Linear Combination Of Derivatives Research Articles

The Poisson spectrum of the symmetric algebra of the Virasoro algebra

Let $W = \mathbb {C}[t,t^{-1}]\partial _t$ be the Witt algebra of algebraic vector fields on $\mathbb {C}^\times$ and let $V\!ir$ be the Virasoro algebra, the unique nontrivial central extension of $W$. In this paper, we study the Poisson ideal structure of the symmetric algebras of $V\!ir$ and $W$, as well as several related Lie algebras. We classify prime Poisson ideals and Poisson primitive ideals of $\operatorname {S}(V\!ir)$ and $\operatorname {S}(W)$. In particular, we show that the only functions in $W^*$ which vanish on a nontrivial Poisson ideal (that is, the only maximal ideals of $\operatorname {S}(W)$ with a nontrivial Poisson core) are given by linear combinations of derivatives at a finite set of points; we call such functions local. Given a local function $\chi \in W^*$, we construct the associated Poisson primitive ideal through computing the algebraic symplectic leaf of $\chi$, which gives a notion of coadjoint orbit in our setting. As an application, we prove a structure theorem for subalgebras of $V\!ir$ of finite codimension and show, in particular, that any such subalgebra of $V\!ir$ contains the central element $z$, substantially generalising a result of Ondrus and Wiesner on subalgebras of codimension one. As a consequence, we deduce that $\operatorname {S}(V\!ir)/(z-\zeta )$ is Poisson simple if and only if $\zeta \neq ~0$.

On the zeros of linear combinations of derivatives of the Riemann zeta function, II

The relevant number to the Dirichlet series [Formula: see text], is defined to be the unique integer [Formula: see text] with [Formula: see text], which maximizes the quantity [Formula: see text]. In this paper, we classify the set of all relevant numbers to the Dirichlet [Formula: see text]-functions. The zeros of linear combinations of [Formula: see text] and its derivatives are also studied. We give an asymptotic formula for the supremum of the real parts of zeros of such combinations. We also compute the degree of the largest derivative needed for such a combination to vanish at a certain point.

On the zeros of linear combinations of derivatives of the Riemann zeta function

In this paper, the zeros of linear combinations of the Riemann zeta function and its derivatives are studied. We establish an asymptotic formula for the number of zeros in a rectangle of height [Formula: see text]. We also find a sharp asymptotic formula for the supremum of the real parts of zeros of such combinations in a certain family.

On the Value Distribution of Algebroid Functions

In this paper, we investigate the value distribution of an algebroid function and its linear combination of derivatives, and obtain some relations between them. One of the relations extends the result of H.Milloux. Finally, we estimate the zeros of algebroid functions’ derivatives.

On expansions in orthogonal polynomials

A recently introduced fast algorithm for the computation of the first N terms in an expansion of an analytic function into ultraspherical polynomials consists of three steps: Firstly, each expansion coefficient is represented as a linear combination of derivatives; secondly, it is represented, using the Cauchy integral formula, as a contour integral of the function multiplied by a kernel; finally, the integrand is transformed to accelerate the convergence of the Taylor expansion of the kernel, allowing for rapid computation using Fast Fourier Transform. In the current paper we demonstrate that the first two steps remain valid in the general setting of orthogonal polynomials on the real line with finite support, orthogonal polynomials on the unit circle and Laurent orthogonal polynomials on the unit circle.

Discrete chromatic series

Chromatic series are series expansions in which the coefficients are linear combinations of derivatives of a function. They were introduced by Ignjatovic (Introduction to Signal Processing Based on Differential Operators, Tech. Rep. 1, Kromos Technology, available at http://www.kromos.com, February 2001.) as a replacement for Taylor's series and are based on orthogonal polynomials. However, real data usually involves the values of a function and not its derivatives which are needed in both theories. In this article, we replace the derivatives by discrete values in the calculations of the coefficients.

A fast and simple algorithm for the computation of Legendre coefficients

We present an $${\mathcal{O}({N\,{\rm log}\,N})}$$ algorithm for the calculation of the first N coefficients in an expansion of an analytic function in Legendre polynomials. In essence, the algorithm consists of an integration of a suitably weighted function along an ellipse, a task which can be accomplished with Fast Fourier Transform, followed by some post-processing. The mathematical underpinning of this algorithm is an old formula that expresses expansion coefficients $${\hat{f}_m}$$ as infinite linear combinations of derivatives. We evaluate the latter with the Cauchy theorem, thereby expressing each $${\hat{f}_m}$$ as a scaled integral of $${f(z)\varphi_m(z)/z^{m+1}}$$ along an appropriate contour, where $${\varphi_m}$$ is a slowly converging hypergeometric function. Next, we transform $${\varphi_m}$$ into another hypergeometric function which converges rapidly. Once we replace the latter function by its truncated Taylor expansion and choose an appropriate elliptic contour, we obtain an expression for the $${\hat{f}_m}$$ s which is amenable to rapid computation.

From high oscillation to rapid approximation IV: accelerating convergence

Modified Fourier expansion is a powerful means for the approximation of nonperiodic smooth functions in a univariate or multivariate setting. In the current paper we consider further enhancement of this approach by two techniques familiar from conventional Fourier analysis: the polynomial subtraction and the hyperbolic cross. We demonstrate that, judiciously subtracting simpler functions dependent on linear combinations of derivatives along boundaries, it is possible to accelerate convergence a great deal and this procedure is considerably more efficient than in the case of conventional Fourier expansion. Moreover, examining the pattern of decay of coefficients in a multivariate setting, we demonstrate that most of them can be disregarded without any ill effect on the quality of approximation.

Stability of Roots of Polynomials Under Linear Combinations of Derivatives

Let T=α0I+α1D+⋅⋅⋅+αnDn, where D is the differentiation operator and \(\alpha_{0}\not=0\) , and let f be a square-free polynomial with large minimum root separation. We prove that the roots of Tf are close to the roots of f translated by −α1/α0.

Asymptotic behavior at infinity for solutions of Emden-Fowler type equations

The semilinear equation Δu = |u|σ−1u is considered in the exterior of a ball in ℝn, n ≥ 3. It is shown that if the exponent σ is greater than a “critical” value (= n/n−2), then for x → ∞ the leading term of the asymptotics of any solution is a linear combination of derivatives of the fundamental solution. It is shown that there exist solutions with the indicated leading term of an asymptotics of such a type.

Nonlinear singular multipoint boundary value problems

AbstractLet Λ be a linear differential operator on (a, b) of order 2m with leading coefficient possessing an integrable reciprocal. Global C(m‐1)(a, b) solutions are shown to exist for certain nonlinear multipoint boundary value problems of the form A u = F(u, …, u(m‐1), where linear combinations of derivatives through order m – 1 are specified at points interior to (a, b). The associated linear problem is completely solved through the construction of an appropriate Green's function which determines a compact operator G = A−1. The linear eigenvalue problem is examined and a complete orthonormal system exhibited in L2(a, b). Solutions of the nonlinear problem are obtained by exhibiting fixed points for the operator GF.

Span of linear combinations of derivatives of polynomials

Span of linear combinations of derivatives of polynomials

Green's Distributions and the Cauchy Problem for the Multi-Mass Klein-Gordon Operator

Explicit forms of the Green's functions (which are to be regarded as distributions in the sense of Schwartz) for the multi-mass Klein-Gordon operator in n-dimensional spaces are presented. The homogeneous Green's functions GN(x) and GN1(x), defined in the usual way by independent paths of integration in the k0 plane, are investigated in the neighborhood of the light cone. The parameter N indicates the total number of masses involved. The singularities on the light cone reflect the well-known difference between even- and odd-dimensional wave propagation. It is found that GN(x; odd n) contains a finite jump on the light cone as well as a linear combination of derivatives up to order ½(n − 2N − 1) of δ(x2); the singular part of GN1(x; odd n) consists of a logarithmic singularity ln (|x2|) along with a polynomial in (x2)−1 of degree ½(n − 2N − 1). For even-dimensional spaces, the singular part of both Green's functions consists of a polynomial in (x2)−1/2 of degree n − 2N + 1 vanishing outside the light cone for GN and vanishing inside the light cone for GN1. In all cases no singularities or finite jumps occur when the order 2N of the operator is greater than the number n + 1 of space-time dimensions. The general solution of the Cauchy problem is given both for the data carrying surface t = 0 and for arbitrary spacelike data surfaces.