the obtained theoretical results.     

Product type quadrature formulas

This paper is concerned with the numerical approximation of integrals of the form _a ^b f(x)g(x)dx by means of a product type quadrature formula. In such a formula the functionf (x) is sampled at a set ofn+1 distinct points and the functiong(x) at a (possibly different) set ofm+1 distinct points. These formulas are a generalization of the classical (regular) numerical integration rules. A number of basic results for such formulas are stated and proved. The concept of a symmetric quadrature formula is defined and the connection between such rules and regular quadrature formulas is discussed. Expressions for the error term are developed. These are applied to a specific example.The work of the first author was supported in part by NIH Grant No. FRO 7129-01 and that of the second author in part by U.S. Army Ballistic Research Laboratories Contract DA-18-001-AMC-876 X.     

Lacunary Quadrature Formulae and Interpolation Singularity

Birkholl quadrature formulae (q.f.), which have algebraic degree of precision (ADP) greater than the number of values used, are studied. In particular, we construct a class of quadrature rules of ADP = 2n + 2r + 1 which are based on the information {ƒ^(j)(−1), ƒ^(j)(−1), j = 0, ..., r − 1 ; ƒ(x_i), ƒ^(2m)(x_i), i = 1, ..., n}, where m is a positive integer and r = m, or r = m − 1. It is shown that the corresponding Birkhoff interpolation problems of the same type are not regular at the quadrature nodes. This means that the constructed quadrature formulae are not of interpolatory type. Finally, for each In, we prove the existence of a quadrature formula based on the information {ƒ(x_i), ƒ^(2m)(x_i), i = 1, ..., 2m}, which has algebraic degree of precision 4m + 1.     

Error expansions for multidimensional trapezoidal rules with Sidi transformations

In 1993, Sidi introduced a set of trigonometric transformations x = ψ(t) that improve the effectiveness of the one-dimensional trapezoidal quadrature rule for a finite interval. In this paper, we extend Sidi's approach to product multidimensional quadrature over [0,1] ^N . We establish the Euler–Maclaurin expansion for this rule, both in the case of a regular integrand function f(x) and in the cases when f(x) has homogeneous singularities confined to vertices. This revised version was published online in August 2006 with corrections to the Cover Date.     

Concerning the order of approximation of periodic continuous functions by trigonometric interpolation polynomials

Letx _kn=2θk/n,k=0,1 …n−1 (n odd positive integer). LetR _n(x) be the unique trigonometric polynomial of order 2n satisfying the interpolatory conditions:R _n(x_kn)=f(x_kn),R _n ^(j)(x_kn)=0,j=1,2,4,k=0,1…,n−1. We setw ₂(t,f) as the second modulus of continuity off(x). Then we prove that |R _n(x)-f(x)|=0(nw₂(1/nf)). We also examine the question of lower estimate of ‖R _n-f‖. This generalizes an earlier work of the author.     

An inverse polynomial method for the identification of the leading coefficient in the Sturm–Liouville operator from boundary measurements

An inverse polynomial method of determining the unknown leading coefficient k=k(x) of the linear Sturm–Liouville operator Au=−(k(x)u^′(x))^′+q(x)u(x), x(0,1), is presented. As an additional condition only two measured data at the boundary (x=0,x=1) are used. In absence of a singular point (u^′(x)≠0,u^″(x)≠0,x[0,1]) the inverse problem is classified as a well-conditioned . If there exists at least one singular point, then the inverse problem is classified as moderately ill-conditioned (u^′(x₀)=0,x₀(0,1);u^′(x)≠0,x≠x₀;u^″(x)≠0,x[0,1]) and severely ill-conditioned (u^′(x₀)=u^″(x₀)=0,x₀(0,1);u^′(x)≠0,u^″(x)≠0,x≠x₀). For each of the cases direct problem solution is approximated by corresponding polynomials and the inverse problem is reformulated as a Cauchy problem for to the first order differential equation with respect the unknown function k=k(x). An approximate analytical solution of the each Cauchy problems are derived in explicit form. Numerical simulations all the above cases are given for noise free and noisy data. An accuracy of the presented approach is demonstrated on numerical test solutions.     

CHAOS EXPANSION OF LOCAL TIME OF FRACTIONAL BROWNIAN MOTIONS

We find the chaos expansion of local time ? _T ^(H)(x,·) of fractional Brownian motion with Hurst coefficient H∈(0,1) at a point x∈R ^d . As an application we show that when H ₀ d<1 then ? _T ^(H)(x,·)∈L ²(μ). Here μ denotes the probability law of B ^(H) and H ₀=max{H ₁,…,H _d }. In particular, we show that when d=1 then ? _T ^(H)(x,·)∈L ²(μ) for all H∈(0,1).     

ОптИМАльНыЕ кВАДРАт УРНыЕ ФОРМУлы Дль клА ссОВ ФУНкцИИ с ИНтЕгРИРУЕ МОИ ВL p r-ОИ пРОИжВОДНОИ

On the classW ^r L _p (1≦p≦∞;r=1, 2,…) of 1-periodic functions ?(x) having an absolutely continuous (r? l)st derivative such that $$\parallel f^{(r)} \parallel _{L_p } \leqq 1 (\parallel f^{(r)} \parallel _{L_\infty } = vrai \sup |f^{(r)} (x)|)$$ vrai sup ¦?(^r)(x)¦) an optimal quadrature formula of the form (0 ≦? ≦r?1, 0 ≦x ₀ < x₁ <…< x_m ≦ 1) is found in the cases ?=r?2 and ?=r? 3 (r=3, 5, …). An exact error bound is established for this formula. The statements proved forW ^r L _p allowed us also to obtain, under certain restrictions posed on the coefficientsp _kl, and the nodesx ₀ andx _m, optimal quadrature formulae for the classes $$W_0^r L_p = \{ f:f \in W^r L_p , f^{(i)} (0) = 0 (i = 0,1,...,r - 2)\} $$ and $$W_0^r L_p = \{ f:f \in \tilde W^r L_p , f^{(i)} (0) = f^{(i)} (1) = 0 (i = 0,1,...,r - 2)\} $$ for the same values ofp andr as above.     

Extension Problem and Harnack's Inequality for Some Fractional Operators

The fractional Laplacian can be obtained as a Dirichlet-to-Neumann map via an extension problem to the upper half space. In this paper we prove the same type of characterization for the fractional powers of second order partial differential operators in some class. We also get a Poisson formula and a system of Cauchy–Riemann equations for the extension. The method is applied to the fractional harmonic oscillator H ^σ = (? Δ + |x|²)^σ to deduce a Harnack's inequality. A pointwise formula for H ^σ f(x) and some maximum and comparison principles are derived.     

On numerical improvement of Gauss–Legendre quadrature rules

Among all integration rules with n points, it is well-known that n-point Gauss–Legendre quadrature rule∫₋₁¹f(x) dx∑_i=1ⁿw_if(x_i)has the highest possible precision degree and is analytically exact for polynomials of degree at most 2n−1, where nodes x_i are zeros of Legendre polynomial P_n(x), and w_i's are corresponding weights.In this paper we are going to estimate numerical values of nodes x_i and weights w_i so that the absolute error of introduced quadrature rule is less than a preassigned tolerance ε₀, say ε₀=10⁻⁸, for monomial functionsf(x)=x^j, j=0,1,…,2n+1.(Two monomials more than precision degree of Gauss–Legendre quadrature rules.) We also consider some conditions under which the new rules act, numerically, more accurate than the corresponding Gauss–Legendre rules. Some examples are given to show the numerical superiority of presented rules.     

New Coins from Old, Smoothly

Given a (known) function f:[0,1]→(0,1), we consider the problem of simulating a coin with probability of heads f(p) by tossing a coin with unknown heads probability p, as well as a fair coin, N times each, where N may be random. The work of Keane and O’Brien (ACM Trans. Model. Comput. Simul. 4(2):213–219, 1994) implies that such a simulation scheme with the probability ℙ _p (N<∞) equal to 1 exists if and only if f is continuous. Nacu and Peres (Ann. Appl. Probab. 15(1A):93–115, 2005) proved that f is real analytic in an open set S⊂(0,1) if and only if such a simulation scheme exists with the probability ℙ _p (N>n) decaying exponentially in n for every p∈S. We prove that for α>0 noninteger, f is in the space C ^α [0,1] if and only if a simulation scheme as above exists with ℙ _p (N>n)≤C(Δ _n (p)) ^α , where \varDelta _n(x):=max{?{x(1-x)/n},1/n}\varDelta _{n}(x):=\max\{\sqrt{x(1-x)/n},1/n\}. The key to the proof is a new result in approximation theory: Let B⁺_n\mathcal{B}^{+}_{n} be the cone of univariate polynomials with nonnegative Bernstein coefficients of degree n. We show that a function f:[0,1]→(0,1) is in C ^α [0,1] if and only if f has a series representation ?_n=1^￥F_n\sum_{n=1}^{\infty}F_{n} with F_n ? B⁺_nF_{n}\in \mathcal{B}^{+}_{n} and ∑ _k>n F _k (x)≤C(Δ _n (x)) ^α for all x∈[0,1] and n≥1. We also provide a counterexample to a theorem stated without proof by Lorentz (Math. Ann. 151:239–251, 1963), who claimed that if some j_n ? B⁺_n\varphi_{n}\in\mathcal{B}^{+}_{n} satisfy |f(x)−φ _n (x)|≤C(Δ _n (x)) ^α for all x∈[0,1] and n≥1, then f∈C ^α [0,1].     

The Christoffel function for the Hermite weight is bell-shaped

We show that the Christoffel function λ_n associated with the Hermite weight function w_H(x)=exp(−x²) is bell-shaped. As a consequence, we describe completely how the weights in a Gauss-type quadrature formula associated with w_H(x) are arranged in magnitude.     

Unbounded divergence of simple quadrature formulas

Given a sequence of real or complex coefficients c_i and a sequence of distinct nodes t_i in a compact interval T, we prove the divergence and the unbounded divergence on superdense sets in the space C(T) of the simple quadrature formulas ∝_Tx(t)du(t) = Q_n(x) + R_n(x) and ∝_Tw(t)x(t)dt = Q_n(x) + R_n(x), where Q_n(x)=∑_i=1^m_n c_ix(t_i), ε C(T).The divergence (not certainly unbounded) for at most one continuous function of the first simple quadrature formula, with m_n = n and u(t) = t, was established by P. J. Davis in 1953.     

Speed of convergence in nonparametric kernel estimation of a regression function and its derivatives

Summary The objective in nonparametric regression is to infer a functiong(x) and itspth order derivativesg ^(g)(x),p≧1 fixed, on the basis of a finite collection of pairs {x _i, g(x_i)+Z _i} _i=1 ⁿ , where the noise componentsZ _i satisfy certain modest assumptions and the domain pointsx _i are selected non-randomly. This paper exhibits a new class of kernel estimatesg _n ^(p) ,p≧0 fixed. The main theoretical results of this study are the rates of convergence obtained for mean square and strong consistency ofg _n ^(p) each of them being uniform on the (0,1).     

Approximation of fractional-order integrals by algebraic polynomials. I

For functionsf(x) representable by an integral operator of a special form, we investigate the behavior of the second difference Δ _h ² f(x)=f(x+h)-2f(x)+f(x-h),h>0, depending on the location of a pointx on the segment [0,1]. Dnepropetrovsk University, Dnepropetrovsk. Translated from Ukrainskii Matematischeskii Zhurnal, Vol. 51, No. 5, pp. 603–613, May, 1999.     

Zur Existenz optimaler Quadraturformeln mit freien Knoten bei Integration analytischer Funktionen

Summary In the present paper we discuss the optimal quadrature rules for integration with positive continuous weight function in Hardy space H₂ of functions analytic in a circle of the complex plane. The new representations of the optimal weights and the norm of the error functional as functions of the nodes are obtained. On this basis we give an elementary proof for the existence of the optimal quadrature formula with free nodes.     

Dual frames in connected with generalized sampling in shift-invariant spaces

The aim of this article is to derive stable generalized sampling in a shift-invariant space by using some special dual frames in L²(0,1). These sampling formulas involve samples of filtered versions of the functions in the shift-invariant space. The involved samples are expressed as the frame coefficients of an appropriate function in L²(0,1) with respect to some particular frame in L²(0,1). Since any shift-invariant space with stable generator is the image of L²(0,1) by means of a bounded invertible operator, our generalized sampling is derived from some dual frame expansions in L²(0,1).     

The numerical evaluation of the error term in a quadrature formula of Clenshaw-Curtis type for the Gegenbauer weight function

A method is derived for the numerical evaluation of the error term arising in a quadrature formula of Clenshaw-Curtis type for functions of the form (1-x²)^{l- \frac12}f(x)(1-x^{2})^{\lambda - \frac{1}{2}}f(x) over the interval [−1,1]. The method is illustrated by an example.     

Carleman estimates for one-dimensional degenerate heat equations

In this paper, we are interested in controllability properties of parabolic equations degenerating at the boundary of the space domain. We derive new Carleman estimates for the degenerate parabolic equation $$ w_t + \left( {a\left( x \right)w_x } \right)_x = f,\quad \left( {t,x} \right) \in \left( {0,T} \right) \times \left( {0,1} \right), $$ where the function a mainly satisfies $$ a \in \mathcal{C}^0 \left( {\left[ {0,1} \right]} \right) \cap \mathcal{C}^1 \left( {\left( {0,1} \right)} \right),a \gt 0 \hbox{on }\left( {0,1} \right) \hbox{and }\frac{1} {{\sqrt a }} \in L^1 \left( {0,1} \right). $$ We are mainly interested in the situation of a degenerate equation at the boundary i.e. in the case where a(0)=0 and / or a(1)=0. A typical example is a(x)=x^α (1 − x)^β with α, β ∈ [0, 2). As a consequence, we deduce null controllability results for the degenerate one dimensional heat equation $$ u_t - (a(x)u_x )_x = h\chi _w ,\quad (t,x) \in (0,T) \times (0,1),\quad \omega \subset \subset (0,1). $$ The present paper completes and improves previous works [7, 8] where this problem was solved in the case a(x)=x^α with α ∈[0, 2). Dedicated to Giuseppe Da Prato on the occasion of his 70th birthday