scholarly journals Approximation by T-Transformation of Double Walsh-Fourier Series to Multivariable Functions

2014 ◽  
Vol 2014 ◽  
pp. 1-14
Author(s):  
Yi Zhao ◽  
Dansheng Yu
Keyword(s):  
Fourier Series ◽  
Series Expansion ◽  
Bounded Variation ◽  
Uniform Approximation ◽  
Partial Sums ◽  
Walsh Series ◽  
Multivariate Functions ◽  
Summability Methods

We study the Walsh series expansion of multivariate functions in Lp  (1≤p≤∞) and, in particular, in Lip(α,p). The rate of uniform approximation by T-transformation of rectangular partial sums of double Walsh to these functions is investigated. By extending the concepts of rest (head) bounded variation series, which was introduced by Leindler (2004), we generalize the related results of Móricz and Rhoades (1996), Nagy (2012). Our results can be applied to many summability methods, including the Nörlund summability and weighted summability.

scholarly journals Metric Fourier Approximation of Set-Valued Functions of Bounded Variation

2021 ◽  
Vol 27 (2) ◽  
Author(s):  
Elena E. Berdysheva ◽  
Nira Dyn ◽  
Elza Farkhi ◽  
Alona Mokhov
Keyword(s):  
Fourier Series ◽  
Bounded Variation ◽  
Error Bounds ◽  
Metric Spaces ◽  
Hausdorff Metric ◽  
Dirichlet Kernel ◽  
Partial Sums ◽  
Functions Of Bounded Variation ◽  
Moduli Of Continuity ◽  
Fourier Approximation

AbstractWe introduce and investigate an adaptation of Fourier series to set-valued functions (multifunctions, SVFs) of bounded variation. In our approach we define an analogue of the partial sums of the Fourier series with the help of the Dirichlet kernel using the newly defined weighted metric integral. We derive error bounds for these approximants. As a consequence, we prove that the sequence of the partial sums converges pointwisely in the Hausdorff metric to the values of the approximated set-valued function at its points of continuity, or to a certain set described in terms of the metric selections of the approximated multifunction at a point of discontinuity. Our error bounds are obtained with the help of the new notions of one-sided local moduli and quasi-moduli of continuity which we discuss more generally for functions with values in metric spaces.

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)

Convergence of the Hausdorff Means of Double Fourier Series*

1968 ◽  
Vol 11 (4) ◽  
pp. 585-591 ◽  
Author(s):  
Fred Ustina
Keyword(s):  
Fourier Series ◽  
Bounded Variation ◽  
Double Fourier Series ◽  
Partial Sums ◽  
Closed Region ◽  
Hausdorff Means

In this paper we prove that if {sm, n(x, y)} is the sequence of partial sums of the Fourier series of a function f(x, y), which is periodic in each variable and of bounded variation in the sense of Hardy-Krause in the period rectangle, then {sm, n(x, y)} converges uniformly to f(x, y) in any closed region D in which this function is continuous at every point. This result is then used to prove that the regular Hausdorff means of the Fourier series of such a function also converge uniformly in such a region.

On the absolute Nörlund summability of a Fourier series

1967 ◽  
Vol 7 (2) ◽  
pp. 252-256 ◽  
Author(s):  
Fu Cheng Hsiang
Keyword(s):  
Fourier Series ◽  
Bounded Variation ◽  
Infinite Series ◽  
Partial Sums ◽  
The Absolute ◽  
Image Position ◽  
Absolute Nörlund Summability

Let Σn−0∞an, 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 writeIfas n → ∞, then we say that the series is summable by the Nörlund method (N, pn) to σ And the series a,Σan, is said to be absolutely summable (N, pn) or summable |N, Pn| if {σn} is of bounded variation, i.e.,

scholarly journals On the degree of approximation of continuous functions by a specific transform of partial sums of their Fourier series

2021 ◽  
Vol 25 (1) ◽  
pp. 5-19
Author(s):  
Xhevat Krasniqi
Keyword(s):  
Fourier Series ◽  
Continuous Function ◽  
Bounded Variation ◽  
Degree Of Approximation ◽  
Continuous Functions ◽  
Auxiliary Function ◽  
Partial Sums ◽  
The Mean ◽  
Approximation Of Continuous Functions ◽  
Positive Integers

Using the Mean Rest Bounded Variation Sequences or the Mean Head Bounded Variation Sequences, we have proved four theorems pertaining to the degree of approximation in sup-norm of a continuous function f by general means τλn;A(f) of partial sums of its Fourier series. The degree of approximation is expressed via an auxiliary function H(t) ≥ 0 and via entries of a matrix whose indices form a strictly increasing sequence of positive integers λ := {λ(n)}∞n=1.

On the Absolute Nörlund Summability of a Fourier series (II)

1969 ◽  
Vol 9 (1-2) ◽  
pp. 161-166 ◽  
Author(s):  
Fu Cheng Hsiang
Keyword(s):  
Fourier Series ◽  
Bounded Variation ◽  
Partial Sums ◽  
Nörlund Means ◽  
The Absolute ◽  
Image Position ◽  
Absolute Nörlund Summability

Let be a given series with its partial sums {Sn} and {Pn} a sequence of real or complex parameters. Write. The transformation given by defines the Nörlund means of {Sn} generated by {Pn}. The series Σann is said to be absolutely summable (N, pn) or summable ∣N, pn∣, if {tn} is of bounded variation, i.e., Σ|tn—tn−1| converges.

On the non-absolute summability of a Fourier series and the conjugate of a Fourier series by a Nörlund method

1967 ◽  
Vol 63 (2) ◽  
pp. 407-411 ◽  
Author(s):  
R. Mohanty ◽  
B. K. Ray
Keyword(s):  
Fourier Series ◽  
Bounded Variation ◽  
Absolute Summability ◽  
Partial Sums ◽  
Nörlund Means ◽  
Sequence Transformation ◽  
Image Position

Let {Sn} be the sequence of partial sums of the infinite seriesΣαn. Let {pn} be a sequence of constants real or complex and let us setThe sequence {tn} of Nörlund means (5) or simply (N, pn) means of the sequence {Sn} generated by the sequence of coefficients {pn} is defined by the following sequence -to-sequence transformationThe series ∑αn or the sequence {Sn} is said to be summable (N, pn) to the sum S, ifand is said to be absolutely summable (N, pn) or summable |N, pn|, if the sequence {tn} is of bounded variation, that is, the series ∑|tn − tn−1| is convergent (2).

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