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Two theorems on the eigenvalues of differences of idempotent matrices determine the natural occupation numbers and orbitals of electronic detachment, attachment, or excitation that pertain to transitions between wavefunctions that each consist of a single Slater determinant. They are also applicable to spin density matrices associated with Slater determinants. When the ranks of the matrices differ, unit eigenvalues occur. In addition, there are ±w pairs of eigenvalues where |w| ≤ 1, whose values are related to overlaps, t, between the corresponding orbitals of Amos and Hall, and Lowdin by the formula w=±1−t212. Generalized overlap amplitudes, including Dyson orbitals and their probability factors, may be inferred from these eigenvalues, which provide numerical criteria for: classifying transitions according to the number of holes and particles in final states with respect to initial states, identifying the most important effects of orbital relaxation produced by self-consistent fields, and the analysis of Fukui functions. Two similar theorems that apply to sums of idempotent matrices regenerate formulae for the natural orbitals and occupation numbers of an unrestricted Slater determinant that were published first by Amos and Hall.

Let d ≥ 1. For any A ∈ Z d × d such that | det A | = 2 , we construct two families of Parseval wavelet frames with two generators. These generators have compact support, any desired number of vanishing moments, and any given degree of regularity. The first family is real valued while the second family is complex valued. To construct these families we use Daubechies low pass filters to obtain refinable functions, and adapt methods employed by Chui and He and Petukhov for dyadic dilations to this more general case. We also construct several families of Parseval wavelet frames with three generators having various symmetry properties. Our constructions are based on the same refinable functions and on techniques developed by Han and Mo and by Dong and Shen for the univariate case with dyadic dilations.

The line between the displayed formulas (16) and (17) was copied incorrectly from [41, Theorem 1].

Let $$A \in \mathbb {R}^{d \times d}$$ , $$d \ge 1$$ be a dilation matrix with integer entries and $$| \det A|=2$$ . We construct several families of compactly supported Parseval framelets associated to A having any desired number of vanishing moments. The first family has a single generator and its construction is based on refinable functions associated to Daubechies low pass filters and a theorem of Bownik. For the construction of the second family we adapt methods employed by Chui and He and Petukhov for dyadic dilations to any dilation matrix A. The third family of Parseval framelets has the additional property that we can find members of that family having any desired degree of regularity. The number of generators is $$2^d+d$$ and its construction involves some compactly supported refinable functions, the Oblique Extension Principle and a slight generalization of a theorem of Lai and Stockler. For the particular case $$d=2$$ and based on the previous construction, we present two families of compactly supported Parseval framelets with any desired number of vanishing moments and degree of regularity. None of these framelet families have been obtained by means of tensor products of lower-dimensional functions. One of the families has only two generators, whereas the other family has only three generators. Some of the generators associated with these constructions are even and therefore symmetric. All have even absolute values.

We construct several families of tight wavelet frames in L2(R2)L2(R2) associated to the quincunx matrix. A couple of those families has five generators. Moreover, we construct a family of tight wavelet frames with three generators. Finally, we show families with only two generators. The generators have compact support, any given degree of regularity, and any fixed number of vanishing moments. Our construction is made in Fourier space and involves some refinable functions, the Oblique Extension Principle and a slight generalization of a theorem of Lai and Stöckler. In addition, we will use well known results on construction of tight wavelet frames with two generators on RR with the dyadic dilation. The refinable functions we use are constructed from the Daubechies low pass filters and are compactly supported. The main difference between these families is that while the refinable functions associated to the five generators have many symmetries, the refinable functions used in the construction of the others families are merely even. The first author was partially supported by MEC/MICINN grant #MTM2011-27998 (Spain).

We investigate the properties of univariate MRA Riesz wavelets. In particular we obtain a generalization to semiorthogonal MRA wavelets of a well-known representation theorem for orthonormal MRA wavelets.

Borwein, Bojanov, Erdelyi, and others, have studieddensity and approximation properties of those Markov systems of differentiablefunctions on a closed interval [a,b], having the property thatthe system of derivatives is a Markov system on the open interval (a,b).In this paper we show that several of these results are valid for anyMarkov system of continuous functions on a closed interval.

We combine methods of linear algebra and analysis to obtain new results on splicing, domain extension, and integral representation of Tchebycheff and weak Tchebycheff systems.

Let m ∈ Z+ be given. For any e > 0 we construct a function f{e} having the following properties: (a) f{e} has support in [−e, 1 + e]. (b) f{e} ∈ Cm(−∞,∞). (c) If h denotes the Haar function and 0 < δ

Given integers b and d, with d>1 and |b|>1, we construct even nonseparable compactly supported refinable functions with dilation factor bb that generate multiresolution analyses on L2(Rd). These refinable functions are nonseparable, in the sense that they cannot be expressed as the product of two functions defined on lower dimensions. We use these scaling functions and a slight generalization of a theorem of Lai and Stöckler to construct smooth compactly supported tight framelets. Both the refinable functions and the framelets they generate can be made as smooth as desired. Estimates for the supports of these refinable functions and framelets, are given. The first author was partially supported by Spanish Science Ministry grant JC2010-0012.

We use maximum principles and classical estimates for the rate of convergence of orthogonal expansions to study the rate of convergence of spectral approximations to various initial-boundary value problems.

We study the existence, uniqueness and approximation properties of rational complex planar spline interpolants of order (3, 1). We also find sufficient conditions for such interpolants to be quasiregular and quasiconformal. Examples are given.

We demonstrate some new properties of Markov systems, involving generalized divided differences, relative differentiation, and weak nondegeneracy.

Let A be a set of real numbers, and let Y n : = { y 0 ,…, y n } be a Cebysev system on A . Assume, moreover, that if inf A or sup A belongs to A , then it is a point of accumulation of A at which all y j are continuous. We find necessary and sufficient conditions for the existence of a function y n + 1 such that also { y 0 ,…, y n , y n + 1 } is a Cebysev system on A . This theorem generalizes earlier results of Zielke and of the author. The proof is based on an integral representation of Markov systems that slightly extends a previous result of Zielke.

Orthogonal bases provide a classical method to represent an element of a Hilbert space in terms of simpler ones. A Riesz basis is the image of an orthogonal basis under an invertible continuous linear mapping. In practical applications, when using orthogonal bases one introduces small truncation errors. As we shall see in the sequel, perturbing a Riesz bases (in particular, an orthogonal basis) by a small amount, yields another Riesz basis. This is one of the reasons that motivate the study of such bases.

We construct smooth nonseparable compactly supported refinable functions that generate multiresolution analyses on L2(ℝd), d > 1. Using these refinable functions we construct smooth nonseparable compactly supported orthonormal wavelet systems. These systems are nonseparable, in the sense that none of its constituent functions can be expressed as the product of two functions defined on lower dimensions. Both the refinable functions and the wavelets can be made as smooth as desired. Estimates for the supports of these scaling functions and wavelets, are given.

This paper contains a brief introduction to the subject followed by a survey, with a few improvements and minor corrections, of some of the recent advances in the theory. In addition, a number of open problems are discussed.

Page Line Where it says Should say 42 8 f (x) := ∫ Rd e−i2π t·x f (t)dt f (x) := ∫ Rd e−i2πx·t f (t)dt 43 2 ψ(2x)= ei2πxμ(2x)p(2πx+ 1/2)φ(x) ψ(2x)= ei2πxμ(2x)p(x+ 1/2)φ(x) 43 17, 18 called Riesz sequence called a Riesz sequence 44 −9 L2(Rd L2(Rd) 45 −17 It follows that It follows from [25, Theorem 2.3.6] that 46 4 D1 = D1 (t)= 47 14 [1,n] × J I(n)× J 47 −21, −20 k is uniquely determined by j and r k uniquely determines j and r 47 −13 Theorem 3 yields Theorem 1(c) and Theorem 3 yield 49 19 T (u) S(u) 50 12 Note that Note that, except for a special case 51 −11 V j = r< j S(Ar;ψ) V j = ∑ r< j S(A r;ψ) 53 15 orthonormal basis orthonormal basis generator 53 −6 V(x) v(x) 54 4 V (x) := (v ,1(x), . . . , v ,|a|(x)) v (x) := (v ,1(x), . . . , v ,|a|(x)) 54 6 V (x) v (x) 54 −21 {μ1, . . . ,μn} {μ1, . . . ,μm} 54 −13 {μ1, . . . ,μn} {μ1, . . . ,μm} On page 49 line 1 and line −20, the expression “ψ = {ψ1, . . . ,ψr}” is redundant and should be deleted. On page 53 line −14, where it says V(x) := v1,1(x), . . . , v1,|a|(x), . . . , vn,1(x), . . . , vn,|a|(x) )T

For functions f(t) satisfying certain growth conditions, we consider a sequence of the form {ff(c t)), nonfundamental in L2(R), and find a representation for those functions which are in the closure of its linear span. Some theorems concerning degree of approximation are also proved. In [1], we found necessary and sufficient conditions for a sequence of the form {f(cn t)} to be fundamental in L2(R). In this paper, motivated by earlier research of L. Schwartz [2], and I. I. Hirschmann, Jr. [3] (see also J. Korevaar [4], W. A. J. Luxemburg and J. Korevaar [5, p. 35, Theorem 8.2], and Clarkson and Erdos [6]), we consider the nonfundamental case and find a representation of those functions which are in the L2(R) closure of the linear span of {f(cn t)}. Our result applies to a different class of functions than those considered by the above mentioned authors. The techniques developed to attack this problem are also applied to find a lower bound for the L2(R) distance from f(c t) to the linear span of {ff(cr -t); r = 0, . . . , n}, obtaining a result similar to [5, p. 31, Theorem 7.1], [4, p. 363, Theorem 4], or [6, p. 6, Theorem 2]. Finally, we also prove a Jackson type theorem valid for a class of continuous functions defined on a bounded interval. In what follows, { d, } will be a sequence of distinct real numbers, satisfying the following conditions: Ic2cr2I > pin rI (p > 0) and 'IcnK2 K 00 (1) (By , IC",K-2 we denote the sum of all terms of the form indicated, with nonvanishing denominator.) Note that (1) is satisfied if, for instance ICn c, ,I > p Ic. I (P > V-). Given a function f(t), by F(t) we shall denote its Fourier transform. Thus F(t) = (27) 1/21 exp(xti)f(t) dt. We shall assume that there are strictly positive numbers a, a and b, such that for t real, f(t) = O[exp(-at2)], F(t) = O[exp(-at2)], t -* oo, and exp(-bt2)/F(t) is in L2(R). By a theorem of Babenko, later generalized by Gel'fand and Silov, we know that the growth condition on f(t) can be replaced by the assumption that F is an Received by the editors January 19, 1979. AMS (MOS) subject classifications (1970). Primary 30A62, 41A30, 42A64; Secondary 42A68.

= O,..., n, then Z, will be called a (weak) Markov system. A normed (weak) Markov system is a (weak) Markov system Z, for which z0 = 1. Markov systems are also called complete CebySev systems or CT-systems (cf. Karlin & Studden [2]). If every element of Z, is bounded in the intersection of

The solutions of the Jeffcott equations describing the behavior of a rotating shaft are investigated analytically, with a focus on the case where deadband is taken into account. Bounds on the solutions are obtained from those for the linearized equations, and the onset of destructive vibrations is predicted by analyzing the Fourier transforms of the solutions; good agreement with numerical solutions and power-spectrum density plots is demonstrated. It is suggested that the present analytical approach could be applied to determine cryogenic-pump stability margins in flight and hot-fire ground testing of launch vehicles such as the Space Shuttle.

We study the closure properties in various spaces, of certain sequences of the form { f ( t + c k )} or {exp( c k ti ) f ( t )}. As an application, we give another proof of the Muntz-Szasz theorem for L 2 (0, 1).

If ƒ(z) is an entire function of order larger than one that satisfies certain additional conditions, characterizations of the growth of its Fourier transform are given, in terms of the growth of ƒ(z) . The results obtained are extensions of theorems of Paley and Wiener, and are a refinement of prior work of Gel'fand and Silov.

Muntz-type theorems are proven for sequences of the form w ( t ) t α k defined on unbounded intervals.

We find Müntz-type theorems for sequences of the form { f ( t + c n ) } \{ f(t + {c_n})\} or { exp ⁡ ( − c n t ) f ( t ) } \{ \exp ( - {c_n}t)f(t)\} on [ 0 , ∞ ) [0,\infty ) .

Conditions de plongement d'une fonction continue dans un espace de Haar de fonctions continues

Let q,(x) be a polynomial of degree n. The well-known inequality of Markov states that IIqxxllr,,-*,lI

We study the completeness in L 2 ( R ) {L_2}(R) of sequences of the form { f ( c n − t ) } \{ f({c_n} - t)\} , where { c n } \{ {c_n}\} is a sequence of distinct real numbers. A Müntztype theorem is proved, valid for a large class of functions and, in particular, for f ( t ) = exp ⁡ ( − t 2 ) f(t) = \exp ( - {t^2}) .

On etudie le probleme de l'extension du domaine de definition des espaces de Haar (faibles). On donne des caracterisations equivalentes de l'extension

We present a concise and elementary proof of a theorem of Karlin and Studden concerning the smoothness properties of functions belonging to a generalized convexity cone. In [1, Chapter XI], Karlin and Studden showed that a function which is convex with respect to an extended complete Tchebycheff system has a continuous derivative of order n 1, a fact which is of considerable importance in the theory of generalized convexity. Since their original proof is rather technical, we present here an alternative one which has the advantage of being very concise and elementary. Given a function y y(k) will denote its convolution with the Gauss kernel, i.e. Jb y(k)(t) a y(s)Gk(t s)ds, where Gk(s) = (k//2w)exp(-?2k2 a2). The set of functions convex with respect to the system { Yo, .. ,yn} will be denoted by C(y0,...,yn). The abbreviations T, CT, WT and ECT will respectively stand for Tchebycheff, complete Tchebycheff, weak Tchebycheff and extended complete Tchebycheff. For the definition of other terms and symbols employed, the reader is referred to the monograph by Karlin and Studden [1, Chapters I and XI]. LEMMA 1. Let { yi}n 0 be a system of continuous functions of bounded variation on an interval [a, b] such that yo 1, and { y .}7= is a WT-system thereon for r = 1, . . ., n. If P is the set of points of (a, b) at which all the functions yi are differentiable, the system { Y7ni I is a WT-system on P. PROOF. If the functions { jy1}=0 are linearly dependent, the assertion is obvious. Otherwise, from the basic composition formula (cf. [1, pp. 14, 15]), we know that {y [k)}"0 is an ECT-system for any natural number k > 0. From [1, Chapter XI, Theorem 1.2 and Remark 1.2], we conclude that also the reduced system {[flk)/y'k)] }=I is an ECT-system. However, bearing in mind that Yo = 1, from [2, Chapter X, Exercise 9], we conclude that lim1k [y(k)(t) /yk)(t)]' = y'(t) on P, for i = 1, ..., n, whence the conclusion follows. Q.E.D. REMARK. Lemma 1 furnishes a much shorter proof of [3, Theorem 3] in a more general framework. Received by the editors February 20, 1975. AMS (MOS) subject classifications (1970). Primary 26A51; Secondary 41A50. X) American Mathematical Society 1976

We prove that, if f ( z ) is an entire function and ¦f(z)¦ ⩽ (A 1 + A 2 ¦z¦ n ) exp [ax 2 + by 2 + cx + dy] , then there are numbers C 1 , C 2 ⩾ 0, depending only on n , A 1 , A 2 , a , b , c , and d such that ¦f′(z)¦ ⩽ (C 1 + C 2 ¦z¦ n + 1 ) exp (ax 2 + by 2 + cx + dy) .