# Differentiation of Fourier Series via Orthogonal Derivative

@article{CruzSantiago2015DifferentiationOF, title={Differentiation of Fourier Series via Orthogonal Derivative}, author={Rene Cruz-Santiago and Jos{\'e} L. L{\'o}pez-Bonilla and R. L{\'o}pez-V{\'a}zquez}, journal={Journal of the Institute of Science and Technology}, year={2015}, volume={20}, pages={113-114} }

It is very known that if the operator d/dx acts on each term into a convergent Fourier Series (FS), then it may result a divergent series. This situation is remedied applying the symmetric derivative to FS, which implies the existence of the important Fejer-Lanczos Factors. In this paper, we show that the orthogonal derivative also leads to these Factors. Journal of Institute of Science and Technology, 2015, 20(2): 113-114

## 2 Citations

Differentiation of a Fourier series

- Mathematics
- 2018

It is very known that if the operator acts on each term into a convergent Fourier series (FS) then it may result a divergent series. This situation is remedied applying the symmetric derivative to…

An Inequality for the Fejér-Lanczos Factors

- Mathematics
- 2017

We find, via numerical experimentation, an inequality satisfied by the Fejer-Lanczos factors. Let’s remember that those factors are important for a correct differentiation of Fourier series.

## References

SHOWING 1-10 OF 28 REFERENCES

Differentiation by integration using orthogonal polynomials, a survey

- MathematicsJ. Approx. Theory
- 2012

The Least-Squares Property of the Lanczos Derivative

- Mathematics
- 2005

Everyone knows that the derivative of a function at a point measures the slope, or instantaneous rate of change, of the function at that point; this follows from the usual difference-quotient…

On the Gibbs Phenomenon and Its Resolution

- MathematicsSIAM Rev.
- 1997

The Gibbs phenomenon is reviewed from a different perspective and it is shown that the knowledge of the expansion coefficients is sufficient for obtaining the point values of a piecewise smooth function, with the same order of accuracy as in the smooth case.

Lanczos-Like σ-Factors for Reducing the Gibbs Phenomenon in General Orthogonal Expansions and Other Representations

- Mathematics
- 2000

The first attempt for reducing the Gibbs phenomenon in an orthogonalexpansion, besides the usual one of Fourier series, is due to Cooke in1927–1928 for the Fourier Bessel series.However, his work was…

Linear Differential Operators

- Mathematics
- 1961

Preface Bibliography 1. Interpolation. Introduction The Taylor expansion The finite Taylor series with the remainder term Interpolation by polynomials The remainder of Lagrangian interpolation…

Lanczos Derivative via a Quadrature Method

- Mathematics
- 2010

The Lanczos derivative, which performs derivation through integration, is deduced from a quadrature method with boundary values.

The Gibbs' phenomenon

- Mathematics
- 2001

The well-known physicist A. A. Michelson started quite an interesting correspondence in the journal Nature in 1898. He complained about the convergence of continuous Fourier series approximations to…

Lanczos' Generalized Derivative

- Mathematics
- 1998

(1998). Lanczos' Generalized Derivative. The American Mathematical Monthly: Vol. 105, No. 4, pp. 320-326.