Fourier Series Extension in terms of Powers of Sine and Cosine Functions

Abstract Fourier series play a fundamental role in mathematical analysis by providing a powerful tool for representing periodic functions as infinite sums of sine and cosine functions. This classical representation has been extensively studied and successfully applied in various areas of mathematics, physics, and engineering. Despite its wide applicability, the traditional Fourier series relies exclusively on first-order trigonometric basis functions, namely sine and cosine. This naturally raises the question of whether alternative families of trigonometric functions can be employed to construct meaningful and effective expansions of periodic functions. Motivated by this observation, the present paper investigates a new framework for expanding periodic functions using a special class of trigonometric basis functions consisting of odd positive integer powers of sine and cosine. By replacing the standard trigonometric functions with their higher-order odd powers, we introduce a generalized form of Fourier series that extends the classical theory. The proposed expansion preserves the periodic structure of the original function while offering additional flexibility in representation. New definitions of extended Fourier series coefficients associated with this generalized expansion are established. Conditions for convergence of the expansion are discussed, along with illustrative examples. MSC 2020:42A16, 42A10, 42A20.