On the summability of Fourier integrals

1970 ◽  
Vol 67 (1) ◽  
pp. 23-28
Author(s):  
M. K. Nayak
Keyword(s):  
Open Interval ◽  
Finite Limit ◽  
Image Position ◽  
Fourier Integrals

We say a series is summable L iftends to a finite limit s as x → 1 in the open interval (0, 1) where

Absolute summability of infinite series on a scale of Abel type summability methods

1968 ◽  
Vol 64 (2) ◽  
pp. 377-387 ◽  
Author(s):  
Babban Prasad Mishra
Keyword(s):  
Infinite Series ◽  
Open Interval ◽  
Absolute Summability ◽  
Finite Limit ◽  
Summability Methods ◽  
Image Position

Suppose that λ > − 1 and thatIt is easy to show thatWith Borwein(1), we say that the sequence {sn} is summable Aλ to s, and write sn → s(Aλ), if the seriesis convergent for all x in the open interval (0, 1)and tends to a finite limit s as x → 1 in (0, 1). The A0 method is the ordinary Abel method.

Strong summability of infinite series on a scale of Abel type summability methods

1967 ◽  
Vol 63 (1) ◽  
pp. 119-127 ◽  
Author(s):  
Babban Prasad Mishra
Keyword(s):  
Infinite Series ◽  
Open Interval ◽  
Strong Summability ◽  
New Method ◽  
Finite Limit ◽  
Summability Methods ◽  
Image Position

Introduction. In a recent paper, Borwein(1) constructed a new method of summability which would read: Letand let {sn} be any sequence of numbers. If, for λ > − 1,is convergent for all x in the open interval (0,1) and tends to a finite limit s as x → 1 in (0,1), we say that the sequence {sn} is Aλ convergent to s and write sn → s(Aλ). The A0 method is the ordinary Abel method.

On a scale of Abel-type summability methods

1957 ◽  
Vol 53 (2) ◽  
pp. 318-322 ◽  
Author(s):  
D. Borwein
Keyword(s):  
Open Interval ◽  
Finite Limit ◽  
Summability Methods ◽  
Image Position

1. Introduction. In this note Abel-type summability methods (Aλ) are defined and some of their properties investigated.Letand let {sn} be any sequence of numbers. Ifis convergent for all x in the open interval (0,1) and tends to a finite limit s as x → 1 in (0,1), we shall say that the sequence is Aλ-convergent to s and write sn → s (Aλ;). The A0 method is the ordinary Abel method.

Quotient groups and realization of tight Riesz groups

1973 ◽  
Vol 16 (4) ◽  
pp. 416-430 ◽  
Author(s):  
John Boris Miller
Keyword(s):  
Normal Subgroup ◽  
Open Interval ◽  
Canonical Homomorphism ◽  
Interval Topology ◽  
Essential Step ◽  
Image Position ◽  
Quotient Groups

Let (G, ≼) be an l-group having a compatible tight Riesz order ≦ with open-interval topology U, and H a normal subgroup. The first part of the paper concerns the question: Under what conditions on H is the structure of (G, ≼, ∧, ∨, ≦, U) carried over satisfactorily to by the canonical homomorphism; and its answer (Theorem 8°): H should be an l-ideal of (G, ≼) closed and not open in (G, U). Such a normal subgroup is here called a tangent. An essential step is to show that ≼′ is the associated order of ≦′.

Rate of growth and convergence factors for power methods of limitation

1974 ◽  
Vol 76 (1) ◽  
pp. 241-246 ◽  
Author(s):  
Abraham Ziv
Keyword(s):  
Radius Of Convergence ◽  
Complex Numbers ◽  
Finite Limit ◽  
Power Method ◽  
Real Constant ◽  
Rate Of Growth ◽  
Image Position

Let , where pk are complex numbers, have 0 < ρ ≤ ∞ for radius of convergence and assume that P(x) ≠ 0 for α ≤ x < ρ (α < ρ is some real constant). Assuming that is convergent for all (x ∈ [0, ρ), we define the P-limit of the sequence s = {sk} byThis, so called, power method of limitation (see (3), Definition 9 and (1) Definition 6) will be denoted by P. The best known power methods are Abel's (P(x) = 1/(1 – x), α = 0, ρ = 1) and Borel's (P(x) = ex, α = 0, ρ = ∞). By Cp we denote the set of all sequences, P-limitable to a finite limit and by the set of all sequences, P-limitable to zero.

Linear measures of some sets of the Cantor type

1957 ◽  
Vol 53 (2) ◽  
pp. 312-317 ◽  
Author(s):  
Trevor J. Mcminn
Keyword(s):  
Lebesgue Measure ◽  
Measure Zero ◽  
Cartesian Product ◽  
Open Interval ◽  
Cantor Set ◽  
Unit Interval ◽  
Linear Measure ◽  
Closed Intervals ◽  
Image Position ◽  
Dense Set

1. Introduction. Let 0 < λ < 1 and remove from the closed unit interval the open interval of length λ concentric with the unit interval. From each of the two remaining closed intervals of length ½(1 − λ) remove the concentric open interval of length ½λ(1 − λ). From each of the four remaining closed intervals of length ¼λ(1 − λ)2 remove the concentric open interval of length ¼λ(l − λ)2, etc. The remaining set is a perfect non-dense set of Lebesgue measure zero and is the Cantor set for λ = ⅓. Let Tλr be the Cartesian product of this set with the set similar to it obtained by magnifying it by a factor r > 0. Letting L be Carathéodory linear measure (1) and letting G be Gillespie linear square(2), Randolph(3) has established the following relations:

A note on a singular integral equation

1950 ◽  
Vol 46 (2) ◽  
pp. 268-271
Author(s):  
Albert E. Heins
Keyword(s):  
Integral Equation ◽  
Fourier Transform ◽  
Singular Integral Equation ◽  
Singular Integral ◽  
Legendre Function ◽  
Actual Calculation ◽  
Image Position ◽  
Fourier Integrals

In his treatise entitled Introduction to the theory of Fourier integrals, E. C. Titchmarsh derives a Fourier transform of the solution of the integral equationThe transform of the solution given by him does not satisfy the Riemann-Lebesgue lemma, and further suffers from the defect that it has certain exponential properties which prevent the actual calculation of the function φ(ξ), since the inversion integral does not exist for all ξ > 0. We shall correct these defects and show further that the solution of (1) is particularly simple and that it can be expressed as the Legendre function of an order which depends on λ.

On the Parseval formulae for Fourier transforms

1942 ◽  
Vol 38 (1) ◽  
pp. 1-19 ◽  
Author(s):  
Sheila M. Edmonds
Keyword(s):  
Fourier Transforms ◽  
Image Position ◽  
Convergence In Mean ◽  
Fourier Integrals

1. The object of this paper is to discuss conditions of validity of the Parseval formulae for Fourier integrals:where the transforms are defined by ordinary convergence; we shall not be concerned with the more elegant theory in which they are given by convergence in mean.

Some applications of renewal theory on the whole line

1970 ◽  
Vol 7 (03) ◽  
pp. 734-746
Author(s):  
Kenny S. Crump ◽  
David G. Hoel
Keyword(s):  
Distribution Function ◽  
Real Line ◽  
Renewal Theory ◽  
Open Interval ◽  
One Dimensional ◽  
The Real ◽  
Half Open Interval ◽  
Negative Function ◽  
Dimensional Distribution ◽  
Image Position

Suppose F is a one-dimensional distribution function, that is, a function from the real line to the real line that is right-continuous and non-decreasing. For any such function F we shall write F{I} = F(b)– F(a) where I is the half-open interval (a, b]. Denote the k-fold convolution of F with itself by Fk* and let Now if z is a non-negative function we may form the convolution although Z may be infinite for some (and possibly all) points.

A note on the representation of functions by absolutely convergent fourier integrals

1948 ◽  
Vol 44 (1) ◽  
pp. 8-12 ◽  
Author(s):  
H. R. Pitt
Keyword(s):  
Bounded Variation ◽  
Functions Of Bounded Variation ◽  
Integrable Functions ◽  
Image Position ◽  
Class Of Functions ◽  
Fourier Integrals

1. We write L for the class of integrable functions in (− ∞, ∞), V for the class of functions of bounded variation, and define A, A to be the classes of functions F(x) which may be expressed in the formsrespectively.

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