Saturation results for a class of linear operators

1983 ◽  
Vol 94 (1) ◽  
pp. 133-148 ◽  
Author(s):  
B. Kuttner ◽  
R. N. Mohapatra ◽  
B. N. Sahney
Keyword(s):  
Fourier Series ◽  
Linear Operators ◽  
Infinite Matrix ◽  
Measurable Functions ◽  
Image Position

Let B denote the space of bounded measurable functions with period 2π. We will suppose throughout that f(x) ∈ B. All norms considered are essential sup norms. Let the Fourier series of f(x) be given byLet D = (dnk) (n, k = 0, 1, …) be an infinite matrix.Let Ln(f; x) be the D transform of the Fourier series of f(t) at t = x, i.e.where Sk(x) = A0(x) + A1(x) + … + Ak(x). Let us write

An elementary proof of the Birkhoff-Hopf theorem

1995 ◽  
Vol 117 (1) ◽  
pp. 31-55 ◽  
Author(s):  
Simon P. Eveson ◽  
Roger D. Nussbaum
Keyword(s):  
Elementary Proof ◽  
Linear Operators ◽  
Positive Linear Operators ◽  
Important Work ◽  
Large Classes ◽  
Contraction Mappings ◽  
Measurable Functions ◽  
Almost Everywhere ◽  
Image Position ◽  
Hopf Theorem

In important work some thirty years ago, G. Birkhoff[2, 3] and E. Hopf [16, 17] showed that large classes of positive linear operators behave like contraction mappings with respect to certain ‘almost’ metrics. Hopf worked in a space of measurable functions and took as his ‘almost’ metric the oscillation ω(y/x) of functions y and x with x(t) > 0 almost everywhere, defined by

Compact embeddings of some weighted Sobolev spaces on ℝN

1995 ◽  
Vol 117 (2) ◽  
pp. 333-338 ◽  
Author(s):  
Raffaele Chiappinelli
Keyword(s):  
Sobolev Spaces ◽  
Weighted Sobolev Spaces ◽  
Compact Embeddings ◽  
Measurable Functions ◽  
Image Position

Let ρ,ρ0,ρ1 be positive, measurable functions on ℝN. For 1 ≤ t < ∞, consider the weighted Lebesgue and Sobolev spaces

Proof that there are infinitely many modalities in Lewis's system S2

1940 ◽  
Vol 5 (3) ◽  
pp. 110-112 ◽  
Author(s):  
J. C. C. McKinsey
Keyword(s):  
Finite Number ◽  
Infinite Matrix ◽  
Image Position

In this note I show, by means of an infinite matrix M, that the number of irreducible modalities in Lewis's system S2 is infinite. The result is of some interest in view of the fact that Parry has recently shown that there are but a finite number of modalities in the system S2 (which is the next stronger system than S2 discussed by Lewis).I begin by introducing a function θ which is defined over the class of sets of signed integers, and which assumes sets of signed integers as values. If A is any set of signed integers, then θ(A) is the set of all signed integers whose immediate predecessors are in A; i.e., , so that n ϵ θ(A) is true if and only if n − 1 ϵ A is true.Thus, for example, θ({−10, −1, 0, 3, 14}) = {−9, 0, 1, 4, 15}. In particular we notice that θ(V) = V and θ(Λ) = Λ, where V is the set of all signed integers, and Λ is the empty set of signed integers.It is clear that, if A and B are sets of signed integers, then θ(A+B) = θ(A)+θ(B).It is also easily proved that, for any set A of signed integers we have . For, if n is any signed integer, then

A mini-gap theorem for Fourier series

1968 ◽  
Vol 64 (1) ◽  
pp. 61-66 ◽  
Author(s):  
W. K. Hayman
Keyword(s):  
Fourier Series ◽  
Gap Theorem ◽  
Image Position

Suppose thatbelongs to L2( − π, π). If most of the coefficients vanish then f (x) cannot be too small in a certain interval without being small generally. More precisely Ingham ((2), Theorem 1) has proved the followingTHEOREM A. Suppose that f (x) is given by (1·1) and that an = 0, except for a sequence n = nν, where nν+1 − nν ≥ C. Then given ∈ > 0 there exists a constant A (∈), such that we have for any real x1

Non-Local Elliptic Boundary-Value Problems

1968 ◽  
Vol 20 ◽  
pp. 1365-1382 ◽  
Author(s):  
Bui An Ton
Keyword(s):  
Differential Operators ◽  
Smooth Boundary ◽  
Elliptic Boundary ◽  
Boundary Value ◽  
Linear Operators ◽  
Elliptic Boundary Value Problem ◽  
Elliptic Boundary Value ◽  
Open Set ◽  
Non Local ◽  
Image Position

Let G be a bounded open set of Rn with a smooth boundary ∂G. We consider the following elliptic boundary-value problem:where A and Bj are, respectively singular integro-differential operators on G and on ∂G, of orders 2m and rj with rj < 2m; Ck are boundary differential operators, and Ljk are linear operators, bounded in a sense to be specified.

Generalized Resolvent Equations and Unsymmetric Dirichlet Spaces

1981 ◽  
Vol 33 (5) ◽  
pp. 1111-1141
Author(s):  
Joanne Elliott
Keyword(s):  
Linear Operators ◽  
Dirichlet Spaces ◽  
Resolvent Equation ◽  
Continuous Linear ◽  
Generalized Resolvent ◽  
Resolvent Equations ◽  
Measure Spaces ◽  
Image Position ◽  
Generalized Resolvent Equations ◽  
Continuous Linear Operators

Let (X, , μ) and (X, , μ′) be measure spaces with the measures μ and μ′ totally finite. Suppose {Uλ: λ > 0} is a family of positive (i.e., ϕ ≧ 0 ⇒ Uλϕ ≧ 0) continuous linear operators from L2(X, dμ′) to L2(X,dμ) with the following additional properties: if ϕ ≧ 0 then Uλϕ is non-decreasing as λ increases, while λ−1Uλϕ is nonincreasing.A family {Mλ:λ > 0} of continuous linear operators from L2(X, dμ) to L2(X, dμ′) satisfies the “generalized resolvent equation” relative to {Uλ} if(0.1)for positive λ and v. If Uλ = λI, then (0.1) is just the well-known resolvent equation. The family {Mλ} is called submarkov if Mλ is a positive operator and(0.2)it is conservative if(0.3)

On the Absolute Nörlund Summability of a Fourier Series

1973 ◽  
Vol 16 (4) ◽  
pp. 599-602
Author(s):  
D. S. Goel ◽  
B. N. Sahney
Keyword(s):  
Fourier Series ◽  
Bounded Variation ◽  
Infinite Series ◽  
Partial Sums ◽  
The Absolute ◽  
Image Position ◽  
Absolute Nörlund Summability

Let be a given infinite series and {sn} the sequence of its partial sums. Let {pn} be a sequence of constants, real or complex, and let us write(1.1)If(1.2)as n→∞, we say that the series is summable by the Nörlund method (N,pn) to σ. The series is said to be absolutely summable (N,pn) or summable |N,pn| if σn is of bounded variation, i.e.,(1.3)

scholarly journals On Riesz operators

1969 ◽  
Vol 16 (3) ◽  
pp. 227-232 ◽  
Author(s):  
J. C. Alexander
Keyword(s):  
Linear Operator ◽  
Banach Spaces ◽  
Linear Operators ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Riesz Operators ◽  
Image Position ◽  
Schauder’S Theorem

In (4) Vala proves a generalization of Schauder's theorem (3) on the compactness of the adjoint of a compact linear operator. The particular case of Vala's result that we shall be concerned with is as follows. Let t1 and t2 be non-zero bounded linear operators on the Banach spaces Y and X respectively, and denote by 1T2 the operator on B(X, Y) defined by

scholarly journals Multiparameter spectral theory and Taylor's joint spectrum in Hilbert space

1988 ◽  
Vol 31 (1) ◽  
pp. 127-144 ◽  
Author(s):  
B. P. Rynne
Keyword(s):  
Hilbert Space ◽  
Spectral Theory ◽  
Linear Operators ◽  
Coupled Systems ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Joint Spectrum ◽  
Image Position

Let n≧1 be an integer and suppose that for each i= 1,…,n, we have a Hilbert space Hi and a set of bounded linear operators Ti, Vij:Hi→Hi, j=1,…,n. We define the system of operatorswhere λ=(λ1,…,λn)∈ℂn. Coupled systems of the form (1.1) are called multiparameter systems and the spectral theory of such systems has been studied in many recent papers. Most of the literature on multiparameter theory deals with the case where the operators Ti and Vij are self-adjoint (see [14]). The non self-adjoint case, which has received relatively little attention, is discussed in [12] and [13].

The Isometries of Hp

1964 ◽  
Vol 16 ◽  
pp. 721-728 ◽  
Author(s):  
Frank Forelli
Keyword(s):  
Unit Circle ◽  
Lebesgue Measure ◽  
Measurable Functions ◽  
Image Position ◽  
Complex Valued

Let a be the Lebesgue measure on the unit circle |z| = 1 withand let Lp be the space of complex-valued σ-measurable functions f such thatis finite. Hp is the closure in Lp of the algebra of analytic polynomials

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