A novel extension to Fourier series for representing combined functions and extension to high precision alternate functions

Sankaralingam Lakshmanaraj

doi:https://doi.org/10.14445/22315373/IJMTT-V66I11P502

International Journal of Mathematics Trends and Technology

Research Article | Open Access | Download PDF

Volume 66 | Issue 11 | Year 2020 | Article Id. IJMTT-V66I11P502 | DOI : https://doi.org/10.14445/22315373/IJMTT-V66I11P502

A novel extension to Fourier series for representing combined functions and extension to high precision alternate functions

Sankaralingam Lakshmanaraj

Abstract

This is a new concept to transform a function to a series combining Cosine and Sine functions with Polynomial, Geometric, Sign and Matrix harmonic functions, unlike Fourier transformation which has only Cosine and Sine functions. The inherent nature of the functions used in synthesizing the target functions finds application in data reduction techniques without compromising fidelity and integrity of function. Here, I demonstrate the efficiency of extension in precision error, compression ratio and implementation complexity while applying it to real-world problems such as faster live streaming, prediction of stock market data, and storage of medical imaging data. I have also found methods to transform any discrete function to a continuous function or a continuous function to another function with less error-rate. This is useful in finding interpolation, smoothing or coil of rough functions, where the nature of curves is not known and is also useful in functions in a certain range such as hearing frequency, visible wave-lengths. Further, I have found smoothing transformation which is useful in finding both accurate values and in finding roots and maximum, minimum turning points of discrete points. Also one of the methods is useful in constructing a decorative curve from a given path.

Keywords

Function transformation, Fourier series, trigonometric functions, polynomial functions, geometric functions, matrix functions, matrix harmonic functions, prediction, data reduction, faster approximations, extrapolation, interpolation, encryption, function compression, image compression, transformation, compression, smoothness, coil, polygon, quadratic, parabolic, quartic, polynomial, decorative, decorative curve, roots, maximum, minimum, turning points.

References

[1] This paper is an original illustration consisting of 24 concepts, all of which are discovered by me (10 extensions to Fourier Series, 7 varieties of high precision functions, 5 categories of smoothing curves, and 2 types of decorating paths). Since I have neither referred to nor copied from any articles, there are no reference links mentioned here.

Citation :

Sankaralingam Lakshmanaraj, "A novel extension to Fourier series for representing combined functions and extension to high precision alternate functions," International Journal of Mathematics Trends and Technology (IJMTT), vol. 66, no. 11, pp. 13-51, 2020. Crossref, https://doi.org/10.14445/22315373/IJMTT-V66I11P502