共查询到20条相似文献,搜索用时 46 毫秒
1.
F.Alberto Grünbaum 《Journal of Mathematical Analysis and Applications》1983,95(2):491-500
We exhibit a second-order differential operator commuting with the reproducing kernel each time that {φn(λ)} is one of the classical orthogonal polynomials: Jacobi, Laguerre, Hermite and Bessel. This is the analog of a known property in the study of time and band-limited signals. 相似文献
2.
Stanisław Lewanowicz 《Journal of Computational and Applied Mathematics》1979,5(3):193-206
In this paper we are constructing a recurrence relation of the form for integrals (called modified moments) in which Ck(λ) is the k-th Gegenbauer polynomial of order , and f is a function satisfying the differential equation of order n, where p0, p1, …, pn ? 0 are polynomials, and mk〈λ〉[p] is known for every k. We give three methods of construction of such a recurrence relation. The first of them (called Method I) is optimum in a certain sense. 相似文献
3.
J Bustoz 《Journal of Mathematical Analysis and Applications》1981,79(1):71-79
It is known that the classical orthogonal polynomials satisfy inequalities of the form Un2(x) ? Un + 1(x) Un ? 1(x) > 0 when x lies in the spectral interval. These are called Turan inequalities. In this paper we will prove a generalized Turan inequality for ultraspherical and Laguerre polynomials. Specifically if Pnλ(x) and Lnα(x) are the ultraspherical and Laguerre polynomials and . We also prove the inequality is a positive constant depending on α and β. 相似文献
4.
Arthur Lubin 《Journal of Functional Analysis》1974,17(4):388-394
Let m and vt, 0 ? t ? 2π be measures on T = [0, 2π] with m smooth. Consider the direct integral = ⊕L2(vt) dm(t) and the operator on , where e(s, t) = exp ∫st ∫Tdvλ(θ) dm(λ). Let μt be the measure defined by for all continuous ?, and let ?t(z) = exp[?∫ (eiθ + z)(eiθ ? z)?1dμt(gq)]. Call {vt} regular iff for all for 1 a.e. 相似文献
5.
Real constant coefficient nth order elliptic operators, Q, which generate strongly continuous semigroups on L2(k) are analyzed in terms of the elementary generator, , for n even. Integral operators are defined using the fundamental solutions pn(x, t) to ut = Au and using real polynomials ql,…, qk on m by the formula, for q = (ql,…, qk), m. It is determined when, strongly on L2(k), . If n = 2 or k = 1, this can always be done. Otherwise the symbol of Q must have a special form. 相似文献
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7.
Hermann König 《Journal of Functional Analysis》1977,24(1):32-51
For an open set Ω ? N, 1 ? p ? ∞ and λ ∈ +, let denote the Sobolev-Slobodetzkij space obtained by completing in the usual Sobolev-Slobodetzkij norm (cf. A. Pietsch, “r-nukleare Sobol. Einbett. Oper., Ellipt. Dgln. II,” Akademie-Verlag, Berlin, 1971, pp. 203–215). Choose a Banach ideal of operators , 1 ? p, q ? ∞ and a quasibounded domain Ω ? N. Theorem 1 of the note gives sufficient conditions on λ such that the Sobolev-imbedding map exists and belongs to the given Banach ideal : Assume the quasibounded domain fulfills condition Ckl for some l > 0 and 1 ? k ? N. Roughly this means that the distance of any to the boundary ?Ω tends to zero as for , and that the boundary consists of sufficiently smooth ?(N ? k)-dimensional manifolds. Take, furthermore, 1 ? p, q ? ∞, p > k. Then, if μ, ν are real positive numbers with λ = μ + v ∈ , μ > λ S(; p,q:N) and v > N/l · λD(;p,q), one has that belongs to the Banach ideal . Here λD(;p,q;N)∈+ and λS(;p,q;N)∈+ are the D-limit order and S-limit order of the ideal , introduced by Pietsch in the above mentioned paper. These limit orders may be computed by estimating the ideal norms of the identity mappings lpn → lqn for n → ∞. Theorem 1 in this way generalizes results of R. A. Adams and C. Clark for the ideals of compact resp. Hilbert-Schmidt operators (p = q = 2) as well as results on imbeddings over bounded domains.Similar results over general unbounded domains are indicated for weighted Sobolev spaces.As an application, in Theorem 2 an estimate is given for the rate of growth of the eigenvalues of formally selfadjoint, uniformly strongly elliptic differential operators with Dirichlet boundary conditions in , where Ω fulfills condition C1l.For an open set Ω in N, let denote the Sobolev-Slobodetzkij space obtained by completing in the usual Sobolev-Slobodetzkij norm, see below. Taking a fixed Banach ideal of operators and 1 ? p, q ? ∞, we consider quasibounded domains Ω in N and give sufficient conditions on λ such that the Sobolev imbedding operator exists and belongs to the Banach ideal. This generalizes results of C. Clark and R. A. Adams for compact, respectively, Hilbert-Schmidt operators (p = q = 2) to general Banach ideals of operators, as well as results on imbeddings over bounded domains. Similar results over general unbounded domains may be proved for weighted Sobolev spaces. As an application, we give an estimate for the rate of growth of the eigenvalues of formally selfadjoint, uniformly strongly elliptic differential operators with Dirichlet boundary conditions in , where Ω is a quasibounded open set in N. 相似文献
8.
Matania Ben-Artzi 《Journal of Differential Equations》1980,38(1):51-60
Let H = ?Δ + V, where the potential V is spherically symmetric and can be decomposed as a sum of a short-range and a long-range term, V(r) = VS(r) + VL. Let λ = lim supr→∞VL(r) < ∞ (we allow λ = ? ∞) and set λ+ = max(λ, 0). Assume that for some r0, VL(r) ?C2k(r0, ∞) and that there exists δ > 0 such that . Assume further that and that 2kδ > 1. It is shown that: (a) The restriction of H to C∞(Rn) is essentially self-adjoint, (b) The essential spectrum of H contains the closure of (λ, ∞). (c) The part of H over (λ, ∞) is absolutely continuous. 相似文献
9.
Let α(n1, n2) be the probability of classifying an observation from population Π1 into population Π2 using Fisher's linear discriminant function based on samples of size n1 and n2. A standard estimator of α, denoted by T1, is the proportion of observations in the first sample misclassified by the discriminant function. A modification of T1, denoted by T2, is obtained by eliminating the observation being classified from the calculation of the discriminant function. The UMVU estimators, and , of ET1 = τ1(n1, n2) and ET2 = τ2(n1, n2) = α(n1 ? 1, n2) are derived for the case when the populations have multivariate normal distributions with common dispersion matrix. It is shown that and are nonincreasing functions of D2, the Mahalanobis sample distance. This result is used to derive the sampling distributions and moments of and . It is also shown that α is a decreasing function of Δ2 = (μ1 ? μ2)′Σ?1(μ1 ? μ2). Hence, by truncating and (or any estimator) at the value of α for Σ = 0, new estimators are obtained which, for all samples, are as close or closer to α. 相似文献
10.
This paper is a study of the distribution of eigenvalues of various classes of operators. In Section 1 we prove that the eigenvalues (λn(T)) of a p-absolutely summing operator, p ? 2, satisfy This solves a problem of A. Pietsch. We give applications of this to integral operators in Lp-spaces, weakly singular operators, and matrix inequalities.In Section 2 we introduce the quasinormed ideal Π2(n), P = (p1, …, pn) and show that for T ∈ Π2(n), 2 = (2, …, 2) ∈ Nn, the eigenvalues of T satisfy More generally, we show that for T ∈ Πp(n), P = (p1, …, pn), pi ? 2, the eigenvalues are absolutely p-summable, We also consider the distribution of eigenvalues of p-nuclear operators on Lr-spaces.In Section 3 we prove the Banach space analog of the classical Weyl inequality, namely , 0 < p < ∞, where αn denotes the Kolmogoroff, Gelfand of approximation numbers of the operator T. This solves a problem of Markus-Macaev.Finally we prove that Hilbert space is (isomorphically) the only Banach space X with the property that nuclear operators on X have absolutely summable eigenvalues. Using this result we show that if the nuclear operators on X are of type l1 then X must be a Hilbert space. 相似文献
11.
It is shown that if satisfies , where σk(A) denotes the sum of all kth order subpermanent of A, then Per[λJn+(1?λ)A] is strictly decreasing in the interval 0<λ<1. 相似文献
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14.
Teruo Ikebe 《Journal of Functional Analysis》1975,20(2):158-177
A spectral representation for the self-adjoint Schrödinger operator H = ?Δ + V(x), x? R3, is obtained, where V(x) is a long-range potential: , grad , being the Laplace-Beltrami operator on the unit sphere Ω. Namely, we shall construct a unitary operator from PL2(R3) onto being the orthogonal projection onto the absolutely continuous subspace for H, such that for any Borel function α(λ), . 相似文献
15.
We study the weight distribution of irreducible cyclic (n, k) codeswith block lengths n = n1((q1 ? 1)/N), where N|q ? 1, gcd(n1,N) = 1, and gcd(l,N) = 1. We present the weight enumerator polynomial, A(z), when k = n1l, k = (n1 ? 1)l, and k = 2l. We also show how to find A(z) in general by studying the generator matrix of an (n1, m) linear code, over GF(qd) where d = gcd (ordn1(q), l). Specifically we study A(z) when is a maximum distance separable code, a maximal shiftregister code, and a semiprimitive code. We tabulate some numbers Aμ which completely determine the weight distributionof any irreducible cyclic (n1(21 ? 1), k) code over GF(2) for all n1 ? 17. 相似文献
16.
D Zwick 《Journal of Mathematical Analysis and Applications》1984,104(2):435-436
For a(1) ? a(2) ? ··· ? a(n) ? 0, b(1) ? b(2) ? ··· ? b(n) ? 0, the ordered values of ai, bi, i = 1, 2,…, n, m fixed, m ? n, and p ? 1 it is shown that where is the integer such that and . The inequality is shown to be sharp. When p < 1 and a(i)'s are in increasing order then the inequality is reversed. 相似文献
17.
Daniel J. Madden 《Journal of Number Theory》1978,10(3):303-323
If k is a perfect field of characteristic p ≠ 0 and k(x) is the rational function field over k, it is possible to construct cyclic extensions Kn over k(x) such that [K : k(x)] = pn using the concept of Witt vectors. This is accomplished in the following way; if [β1, β2,…, βn] is a Witt vector over k(x) = K0, then the Witt equation generates a tower of extensions through where . In this paper, it is shown that there exists an alternate method of generating this tower which lends itself better for further constructions in Kn. This alternate generation has the form Ki = Ki?1(yi); yip ? yi = Bi, where, as a divisor in Ki?1, Bi has the form . In this form q is prime to Πpjλj and each λj is positive and prime to p. As an application of this, the alternate generation is used to construct a lower-triangular form of the Hasse-Witt matrix of such a field Kn over an algebraically closed field of constants. 相似文献
18.
A.M Fink 《Journal of Mathematical Analysis and Applications》1977,61(2):404-408
We show how inequalities of the type when F(0) = 0 can be used to find lower bounds of the first eigenvalue of the integral equation F(z) = λ ∝0ak(s, z)F(s) ds. 相似文献
19.
Woody Lichtenstein 《Journal of Functional Analysis》1979,34(3):433-455
For a symmetric space of compact type, the highest-weight vectors for representations of G occurring in become heavily concentrated near certain submanifolds of as the highest weight goes to infinity. This fact is applied to obtain estimates for the spectral measures of the operators qλ = PλqPλ, where is an orthogonal projection onto a G-irreducible summand, and q: G/K → is a continuous function acting on by multiplication. 相似文献
20.
Morris L Eaton 《Journal of multivariate analysis》1976,6(3):422-425
Let Σ be an n × n positive definite matrix with eigenvalues λ1 ≥ λ2 ≥ … ≥ λn > 0 and let M = {x, y | x?Rn, y?Rn, x ≠ 0, y ≠ 0, x′y = 0}. Then for x, y in M, we have that and the inequality is sharp. If is a partitioning of Σ, let θ1 be the largest canonical correlation coefficient. The above result yields . 相似文献