共查询到20条相似文献,搜索用时 329 毫秒
1.
David Gómez-Ullate Niky Kamran Robert Milson 《Foundations of Computational Mathematics》2013,13(4):615-666
Exceptional orthogonal polynomial systems (X-OPSs) arise as eigenfunctions of Sturm–Liouville problems, but without the assumption that an eigenpolynomial of every degree is present. In this sense, they generalize the classical families of Hermite, Laguerre, and Jacobi, and include as a special case the family of CPRS orthogonal polynomials introduced by Cariñena et al. (J. Phys. A 41:085301, 2008). We formulate the following conjecture: every exceptional orthogonal polynomial system is related to a classical system by a Darboux–Crum transformation. We give a proof of this conjecture for codimension 2 exceptional orthogonal polynomials (X2-OPs). As a by-product of this analysis, we prove a Bochner-type theorem classifying all possible X2-OPSs. The classification includes all cases known to date plus some new examples of X2-Laguerre and X2-Jacobi polynomials. 相似文献
2.
Th. Stoll 《Indagationes Mathematicae》2003,14(2):263-274
Let A, B, C denote rational numbers with AB ≠ 0 and m > n ≥ 3 arbitrary rational integers. We study the Diophantine equation APm(x) + Bpn(y) = C, in x, y ? , where {Pk(x)}I is one of the three classical continuous orthogonal polynomial families, i.e. Laguerre polynomials, Jacobi polynomials (including Gegenbauer, Legendre or Chebyshev polynomials) and Hermite polynomials. We prove that with exception of the Chebyshev polynomials for all such polynomial families there are at most finitely many solutions (x, y) ? 2 provided n > 4. The tools are besides the criterion [3], a theorem of Szeg— [14] on monotonicity of stationary points of polynomials which satisfy a second order Sturm-Liouville differential equation,
3.
We consider the numerical solution of Sturm-Liouville eigenvalue problems by Legendre-Gauss Tau method. The latter approximates the solution of differential equations as a finite sum of Legendre polynomials . We propose an improved approach which seeks approximants in terms of a finite sum of exponentially weighted Legendre polynomials for some real or complex frequencies {ωk}. With the introduction of such exponentials, Legendre-Gauss Tau method can detect the sharp variations exhibited by the highly indexed Sturm-Liouville eigenfunctions. The efficiency of our results is illustrated through numerical examples. 相似文献
4.
Marina Chugunova 《Journal of Mathematical Analysis and Applications》2008,342(2):970-988
We study the spectrum of the linear operator L=−θ∂−?θ∂(sinθθ∂) subject to the periodic boundary conditions on θ∈[−π,π]. We prove that the operator is closed in with the domain in for |?|<2, its spectrum consists of an infinite sequence of isolated eigenvalues and the set of corresponding eigenfunctions is complete. By using numerical approximations of eigenvalues and eigenfunctions, we show that all eigenvalues are simple, located on the imaginary axis and the angle between two subsequent eigenfunctions tends to zero for larger eigenvalues. As a result, the complete set of linearly independent eigenfunctions does not form a basis in . 相似文献
5.
P. Shvartsman 《Journal of Functional Analysis》2004,210(1):1-42
Let F be a mapping from a metric space into the family of all m-dimensional affine subsets of a Banach space X. We present a Helly-type criterion for the existence of a Lipschitz selection f of the set-valued mapping F, i.e., a Lipschitz continuous mapping satisfying . The proof of the main result is based on an inductive geometrical construction which reduces the problem to the existence of a Lipschitz (with respect to the Hausdorff distance) selector SX(m) defined on the family of all convex compacts in X of dimension at most m. If X is a Hilbert space, then the classical Steiner point of a convex body provides such a selector, but in the non-Hilbert case there is no known way of constructing such a point. We prove the existence of a Lipschitz continuous selector for an arbitrary Banach space X. The proof is based on a new result about Lipschitz properties of the center of mass of a convex set. 相似文献
6.
Let f, g be entire functions. If there exist M1,M2>0 such that |f(z)|?M1|g(z)| whenever |z|>M2 we say that f?g. Let X be a reproducing Hilbert space with an orthogonal basis . We say that X is an ordered reproducing Hilbert space (or X is ordered) if f?g and g∈X imply f∈X. In this note, we show that if then X is ordered; if then X is not ordered. In the case , there are examples to show that X can be of order or opposite. 相似文献
7.
Construction of differential operators having Bochner-Krall orthogonal polynomials as eigenfunctions
K.H. Kwon 《Journal of Mathematical Analysis and Applications》2006,324(1):285-303
Suppose is a sequence of polynomials orthogonal with respect to the moment functional τ=σ+ν, where σ is a classical moment functional (Jacobi, Laguerre, Hermite) and ν is a point mass distribution with finite support. In this paper, we develop a new method for constructing a differential equation having as eigenfunctions. 相似文献
8.
Stefan Gille 《Journal of Pure and Applied Algebra》2007,208(2):391-419
We construct for any scheme X with a dualizing complex I• a Gersten-Witt complex and show that the differential of this complex respects the filtration by the powers of the fundamental ideal. To prove this we introduce second residue maps for one-dimensional local domains which have a dualizing complex. This residue maps generalize the classical second residue morphisms for discrete valuation rings. For the cohomology of the quotient complexes of this filtration we prove , where μI is the codimension function of the dualizing complex I• and denotes the Chow group of μI-codimension p-cycles modulo rational equivalence. 相似文献
9.
Stavros Iliadis 《Topology and its Applications》2010,157(4):752-759
In Iliadis (2005) [13] for an ordinal α the notion of the so-called (bn-Ind?α)-dimensional normal base C for the closed subsets of a space X was introduced. This notion is defined similarly to the classical large inductive dimension Ind. In this case we shall write here I(X,C)?α and say that the base dimension I of the space X by the normal base C is less than or equal to α. The classical large inductive dimension Ind of a normal space X, the large inductive dimension Ind0 of a Tychonoff space X defined independently by Charalambous and Filippov, as well as, the relative inductive dimension defined by Chigogidze for a subspace X of a Tychonoff space Y may be considered as the base dimension I of X by normal bases Z(X) (all closed subsets of X), Z(X) (all functionally closed subsets of X), and , respectively.In the present paper, we shall consider normal bases of spaces consisting of functionally closed subsets. In particular, we introduce new dimension invariant : for a space X, is the minimal element α of the class O∪{−1,∞}, where O is the class of all ordinals, for which there exists a normal base C on X consisting of functionally closed subsets such that I(X,C)?α. We prove that in the class of all completely regular spaces X of weight less than or equal to a given infinite cardinal τ such that there exist universal spaces. However, the following questions are open.(1) Are there universal elements in the class of all normal (respectively, of all compact) spaces X of weight ?τ with ?(2) Are there universal elements in the class of all Tychonoff (respectively, of all normal) spaces X of weight ?τ with Ind0(X)?n∈ω? (Note that for a compact space X.) 相似文献
10.
Yuexu Zhao 《Journal of Mathematical Analysis and Applications》2008,339(1):553-565
Let X1,X2,… be a strictly stationary sequence of ρ-mixing random variables with mean zeros and positive, finite variances, set Sn=X1+?+Xn. Suppose that , , where q>2δ+2. We prove that, if for any 0<δ?1, then
11.
H. Movahedi-Lankarani R. Wells 《Journal of Mathematical Analysis and Applications》2003,285(1):299-320
The C1-Weierstrass approximation theorem is proved for any compact subset X of a Hilbert space . The same theorem is also proved for Whitney 1-jets on X when X satisfies the following further condition: There exist finite dimensional linear subspaces such that ?n?1Hn is dense in and πn(X)=X∩Hn for each n?1. Here, is the orthogonal projection. It is also shown that when X is compact convex with and satisfies the above condition, then C1(X) is complete if and only if the C1-Whitney extension theorem holds for X. Finally, for compact subsets of , an extension of the C1-Weierstrass approximation theorem is proved for C1 maps with compact derivatives. 相似文献
12.
W.A. Webb 《Journal of Number Theory》2004,106(1):128-141
The polynomial Pell's equation is X2−DY2=1, where D is a polynomial with integer coefficients and the solutions X,Y must be polynomials with integer coefficients. Let D=A2+2C be a polynomial in , where . Then for a prime, a necessary and sufficient condition for which the polynomial Pell's equation has a nontrivial solution is obtained. Furthermore, all solutions to the polynomial Pell's equation satisfying the above condition are determined. 相似文献
13.
Let {f0,…,fn;g0,…,gn} be a sequence of homogeneous polynomials in 2n+2 variables with no common zeros in P2n+1 and suppose that the degrees of the polynomials are such that is a homogeneous polynomial. We shall refer to the hypersurface X defined by Q as a generalized quadric. In this note, we prove that generalized quadrics in for n≥1 are reduced. 相似文献
14.
Abey López García 《Advances in Mathematics》2008,218(4):1081-1106
We prove ratio asymptotic for sequences of multiple orthogonal polynomials with respect to a Nikishin system of measures N(σ1,…,σm) such that for each k, σk has constant sign on its support consisting on an interval , on which almost everywhere, and a set without accumulation points in . 相似文献
15.
16.
Salah Mecheri Messaoud Bounkhel 《Journal of Mathematical Analysis and Applications》2003,287(1):51-60
In this paper we characterize the global minimum of an arbitrary function defined on a Banach space, in terms of a new concept of derivatives adapted for our case from a recent work due to D.J. Keckic (J. Operator Theory, submitted for publication). Using these results we establish several new characterizations of the global minimum of the map defined by Fψ(X)=‖ψ(X)‖1, where is a map defined by ψ(X)=S+φ(X) and φ:B(H)→B(H) is a linear map, S∈C1, and . Further, we apply these results to characterize the operators which are orthogonal to the range of elementary operators. 相似文献
17.
Kaori Yamazaki 《Topology and its Applications》2006,153(15):2805-2814
It is well known that every pair of disjoint closed subsets F0,F1 of a normal T1-space X admits a star-finite open cover U of X such that, for every U∈U, either or holds. We define a T1-space X to be strongly base-normal if there is a base B for X with |B|=w(X) satisfying that every pair of disjoint closed subsets F0,F1 of X admits a star-finite cover B′ of X by members of B such that, for every B∈B′, either or holds. We prove that there is a base-normal space which is not strongly base-normal. Moreover, we show that Rudin's Dowker space is strongly base-(collectionwise)normal. Strong zero-dimensionality on base-normal spaces are also studied. 相似文献
18.
Rita Giuliano 《Journal of Mathematical Analysis and Applications》2011,378(2):555-570
In this paper we consider several examples of sequences of partial sums of triangular arrays of random variables {Xn:n?1}; in each case Xn converges weakly to an infinitely divisible distribution (a Poisson distribution or a centered Normal distribution). For each sequence we prove large deviation results for the logarithmically weighted means with speed function . We also prove a sample path large deviation principle for {Xn:n?1} defined by , where σ2∈(0,∞) and {Un:n?1} is a sequence of independent standard Brownian motions. 相似文献
19.
John A. Rhodes 《Journal of Number Theory》2003,102(2):278-297
We define n families of Hecke operators for GLn whose generating series are rational functions of the form qk(u)−1 where qk is a polynomial of degree , and whose form is that of the kth exterior product. This work can be viewed as a refinement of work of Andrianov (Math. USSR Sb. 12(3) (1970)), in which he defined Hecke operators the sum of whose generating series was a rational function with nontrivial numerator and whose denominator was essentially .By a careful analysis of the Satake map which defines an isomorphism between a local Hecke algebra and a ring of symmetric polynomials, we define n families of (polynomial) Hecke operators and characterize their generating series as rational functions. We then give an explicit means by which to locally invert the Satake isomorphism, and show how to translate these polynomial operators back to the classical double coset setting. The classical Hecke operators have generating series of exactly the same form as their polynomial counterparts, and hence are of number-theoretic interest. We give explicit examples for GL3 and GL4. 相似文献