The Structure of the Algebra of Hankel Transforms and the Algebra of Hankel-Stieltjes Transforms

1971 ◽  
Vol 23 (2) ◽  
pp. 236-246 ◽  
Author(s):  
Alan Schwartz
Keyword(s):  
Banach Space ◽  
Entire Function ◽  
Power Series ◽  
Real Number ◽  
Stieltjes Transform ◽  
Borel Measures ◽  
Definition Of ◽  
Image Position ◽  
Complex Valued ◽  
Stieltjes Transforms

Let M be the space of all bounded regular complex-valued Borel measures defined on I = [0, ∞). M is a Banach space with ‖μ‖ = ∫d|μ|(x) (μ ∈ M). (Integrals in this paper extend over all of I unless otherwise specified.) Let v be a fixed real number no smaller than and let if z ≠ 0 and , where Jv, is the Bessel function of the first kind of order v and cv =[2vΓ(v + 1)]–1; is an entire function, as can be seen from the power series definition ofThe Hankel-Stieltjes transform of order v is given by .

scholarly journals A Theorem on Algebras of Measures on Topological Groups

1959 ◽  
Vol 11 (4) ◽  
pp. 195-206 ◽  
Author(s):  
J. H. Williamson
Keyword(s):  
Banach Space ◽  
Abelian Group ◽  
Compact Abelian Group ◽  
Locally Compact Abelian Group ◽  
Topological Groups ◽  
Finite Sets ◽  
Locally Compact ◽  
Borel Measures ◽  
Bounded Complex ◽  
Image Position

Let G be a locally compact Abelian group, and the set of bounded complex (regular countably-additive Borel) measures on G. It is well known that becomes a Banach space if the norm is defined bythe supremum being over all finite sets of disjoint Borel subsets of G.

Compound Invariants and Mixed F-, DF-Power Spaces

1998 ◽  
Vol 50 (6) ◽  
pp. 1138-1162 ◽  
Author(s):  
P. A. Chalov ◽  
T. Terzioğlu ◽  
V. P. Zahariuta
Keyword(s):  
Power Series ◽  
Tensor Products ◽  
Main Tool ◽  
Proper Definition ◽  
Definition Of ◽  
Image Position ◽  
Isomorphic Classification

AbstractThe problems on isomorphic classification and quasiequivalence of bases are studied for the class of mixed F-, DF-power series spaces, i.e. the spaces of the following kind where ai (p, q) = exp((p - λiq)ai), p,q ∈ ℕ, and λ = (λi)i∈ℕ, a = (ai)i∈ℕ are some sequences of positive numbers. These spaces, up to isomorphisms, are basis subspaces of tensor products of power series spaces of F- and DF-types, respectively. The mrectangle characteristic of the space G(λ a) is defined as the number of members of the sequence (ïiÒ ai)i2N which are contained in the union of m rectangles Pk = (δk, εk] ✗ (τk, tk], k = 1, 2 , . . . , m. It is shown that each m-rectangle characteristic is an invariant on the considered class under some proper definition of an equivalency relation. The main tool are new compound invariants, which combine some version of the classical approximative dimensions (Kolmogorov, Pełczynski) with appropriate geometrical and interpolational operations under neighborhoods of the origin (taken from a given basis).

scholarly journals CONDITIONAL FEYNMAN INTEGRAL AND SCHRÖDINGER INTEGRAL EQUATION ON A FUNCTION SPACE

2009 ◽  
Vol 79 (1) ◽  
pp. 1-22 ◽  
Author(s):  
DONG HYUN CHO
Keyword(s):  
Integral Equation ◽  
Function Space ◽  
Feynman Integral ◽  
Borel Measures ◽  
Analytic Feynman Integral ◽  
Conditional Expectations ◽  
Image Position ◽  
Vector Valued ◽  
Complex Borel Measures ◽  
Stieltjes Transforms

AbstractLet Cr[0,t] be the function space of the vector-valued continuous paths x:[0,t]→ℝr and define Xt:Cr[0,t]→ℝ(n+1)r by Xt(x)=(x(0),x(t1),…,x(tn)), where 0<t1<⋯<tn=t. In this paper, using a simple formula for the conditional expectations of the functions on Cr[0,t] given Xt, we evaluate the conditional analytic Feynman integral Eanfq[Ft∣Xt] of Ft given by where θ(s,⋅) are the Fourier–Stieltjes transforms of the complex Borel measures on ℝr, and provide an inversion formula for Eanfq[Ft∣Xt]. Then we present an existence theorem for the solution of an integral equation including the integral equation which is formally equivalent to the Schrödinger differential equation. We show that the solution can be expressed by Eanfq[Ft∣Xt] and a probability distribution on ℝr when Xt(x)=(x(0),x(t)).

scholarly journals Complex inversion theorems for generalized Stieltjes transforms

1974 ◽  
Vol 18 (3) ◽  
pp. 328-358 ◽  
Author(s):  
Angelina Byrne ◽  
E. R. Love
Keyword(s):  
Complex Number ◽  
Complex Numbers ◽  
Stieltjes Transform ◽  
Image Position ◽  
Stieltjes Transforms

In this paper we seek to establish some “complex inversion formulae” for the generalized Stieltjes transform for all s in the cut plane, supposing that p is any complex number except zero and the negative integers. The “cut plane” means all complex numbers except those which are negative real or zero.

On Generalized Powers of the Difference Operator

1965 ◽  
Vol 17 ◽  
pp. 124-129
Author(s):  
K. R. Unni
Keyword(s):  
Entire Function ◽  
Exponential Type ◽  
Difference Operator ◽  
Alternative Definition ◽  
Averaging Operator ◽  
The Difference ◽  
Definition Of ◽  
Image Position

Sumner (3) discussed for arbitrary real λ and h, where the averaging operator ∇h is defined by(1.1)when f(z) is an entire function of exponential type <2π/|h|. Boas (2) gave an alternative definition of ∇h which gave Sumner's results quickly and showed that his definition is equivalent to that of Sumner.

A finite set of generators for the homeotopy group of a 2-manifold

1964 ◽  
Vol 60 (4) ◽  
pp. 769-778 ◽  
Author(s):  
W. B. R. Lickorish
Keyword(s):  
Real Number ◽  
Euclidean Plane ◽  
Piecewise Linear ◽  
Linear Homeomorphism ◽  
Finite Set ◽  
Definition Of ◽  
Image Position

The homeotopy group Λx of a space X is the group of all homeomorphisms of X to itself, modulo the subgroup of those homeomorphisms that are isotopic to the identity. In this paper X will be taken to be a closed oriented 2-manifold, together with a polyhedral structure, and the definition of Λx is then restricted to the consideration of piecewise-linear homeomorphisms and isotopies. Although this restriction to the polyhedral category is not really essential to what follows, it does tend to simplify some of the arguments. In (2) a homeomorphism of X was associated with every simple closed (polyhedral) curve c in X in the following way. First, let A be an annulus in the Euclidean plane parametrized by (r, θ) where 1 ≤ r ≤ 2 and θ is a real number mod 2 π. We define a homeomorphism H: A → A byH is then fixed on the boundary of A. If now e: A → X is an orientation-preserving embedding, and eA is a neighbourhood of c in X, then eHe−1|eA can be extended by the identity on X − eA to a homeomorphism h:X → X. Any piecewise linear homeomorphism hc which is isotopic to h will be called a twist about c or, if c is not specified, just a twist.

On Generalized Averaging Operators

1958 ◽  
Vol 10 ◽  
pp. 122-126 ◽  
Author(s):  
R. P. Boas
Keyword(s):  
Entire Function ◽  
Exponential Type ◽  
Alternative Definition ◽  
Averaging Operators ◽  
Definition Of ◽  
Image Position ◽  
Class Of Functions

Let Sumner (4) has discussed for arbitrary real λ and h, whereƒ(Z) is an entire function of exponential type I shall show that in this case an alternative definition of ∇λλ, which leads to Sumner's results more quickly, is equivalent to Sumner's. (However, Sumner's definition is, in principle, applicable to a wider class of functions.)

Functions of Bounded mean Square, and Generalized Fourier-Stieltjes Transforms

1970 ◽  
Vol 22 (5) ◽  
pp. 1016-1034 ◽  
Author(s):  
J. Henniger
Keyword(s):  
Banach Space ◽  
Finite Interval ◽  
Complex Function ◽  
Complex Numbers ◽  
Mean Square ◽  
Convolution Algebra ◽  
Algebra Of Functions ◽  
Image Position ◽  
Stieltjes Transforms

A complex function on the real line is said to be bounded in mean square if it is locally in L2 (i.e. on each finite interval) and satisfies(1.1)The set of all such functions clearly forms a linear space over the complex numbers and is a Banach space B under the norm ‖·‖B defined by (1.1). This space, among others, has been discussed by Beurling in [1], where it was shown to be the dual, in the Banach space sense, of a certain Banach (convolution) algebra of functions. We have used Beurling's characterization of B and others of his results throughout this paper, and indeed the essence of one or two of the proofs has been derived from his theorems.

scholarly journals The minimum modulus of gap power series

1978 ◽  
Vol 21 (1) ◽  
pp. 49-54 ◽  
Author(s):  
P. C. Fenton
Keyword(s):  
Entire Function ◽  
Power Series ◽  
Maximum Modulus ◽  
Minimum Modulus ◽  
Image Position

Letbe an entire function, where (αn) is a strictly increasing sequence of non-negativeintegers. The maximum modulus, M(r), the minimum modulus, m(r), and the maxi-mum term, μ(r), of f defined by

Absolute Convergence Factors for Hp Series

1969 ◽  
Vol 21 ◽  
pp. 187-195 ◽  
Author(s):  
James Caveny
Keyword(s):  
Banach Space ◽  
Fourier Coefficients ◽  
Absolute Convergence ◽  
Famous Theorem ◽  
Conjugate Space ◽  
Measurable Functions ◽  
Image Position ◽  
Complex Valued

A famous theorem of Hardy asserts that if f ∊ H1, then the sequence of Fourier coefficients satisfies . For this reason we say that the sequence (1, 1/2, 1/3, …) belongs to the multiplier class (H1, l1). In this paper, we investigate the multiplier classes (Hp, l1) for 1 ≧ p ≧ ∞. Our observations are based on the fact that a sequence (λ(0), λ(l), …) belongs to (Hp, l1) independent of the arguments of its terms. We also show that (Hp, l1) may be thought of as the conjugate space of a certain Banach space.1. Preliminaries.Lp denotes the space of complex-valued Lebesgue measurable functions f defined on the circle |z| = 1 such thatis finite.

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