Generalization of the Hausdorff Moment Problem

1981 ◽  
Vol 33 (4) ◽  
pp. 946-960 ◽  
Author(s):  
David Borwein ◽  
Amnon Jakimovski
Keyword(s):  
Moment Problem ◽  
Sufficient Conditions ◽  
Natural Generalization ◽  
Necessary And Sufficient Conditions ◽  
Real Numbers ◽  
Image Position ◽  
Necessary And Sufficient ◽  
Hausdorff Moment Problem ◽  
Positive Integers ◽  
Classical Moment Problem

Suppose throughout that {kn} is a sequence of positive integers, thatthat k0 = 1 if l0 = 1, and that {un(r)}; (r = 0, 1, …, kn – 1, n = 0, 1, …) is a sequence of real numbers. We shall be concerned with the problem of establishing necessary and sufficient conditions for there to be a function a satisfying(1)and certain additional conditions. The case l0 = 0, kn = 1 for n = 0, 1, … of the problem is the version of the classical moment problem considered originally by Hausdorff [5], [6], [7]; the above formulation will emerge as a natural generalization thereof.

scholarly journals The Hausdorff Moment Problem

1978 ◽  
Vol 21 (3) ◽  
pp. 257-265
Author(s):  
David Borwein
Keyword(s):  
Moment Problem ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Real Numbers ◽  
Image Position ◽  
Necessary And Sufficient ◽  
Hausdorff Moment Problem

Suppose throughout thatand that {μn}(n≥ 0) is a sequence of real numbers. The (generalized) Hausdorff moment problem is to determine necessary and sufficient conditions for there to be a function x in some specified class satisfying.

On the Hausdorff and Trigonometric Moment Problems

1961 ◽  
Vol 13 ◽  
pp. 454-461
Author(s):  
P. G. Rooney
Keyword(s):  
Moment Problem ◽  
Closed Interval ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Functions Of Bounded Variation ◽  
Moment Problems ◽  
Image Position ◽  
Necessary And Sufficient ◽  
Hausdorff Moment Problem

Let K be a subset of BV(0, 1)—the space of functions of bounded variation on the closed interval [0, 1]. By the Hausdorff moment problem for K we shall mean the determination of necessary and sufficient conditions that corresponding to a given sequence μ = {μn|n = 0, 1, 2, …} there should be a function α ∈ K so that(1)For various collections K this problem has been solved—see (3, Chapter III)By the trigonometric moment problem for K we shall mean the determination of necessary and sufficient conditions that corresponding to a sequence c = {cn|n = 0, ± 1, ± 2, …} there should be a function α ∈ K so that(2)For various collections K this problem has also been solved—see, for example (4, Chapter IV, § 4). It is noteworthy that these two problems have been solved for essentially the same collections K.

On a certain kind of reducible rational fractions

1980 ◽  
Vol 21 (3) ◽  
pp. 321-328
Author(s):  
Mordechai Lewin
Keyword(s):  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Rational Fraction ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Image Position ◽  
Necessary And Sufficient ◽  
Positive Coefficients ◽  
Positive Integers ◽  
Exact Positions

The rational fractiona, c, p, q positive integers, reduces to a polynomial under conditions specified in a result of Grosswald who also stated necessary and sufficient conditions for all the coefficients to tie nonnegative.This last result is given a different proof using lemmas interesting in themselves.The method of proof is used in order to give necessary and sufficient conditions for the positive coefficients to be equal to one. For a < 2pq, a = αp + βq, α, β nonnegative integers, c > 1, the exact positions of the nonzero coefficients are established. Also a necessary and sufficient condition for the number of vanishing coefficients to be minimal is given.

On the Fourier Constants of a Bounded Function

1936 ◽  
Vol 32 (2) ◽  
pp. 201-211 ◽  
Author(s):  
S. Verblunsky
Keyword(s):  
Bounded Function ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Real Numbers ◽  
Image Position ◽  
Necessary And Sufficient

1. In a former paper we solved the following problem (Problem I). Given k + 1 real numbers co, …, ck, to find necessary and sufficient conditions that there shall exist a function f(x) in (0, 1) which satisfies the conditions

Joins of paths and cycles of equal order with sigma chromatic number 2

2021 ◽  
pp. 2250006
Author(s):  
Agnes D. Garciano ◽  
Maria Czarina T. Lagura ◽  
Reginaldo M. Marcelo
Keyword(s):  
Chromatic Number ◽  
Connected Graph ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Odd Cycle ◽  
Necessary And Sufficient ◽  
Positive Integers ◽  
Simple Connected Graph

For a simple connected graph [Formula: see text] let [Formula: see text] be a coloring of [Formula: see text] where two adjacent vertices may be assigned the same color. Let [Formula: see text] be the sum of colors of neighbors of any vertex [Formula: see text] The coloring [Formula: see text] is a sigma coloring of [Formula: see text] if for any two adjacent vertices [Formula: see text] [Formula: see text] The least number of colors required in a sigma coloring of [Formula: see text] is the sigma chromatic number of [Formula: see text] and is denoted by [Formula: see text] A sigma coloring of a graph is a neighbor-distinguishing type of coloring and it is known that the sigma chromatic number of a graph is bounded above by its chromatic number. It is also known that for a path [Formula: see text] and a cycle [Formula: see text] where [Formula: see text] [Formula: see text] and [Formula: see text] if [Formula: see text] is even. Let [Formula: see text] the join of the graphs [Formula: see text], where [Formula: see text] or [Formula: see text] [Formula: see text] and [Formula: see text] is not an odd cycle for any [Formula: see text]. It has been shown that if [Formula: see text] for [Formula: see text] and [Formula: see text] then [Formula: see text]. In this study, we give necessary and sufficient conditions under which [Formula: see text] where [Formula: see text] is the join of copies of [Formula: see text] and/or [Formula: see text] for the same value of [Formula: see text]. Let [Formula: see text] and [Formula: see text] be positive integers with [Formula: see text] and [Formula: see text] In this paper, we show that [Formula: see text] if and only if [Formula: see text] or [Formula: see text] is odd, [Formula: see text] is even and [Formula: see text]; and [Formula: see text] if and only if [Formula: see text] is even and [Formula: see text] We also obtain necessary and sufficient conditions on [Formula: see text] and [Formula: see text], so that [Formula: see text] for [Formula: see text] where [Formula: see text] or [Formula: see text] other than the cases [Formula: see text] and [Formula: see text]

scholarly journals Characterizations and Identities for Isosceles Triangular Numbers

2021 ◽  
Vol 14 (2) ◽  
pp. 380-395
Author(s):  
Jiramate Punpim ◽  
Somphong Jitman
Keyword(s):  
Positive Integer ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Triangular Numbers ◽  
Triangular Number ◽  
Recursive Formulas ◽  
Necessary And Sufficient ◽  
Positive Integers ◽  
Special Case

Triangular numbers have been of interest and continuously studied due to their beautiful representations, nice properties, and various links with other figurate numbers. For positive integers n and l, the nth l-isosceles triangular number is a generalization of triangular numbers defined to be the arithmetic sum of the formT(n, l) = 1 + (1 + l) + (1 + 2l) + · · · + (1 + (n − 1)l).In this paper, we focus on characterizations and identities for isosceles triangular numbers as well as their links with other figurate numbers. Recursive formulas for constructions of isosceles triangular numbers are given together with necessary and sufficient conditions for a positive integer to be a sum of isosceles triangular  numbers. Various identities for isosceles triangular numbers are established. Results on triangular numbers can be viewed as a special case.

scholarly journals An extended Hamburger moment problem

1985 ◽  
Vol 28 (2) ◽  
pp. 167-183 ◽  
Author(s):  
Olav Njåstad
Keyword(s):  
Distribution Function ◽  
Moment Problem ◽  
Sufficient Conditions ◽  
Positive Definite ◽  
Moment Problems ◽  
Real Numbers ◽  
Orthogonal Functions ◽  
Hamburger Moment Problem ◽  
Hankel Determinants ◽  
Necessary And Sufficient

The classical Hamburger moment problem can be formulated as follows: Given a sequence {cn:n=0,1,2,…} of real numbers, find necessary and sufficient conditions for the existence of a distribution function ψ (i.e. a bounded, real-valued, non-decreasing function) on (– ∞,∞) with infinitely many points of increase, such that , n = 0,1,2, … This problem was posed and solved by Hamburger [5] in 1921. The corresponding problem for functions ψ on the interval [0,∞) had already been treated by Stieltjes [15] in 1894. The characterizations were in terms of positivity of Hankel determinants associated with the sequence {cn}, and the original proofs rested on the theory of continued fractions. Much work has since been done on questions connected with these problems, using orthogonal functions and extension of positive definite functionals associated with the sequence. Accounts of the classical moment problems with later developments can be found in [1,4,14]. Good modern accounts of the theory of orthogonal polynomials can be found in [2,3].

Necessary and sufficient conditions for asymptotic decay of oscillations in delayed functional equations

1981 ◽  
Vol 91 (1-2) ◽  
pp. 135-145
Author(s):  
Lu-San Chen ◽  
Cheh-Chih Yeh
Keyword(s):  
Differential Operator ◽  
Functional Equations ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Oscillatory Solutions ◽  
Asymptotic Decay ◽  
Image Position ◽  
Necessary And Sufficient

SynopsisThis paper studies the equationwhere the differential operator Ln is defined byand a necessary and sufficient condition that all oscillatory solutions of the above equation converge to zero asymptotically is presented. The results obtained extend and improve previous ones of Kusano and Onose, and Singh, even in the usual case wherewhere N is an integer with l≦N≦n–1.

Coefficient Regions for Univalent Trinomials

1980 ◽  
Vol 32 (1) ◽  
pp. 1-20 ◽  
Author(s):  
Q. I. Rahman ◽  
J. Waniurski
Keyword(s):  
Unit Disk ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Image Position ◽  
Necessary And Sufficient

The problem of determining necessary and sufficient conditions bearing upon the numbers a2 and a3 in order that the polynomial z + a2z2 + a3z3 be univalent in the unit disk |z| < 1 was solved by Brannan ([3], [4]) and by Cowling and Royster [6], at about the same time. For his investigation Brannan used the following result due to Dieudonné [7] and the well-known Cohn rule [9].THEOREM A (Dieudonné criterion). The polynomial1is univalent in |z| < 1 if and only if for every Θ in [0, π/2] the associated polynomial2does not vanish in |z| < 1. For Θ = 0, (2) is to be interpreted as the derivative of (1).The procedure of Cowling and Royster was based on the observation that is univalent in |z| < 1 if and only if for all α such that 0 ≧ |α| ≧ 1, α ≠ 1 the functionis regular in the unit disk.

Traces of Matrices of Zeros and Ones

1960 ◽  
Vol 12 ◽  
pp. 463-476 ◽  
Author(s):  
H. J. Ryser
Keyword(s):  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Critical Survey ◽  
Combinatorial Properties ◽  
Image Position ◽  
Necessary And Sufficient ◽  
Sum Vector

This paper continues the study appearing in (9) and (10) of the combinatorial properties of a matrix A of m rows and n columns, all of whose entries are 0's and l's. Let the sum of row i of A be denoted by ri and let the sum of column j of A be denoted by Sj. We call R = (r1, … , rm) the row sum vector and S = (s1 . . , sn) the column sum vector of A. The vectors R and S determine a class1.1consisting of all (0, 1)-matrices of m rows and n columns, with row sum vector R and column sum vector S. The majorization concept yields simple necessary and sufficient conditions on R and S in order that the class 21 be non-empty (4; 9). Generalizations of this result and a critical survey of a wide variety of related problems are available in (6).

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