On Uniqueness Sets for Expansions in Sequences of Functions Arising from Singular Generating Functions

1981 ◽  
Vol 33 (4) ◽  
pp. 803-816
Author(s):  
Jet Wimp
Keyword(s):  
Generating Functions ◽  
Trivial Representation ◽  
Linearly Independent ◽  
Image Position ◽  
Uniqueness Sets ◽  
Uniqueness Set ◽  
Sequence Of Functions

Let {pn(z)}; be a sequence of functions analytic in a region D. A problem in analysis which has received much attention is the following: describe the sets Z ⊂ D for which(1)implies hn is 0 for all n, (To make the problem interesting, only those situations are studied where finite subsets of the pn(z) are linearly independent in D.) Another way of phrasing this is: Characterize the uniqueness sets of pn(z), a uniqueness set Z being a set in D such that the restriction of {pn(z)}; to Z is linearly independent. If Z is not a uniqueness set then for some {hn}; not all 0, we have(2)This formula is called a non-trivial representation of 0 (on Z).

scholarly journals On Generating Functions

1930 ◽  
Vol 2 (2) ◽  
pp. 71-82 ◽  
Author(s):  
W. L. Ferrar
Keyword(s):  
Differential Equation ◽  
Recurrence Relation ◽  
Generating Function ◽  
Generating Functions ◽  
Order Differential Equation ◽  
Second Order ◽  
Second Order Differential Equation ◽  
Image Position ◽  
Three Term Recurrence Relation ◽  
Sequence Of Functions

It is well known that the polynomial in x,has the following properties:—(A) it is the coefficient of tn in the expansion of (1–2xt+t2)–½;(B) it satisfies the three-term recurrence relation(C) it is the solution of the second order differential equation(D) the sequence Pn(x) is orthogonal for the interval (— 1, 1),i.e. whenSeveral other familiar polynomials, e.g., those of Laguerre Hermite, Tschebyscheff, have properties similar to some or all of the above. The aim of the present paper is to examine whether, given a sequence of functions (polynomials or not) which has one of these properties, the others follow from it : in other words we propose to examine the inter-relation of the four properties. Actually we relate each property to the generating function.

On calculating extinction probabilities for branching processes in random environments

1969 ◽  
Vol 6 (03) ◽  
pp. 478-492 ◽  
Author(s):  
William E. Wilkinson
Keyword(s):  
Generating Functions ◽  
Infinite Sequence ◽  
Branching Processes ◽  
Transition Probability ◽  
Transition Probability Matrix ◽  
Random Environments ◽  
Real Numbers ◽  
Discrete Time Markov Chain ◽  
Extinction Probabilities ◽  
Image Position

Consider a discrete time Markov chain {Zn } whose state space is the non-negative integers and whose transition probability matrix ║Pij ║ possesses the representation where {Pr }, r = 1,2,…, is a finite or denumerably infinite sequence of non-negative real numbers satisfying , and , is a corresponding sequence of probability generating functions. It is assumed that Z 0 = k, a finite positive integer.

The Generating Functions of Certain Special Determinants

1904 ◽  
Vol 24 ◽  
pp. 387-392 ◽  
Author(s):  
Thomas Muir
Keyword(s):  
Generating Functions ◽  
General Theorem ◽  
Image Position

(1) From a general theorem, known since 1855, and perhaps earlier, regarding the reciprocal of the seriesit follows thatwhere β0 = 1 and

An Extremum Result

1962 ◽  
Vol 14 ◽  
pp. 597-601 ◽  
Author(s):  
J. Kiefer
Keyword(s):  
Probability Measure ◽  
Compact Space ◽  
Ad Hoc ◽  
Continuous Functions ◽  
Orthogonality Condition ◽  
Technical Detail ◽  
Real Numbers ◽  
Discrete Probability ◽  
Linearly Independent ◽  
Image Position

The main object of this paper is to prove the following:Theorem. Let f1, … ,fk be linearly independent continuous functions on a compact space. Then for 1 ≤ s ≤ k there exist real numbers aij, 1 ≤ i ≤ s, 1 ≤ j ≤ k, with {aij, 1 ≤ i, j ≤ s} n-singular, and a discrete probability measure ε*on, such that(a) the functions gi = Σj=1kaijfj 1 ≤ i ≤ s, are orthonormal (ε*) to the fj for s < j ≤ k;(b)The result in the case s = k was first proved in (2). The result when s < k, which because of the orthogonality condition of (a) is more general than that when s = k, was proved in (1) under a restriction which will be discussed in § 3. The present proof does not require this ad hoc restriction, and is more direct in approach than the method of (2) (although involving as much technical detail as the latter in the case when the latter applies).

A Note on Coverings of En by Convex Sets

1967 ◽  
Vol 10 (5) ◽  
pp. 669-673 ◽  
Author(s):  
J.H.H. Chalk
Keyword(s):  
Convex Sets ◽  
Independent Set ◽  
Linearly Independent ◽  
Image Position ◽  
Take All

Let (x1, x2, …, xn) denote the coordinates of a point of Euclidean n-space En. Let be a set of n+1 points of En with the property thatform a linearly independent set and define a lattice Λ of pointsby allowing u1, …, un to take all integer values.

26.—The Strong Limit-2 Case of Fourth-order Differential Equations

1974 ◽  
Vol 71 (4) ◽  
pp. 297-304
Author(s):  
V. Krishna Kumar
Keyword(s):  
Differential Equation ◽  
Fourth Order ◽  
Order Equation ◽  
Complex Numbers ◽  
Strong Limit ◽  
Fourth Order Equation ◽  
Linearly Independent ◽  
Image Position ◽  
Adjoint Operators ◽  
Bounded Below

SynopsisThe fourth-order equation considered isConditions are given on the coefficients r, p and q which ensure that this differential equation (*) is in the strong limit-2 case at ∞, i.e. is limit-2 at ∞. This implies that (*) has exactly two linearly independent solutions which are in the integrable-square space ℒ2(0, ∞) for all complex numbers λ with im [λ] ≠ 0. Additionally the conditions imply that self-adjoint operators generated by M[·] in ℒ2(0, ∞) are semi-bounded below. The results obtained are applied to the case when the coefficients r, p and q are powers of x ∈ [0, ∞).

A direct derivation of the Catalan formula

2013 ◽  
Vol 97 (538) ◽  
pp. 53-60 ◽  
Author(s):  
Gerry Leversha
Keyword(s):  
Generating Functions ◽  
Recurrence Relations ◽  
Reflection Principle ◽  
Catalan Numbers ◽  
Equal Size ◽  
Alternative Form ◽  
Direct Derivation ◽  
Image Position

Many readers will be familiar with the sequence of Catalan numbers {Cn: n ≥ 0} and the formulawith its alternative formThese can be proved by using recurrence relations, generating functions or André's reflection principle. A good reference for all of these methods is Martin Griffiths' book [1].However, none of these approaches strikes me as being naturally combinatorial. A formula such as (1) is often derived by making a list of all the ways of doing something, and then subdividing this list into classes of equal size, so that either one class consists entirely of ‘valid’ cases or there is exactly one ‘valid’ case in each list.

The Schwarzian Derivative and Disconjugacy of nth order Linear Differential Equations

1969 ◽  
Vol 21 ◽  
pp. 235-249 ◽  
Author(s):  
Meira Lavie
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Schwarzian Derivative ◽  
Order Linear ◽  
Second Order Differential Equations ◽  
Number Of Zeros ◽  
Linearly Independent ◽  
Basic Relationship ◽  
Linear Differential ◽  
Image Position

In this paper we deal with the number of zeros of a solution of the nth order linear differential equation1.1where the functions pj(z) (j = 0, 1, …, n – 2) are assumed to be regular in a given domain D of the complex plane. The differential equation (1.1) is called disconjugate in D, if no (non-trivial) solution of (1.1) has more than (n – 1) zeros in D. (The zeros are counted by their multiplicity.)The ideas of this paper are related to those of Nehari (7; 9) on second order differential equations. In (7), he pointed out the following basic relationship. The function1.2where y1(z) and y2(z) are two linearly independent solutions of1.3is univalent in D, if and only if no solution of equation(1.3) has more than one zero in D, i.e., if and only if(1.3) is disconjugate in D.

scholarly journals Exhaustive operators and vector measures

1975 ◽  
Vol 19 (3) ◽  
pp. 291-300 ◽  
Author(s):  
N. J. Kalton
Keyword(s):  
Linear Operator ◽  
Vector Space ◽  
Compact Hausdorff Space ◽  
Hausdorff Space ◽  
Continuous Functions ◽  
Vector Measures ◽  
Borel Functions ◽  
Image Position ◽  
Real Topological Vector Space ◽  
Sequence Of Functions

Let S be a compact Hausdorff space and let Φ: C(S)→E be a linear operator defined on the space of real-valued continuous functions on S and taking values in a (real) topological vector space E. Then Φ is called exhaustive (7) if given any sequence of functions fn ∈ C(S) such that fn ≧ 0 andthen Φ(fn)→0 If E is complete then it was shown in (7) that exhaustive maps are precisely those which possess regular integral extensions to the space of bounded Borel functions on S; this is equivalent to possessing a representationwhere μ is a regular countably additive E-valued measure defined on the σ-algebra of Borel subsets of S.

Generating Functions for a Class of Arithmetic Functions

1966 ◽  
Vol 9 (4) ◽  
pp. 427-431 ◽  
Author(s):  
A. A. Gioia ◽  
M.V. Subbarao
Keyword(s):  
Generating Functions ◽  
Multiplicative Function ◽  
Arithmetic Function ◽  
Arithmetic Functions ◽  
Prime Divisors ◽  
Image Position

In this note the arithmetic functions L(n) and w(n) denote respectively the number and product of the distinct prime divisors of the integer n ≥ 1, and L(l) = 0, w(l) = 1. An arithmetic function f is called multiplicative if f(1) = 1 and f(mn) = f(m)f(n) whenever (m, n) = 1. It is known ([1], [3], [4]) that every multiplicative function f satisfies the identity1.1

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