We give some properties relating the recurrence relations of orthogonal polynomials associated with any two symmetric distributions dφ1(x) and d2(x) such that dφ2(x) = (1 + kx2)d1(x). As applications of properties, recurrence relations for many interesting systems of orthogonal polynomials are obtained. 相似文献
On the ring R = F[x1,..., xn] of polynomials in n variables over a field F special isomorphisms A’s of R into R are defined which preserve the greatest common divisor of two polynomials. The ring R is extended to the ring S: = F[[x1,..., xn]]+ and the ring T: = F[[x1,..., xn]] of generalized polynomials in such a way that the exponents of the variables are non-negative rational numbers and rational numbers, respectively. The isomorphisms A’s are extended to automorphisms B’s of the ring S. Using the property that the isomorphisms A’s preserve GCD it is shown that any pair of generalized polynomials from S has the greatest common divisor and the automorphisms B’s preserve GCD. On the basis of this Theorem it is proved that any pair of generalized polynomials from the ring T = F[[x1,..., xn]] has a greatest common divisor. 相似文献
This paper is devoted to refining the Bernstein inequality. Let D be the differentiation operator. The action of the operator Λ = D/n on the set of trigonometric polynomials Tn is studied: the best constant is sought in the inequality between the measures of the sets {x ∈ T: |Λt(x)| > 1} and {x ∈ T: |t(x)| > 1}. We obtain an upper estimate that is order sharp on the set of uniformly bounded trigonometric polynomials TnC = {t ∈ Tn: ‖t‖ ≤ C}. 相似文献
The authors consider irreducible representations of a nilpotent Lie group and define a Fourier transform for Schwartz class (and other) functions φ on N by forming the kernels Kφ(x, y) of the trace class operations πφ = ∝Nφ(n)πndn, regarding the π as modeled in L2(Rk) for all π in general position. For a special class of groups they show that the models, and parameters λ labeling the representations in general position, can be chosen so the joint behavior of the kernels Kφ(x, y, λ) can be interpreted in a useful way. The variables (x, y, λ) run through a Zariski open set in Rn, n = dim N. The authors show there is a polynomial map u = A(x, y, λ) that is a birational isomorphism A: Rn → Rn with the following properties. The Fourier transforms F1φ = Kφ(x, y, λ) all factor through A to give “rationalized” Fourier transforms Fφ(u) such that Fφ ° A = F1φ. On the rationalized parameter space a function f(u) is of the form Fφ = f ? f is Schwartz class on Rn. If polynomial operators T?P(N) are transferred to operators on Rn such that is transformed isomorphically to P(Rn). 相似文献
It is shown that the (φ2n)2 quantum field theory model is Lorentz covariant in the sense that the Poincaré transformation φ(x, t) → φ({a, Λ}(x, t)) is locally unitarily implementable in Fock space. It follows that the corresponding theory of local observables satisfies all of the Haag-Kastler axioms. The method of proof generalizes that of Cannon and Jaffe for the (φ4)2 model and relies on higher order estimates for the generator of Lorentz rotations. 相似文献
This paper deals with the numerical solution of the general mathematical programming problem of minimizing a scalar functionf(x) subject to the vector constraints φ(x)=0 and ψ(x)≥0. The approach used is an extension of the Hestenes method of multipliers, which deals with the equality constraints only. The above problem is replaced by a sequence of problems of minimizing the augmented penalty function Ω(x, λ, μ,k)=f(x)+λTφ(x)+kφT(x)φ(x) ?μT \(\tilde \psi \) (x)+k \(\tilde \psi \) T(x) \(\tilde \psi \) (x). The vectors λ and μ, μ ≥ 0, are respectively the Lagrange multipliers for φ(x) and \(\tilde \psi \) (x), and the elements of \(\tilde \psi \) (x) are defined by \(\tilde \psi \) (j)(x)=min[ψ(j)(x), (1/2k) μ(j)]. The scalark>0 is the penalty constant, held fixed throughout the algorithm. Rules are given for updating the multipliers for each minimization cycle. Justification is given for trusting that the sequence of minimizing points will converge to the solution point of the original problem. 相似文献
Suppose we have an operator T that maps a set of orthogonal polynomials {Pn(x)}n = o∞ to another set of orthogonal polynomials. We show how such a mapping can be used to derive connection relations as well as bilinear formulas for the pre-images {Pn(x)}n = o∞. This method is carried out in detail for the Jacobi, Laguerre, and Hahn polynomials. 相似文献
Let K be a compact subset of Rd and let m define a semi-dynamical system on M1(K),, the space of probability measures on K, induced by a homogeneous Markov process. Let J(φ) be the prolongational limit set for φ?M1(K). The main result of the paper is the decomposition: support of J(φ) = ∪ {support of J(δx): x? support φ}, where δx is the Dirac measure at the point x. 相似文献
An earlier paper (Mh. Math.51, 278–297 (1949)) exploited the property ‖∣Tx‖∣2≤‖∣x‖∣ ‖∣T2x‖∣, and the same property for polynomials in the operatorT, as an aid in establishing spectral resolutions associated withT. The present paper uses the weaker property ‖∣T‖∣=‖∣T2‖∣1/2=..., and its extension to polynomials, for the same purpose. Also considered are the possibility of equivalence between the two types of conditions, and the use of arithmethical hypotheses concerning the eigenvalues. 相似文献
This paper is concerned with the practical evaluation of the product integral ∫1? 1f(x)k(x)dx for the case when k(x) = In|x - λ|, λ? (?1, +1) and f is bounded in [?1, +1]. The approximation is a quadrature rule where the weights {wn,n,i} are chosen to be exact when f is given by a linear combination of a chosen set of functions {φn,j}. In this paper the functions {φn,j} are chosen to be cubic B-splines. An error bound for product quadrature rules based on cubic splines is provided. Examples that test the performance of the product quadrature rules for different choices of the function are given. A comparison is made with product quadrature rules based on first kind Chebyshev polynomials. 相似文献
Advanced speech information processing systems require further research on speaker-dependent information. Recently, a specific system of discrete orthogonal polynomials {φLr(l); l = 1, 2, …, L} Rr=0 has been encountered to play a dominant role in a segmental probability model recently proposed in the speaker-dependent feature extraction from speech waves and applied to text-independent speaker verification. Here, these speech polynomials are shown to be the shifted Chebyshev polynomials on a discrete variable tr(l − 1, L), whose structural and spectral properties are discussed and reviewed in light of the recent discoveries in the field of discrete orthogonal polynomials. 相似文献
In this paper, we derive an explicit expression for the parameter sequences of a chain sequence in terms of the corresponding orthogonal polynomials and their associated polynomials. We use this to study the orthogonal polynomials Kn(λ,M,k) associated with the probability measure dφ(λ,M,k;x), which is the Gegenbauer measure of parameter λ+1 with two additional mass points at ±k. When k=1 we obtain information on the polynomials Kn(λ,M) which are the symmetric Koornwinder polynomials. Monotonicity properties of the zeros of Kn(λ,M,k) in relation to M and k are also given. 相似文献
The paper investigates the properties of an operator Tφ on the Hilbert space l2(?), which are induced by the mapping φ of the set ? into itself. It is shown if the mapping φ is such that every preimage has finite, but not equipotentionally bounded cardinality, then the operator Tφ allows a closure and can be represented as a countable sum of partial isometries. The C*-algebras Uφ, Pφ and Uφ associated with given mappings and generated by the mentioned partial isometries are considered. Some properties of these algebras and some relations between them are given. 相似文献
Let ∥·∥ be a norm in R2 and let γ be the unit sphere induced by this norm. We call a segment joining points x,y ε R2 rational if (x1 ? y1)/(x2 ? y2) or (x2 ? y2)/(x1 ? y1) is a rational number. Let γ be a convex curve containing no rational segments. Satisfaction of the condition $$T_\nu (x) = \sum\nolimits_{\parallel n\parallel = \nu } {c_n e^{2\pi i(n_1 x_1 + n_2 x_2 )} } \to 0(\nu \to \infty )$$ in measure on the set e? [- 1/2,1/2)×[- 1/2, 1/2) =T2 of positive planar measure implies ∥Tv∥L4 (T2) → 0(v → ∞). if, however, γ contains a rational segment, then there exist a sequence of polynomials {Tv} and a set E ? T2, ¦E¦ > 0, such that Tv(x) → 0(v → ∞) on E; however, ¦cn¦ ? 0 for ∥n∥ → ∞. 相似文献
We give a parametric family of quintic polynomials of the form x5 + ax + b (a, b ∈ ) with dihedral Galois group D5. Some properties of the fields defined by these polynomials are also described. 相似文献
Let x0 be a nonzero vector in \({\mathbb{C}^{n}}\) , and let \({U\subseteq \mathcal{M}_{n}}\) be a domain containing the zero matrix. We prove that if φ is a holomorphic map from U into \({\mathcal{M}_{n}}\) such that the local spectrum of T ∈ U at x0 and the local spectrum of φ(T) at x0 have always a common value, then T and φ(T) have always the same spectrum, and they have the same local spectrum at x0 a.e. with respect to the Lebesgue measure on U. If \({\varphi \colon U\rightarrow \mathcal{M}_{n}}\) is holomorphic with φ(0) = 0 such that the local spectral radius of T at x0 equals the local spectral radius of φ(T) at x0 for all T ∈ U, there exists \({\xi \in \mathbb{C}}\) of modulus one such that ξT and φ(T) have the same spectrum for all T in U. We also prove that if for all T ∈ U the local spectral radius of φ(T) coincides with the local spectral radius of T at each vector x, there exists \({\xi \in \mathbb{C}}\) of modulus one such that φ(T) = ξT on U. 相似文献
LetCub ( $\mathbb{J}$ , X) denote the Banach space of all uniformly continuous bounded functions defined on $\mathbb{J}$ 2 ε {?+, ?} with values in a Banach spaceX. Let ? be a class fromCub( $\mathbb{J}$ ,X). We introduce a spectrumsp?(φ) of a functionφ εCub (?,X) with respect to ?. This notion of spectrum enables us to investigate all twice differentiable bounded uniformly continuous solutions on ? to the abstract Cauchy problem (*)ω′(t) =Aω(t) +φ(t),φ(0) =x,φ ε ?, whereA is the generator of aC0-semigroupT(t) of bounded operators. Ifφ = 0 andσ(A) ∩i? is countable, all bounded uniformly continuous mild solutions on ?+ to (*) are studied. We prove the bound-edness and uniform continuity of all mild solutions on ?+ in the cases (i)T(t) is a uniformly exponentially stableC0-semigroup andφ εCub(?,X); (ii)T(t) is a uniformly bounded analyticC0-semigroup,φ εCub (?,X) andσ(A) ∩i sp(φ) = Ø. Under the condition (i) if the restriction ofφ to ?+ belongs to ? = ?(?+,X), then the solutions belong to ?. In case (ii) if the restriction ofφ to ?+ belongs to ? = ?(?+,X), andT(t) is almost periodic, then the solutions belong to ?. The existence of mild solutions on ? to (*) is also discussed. 相似文献
The equations [gradφ(x)]TF(x)=h(x) and F(ψ(x))–ψ(x) are considered. They arise in the stability theory of differential and difference equations. The scalar function h(x) is a given, and the function ψ(x) an unknown, formal power series in the n indeterminates x=(x1,…,xn)T, and h(0)=ψ=0; the elements of the n×n matrix F(x) are also formal power series in x, F(0)=0. It is shown that the solvability of both equations depends on the eigenvalues of the Jacobian Fx(0). 相似文献