The present paper focuses on three types of fractional convolutions. First consideration is to extend the famous Young's inequality in the context of fractional convolution. Then for these convolutions, weighted iterated L p − L q inequalities have been investigated. Finally, the notion of fractional convolutions between functions and distributions and also between distributions has been defined and various corresponding properties have been studied.

Chirps are familiar in nature, have a built-in resistance to noise and interference, and are connected to a wide range of highly oscillatory processes. Detecting chirp oscillating patterns by traditional Fourier series is challenging because the chirp frequencies constantly change over time. Estimating such types of functions considering the partial sums of a Fourier series in Fourier analysis does not permit an approximate solution, which entails more Fourier coefficients required for signal reconstruction. The standard Fourier series, therefore, has a poor convergence rate and is an inadequate approximation. In this study, we use a parameterized orthonormal basis with an adjustable parameter to match the oscillating behavior of the chirp to approximate linear chirps using the partial sums of a generalized Fourier series known as fractional Fourier series, which gives the best approximation with only a small number of fractional Fourier coefficients. We used the fractional Fourier transform to compute the fractional Fourier coefficients at sample points. Additionally, we discover that the fractional parameter has the best value at which fractional Fourier coefficients of zero degrees have the most considerable magnitude, leading to the rapid decline of fractional Fourier coefficients of high degrees. Furthermore, fractional Fourier series approximation with optimal fractional parameters provides the minimum mean square error over the fractional Fourier parameter domain.

In this paper, we define and investigate fractional cosine and sine transforms for functions as well as distributions. For these transforms, several properties are provided, the representation theorems are proved and the analyticity is established.