This proposal suggests expanding the use of generalized trigonometric functions in Fourier analysis to improve signal compression and reduce artifacts. Currently, Fourier analysis decomposes signals into sine and cosine waves in L2 space, but this causes the Gibbs phenomenon of overshooting when reconstructing non-continuous signals. The proposal aims to study trigonometric functions in different Lp spaces and modify Fourier series accordingly to minimize or eliminate the Gibbs phenomenon. This could enhance applications like image processing, media transfer, and data storage.

1. Proposal
Generalized Trigonometric Functions in Signal and Data Analysis
Improving signal quality and storage space
Signals and information, such as sound waves, can be compressed by breaking them
down into a series of sine and cosine waves. We are conducting research to determine
if this compression can be improved in both quality and size by investigating and
expanding current compression methods.
Introduction:
One of the fundamental methods used for the decomposition of a signal into a
series of harmonic functions (trigonometric functions) is called Fourier series. Fourier
series were introduced by Joseph Fourier in 1822 while studying problems of heat
transfer and vibrations. While doing heat transfer work, Fourier demonstrated that any
periodic signal can be viewed as a linear composition of sine and cosine waves. Fourier
created the following model in 𝐿2
space on some given interval to show how a signal
could be built from a sum of sinusoids and also gave an explicit way to calculate the
coefficients:
𝑓 = ∑ 𝑎𝑖sin(𝑖𝜋𝑥)
∞
𝑖=1
+ ∑ 𝑏𝑖
∞
𝑖=0
cos(𝑖𝜋𝑥)
where 𝑎𝑖 and 𝑏𝑖 are calculated constants. 𝐿2
space can be thought of as the convention
of spaces since in this space we have Euclidean geometry and dot product operations
which give us projections on linear space generated by sin(𝑖𝜋𝑥) or cos(𝑖𝜋𝑥) functions.

2. The ease of obtaining the coefficients is the main advantage of this method.
Unfortunately, this method is not well suited for non-continuous functions (signals in
digital circuits, black and white photographs, etc). In 𝐿2
space, a strange phenomena
called the Gibbs phenomena occurs when creating a Fourier series of a periodic
function. The Gibbs phenomenon is the overshooting of the approximation of a signal
function using Fourier series. The issue is that even as more terms are added in the
series, the overshoot does not go away. In signal processing, the Gibbs phenomenon is
undesirable because it causes artifacts, namely waveform distortion from the overshoot
and undershoot, and ringing artifacts from the oscillations. In MRI, the Gibbs
phenomenon causes artifacts in the presence of adjacent regions of markedly differing
signal intensity. This is most commonly encountered in spinal MR imaging, where the
Gibbs phenomenon may simulate the appearance of syringomyelia. This is one of many
problems the Gibbs phenomenon causes. A possible solution to the Gibbs phenomenon
could be creating a Fourier series of a signal in a space other than 𝐿2
space in 𝐿 𝑃
space.
Let us note that 𝐿2
space, which is a special example of 𝐿 𝑃
space where p is 2, is defined
using a natural generalization of the p-norm for finite-dimensional vector spaces. In 𝐿 𝑝
space, generalized arcsine is expressed by:
∫
1
(1 − 𝑧 𝑝)1/𝑞
𝑥
0
𝑑𝑧 𝑥 ∈ [0,1]
What we will do is study the properties of trigonometric functions in 𝐿 𝑃
space (i.e.
𝑠𝑖𝑛 𝑝 , 𝑠𝑖𝑛 𝑝𝑞 , 𝑐𝑜𝑠 𝑝, 𝑐𝑜𝑠 𝑝𝑞). By experimenting with trigonometric functions in different 𝐿 𝑝
spaces and modifying Fourier series with these functions, we could find a way to
minimalize or eliminate the Gibbs phenomenon. This would lead to an improvement of
image processing and media transfer as well as an increase in data storage capabilities.