scholarly journals Orthorecursive expansion of unity

2020 ◽  
Vol 16 (06) ◽  
pp. 1209-1226
Author(s):  
A. B. Kalmynin ◽  
P. R. Kosenko
Keyword(s):  
Infinite Series ◽  
Recursive Relation ◽  
Partial Sums ◽  
Growth Order ◽  
Alternative Definition ◽  
Harmonic Numbers ◽  
Numerical Computations ◽  
Norm Minimization ◽  
Orthorecursive Expansion

We study the properties of a sequence [Formula: see text] defined by the recursive relation [Formula: see text] for [Formula: see text] and [Formula: see text]. This sequence also has an alternative definition in terms of certain norm minimization in the space [Formula: see text]. We prove estimates on the growth order of [Formula: see text] and the sequence of its partial sums, infinite series identities, connecting [Formula: see text] with the harmonic numbers [Formula: see text] and also formulate some conjectures based on numerical computations.

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)

Possible and Impossible Series

2015 ◽  
Vol 108 (7) ◽  
pp. 560
Author(s):  
Mark MacLean
Keyword(s):  
Infinite Series ◽  
Partial Sums ◽  
Its Sequence

A lesson helps students discern possible relationships between an infinite series, its sequence of terms, and the sequence of partial sums.

Theoretical Investigations and Numerical Evaluations of Wall Effects in Cavity Flows

1975 ◽  
Vol 97 (4) ◽  
pp. 482-491
Author(s):  
S. Popp
Keyword(s):  
Exact Solution ◽  
Infinite Series ◽  
Cavity Flows ◽  
Two Dimensional ◽  
Cavitating Flows ◽  
Wall Effects ◽  
Numerical Computations ◽  
Hodograph Method ◽  
Shaped Body ◽  
Theoretical Investigations

The wall effects for fully and partially cavitating flows are investigated for both compressible and incompressible two-dimensional jets. The exact solution for Roshko’s model in a channel with a wedge shaped body is obtained and some particular models are studied. The hodograph method as developed by S. V. Falkovitch [19] is used and the solutions are given as infinite series of Chaplygin’s functions. The exact expressions of the drag coefficients for the aforementioned configurations are also given. Numerical computations are carried out for wedges of all angles. Tables and diagrams are included.

On Absolute Summability by Riesz and Generalized Cesàro Means. I

1970 ◽  
Vol 22 (2) ◽  
pp. 202-208 ◽  
Author(s):  
H.-H. Körle
Keyword(s):  
Infinite Series ◽  
Absolute Summability ◽  
Partial Sums ◽  
Cesàro Means ◽  
Riesz Means ◽  
Cesaro Means ◽  
Cesáro Means ◽  
Image Position ◽  
Generalized Cesàro Means

1. The Cesàro methods for ordinary [9, p. 17; 6, p. 96] and for absolute [9, p. 25] summation of infinite series can be generalized by the Riesz methods [7, p. 21; 12; 9, p. 52; 6, p. 86; 5, p. 2] and by “the generalized Cesàro methods“ introduced by Burkill [4] and Borwein and Russell [3]. (Also cf. [2]; for another generalization, see [8].) These generalizations raise the question as to their equivalence.We shall consider series(1)with complex terms an. Throughout, we will assume that(2)and we call (1) Riesz summable to a sum s relative to the type λ = (λn) and to the order κ, or summable (R, λ, κ) to s briefly, if the Riesz means(of the partial sums of (1)) tend to s as x → ∞.

On a Tauberian theorem for absolute harmonic summability

1969 ◽  
Vol 65 (1) ◽  
pp. 93-99 ◽  
Author(s):  
O. P. Varshney ◽  
Govind Prasad
Keyword(s):  
Infinite Series ◽  
Tauberian Theorem ◽  
Partial Sums ◽  
Image Position ◽  
Harmonic Summability

Let Σan be a given infinite series with the sequence of partial sums {Sn}. Let the sequence be defined bywhere is given by

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