The problem of L1-convergence of the Fourier cosine series (1.1) has been settled for various special classes of coefficients. Young [6] found that an logn= o(1), n→∞ is a necessary and sufficient condition for cosine series with convex (∆an ≥ 0) coefficients, and Kolmogorov [5] extended this result to the cosine series with quasi-convex ( ∑∞ n=1n|∆an−1| < ∞) coefficients. Later, Garrett and St...

A time-series consisting of white noise plus Brownian motion sampled at equal intervals of time is exactly orthogonalised by a discrete cosine transform (DCTII). This paper explores the properties of a version of spectral analysis based on the discrete cosine transform and its use in distinguishing between a stationary time-series and an integrated (unit root) time-series.

On page 335 in his lost notebook, Ramanujan recorded without proofs two identities involving finite trigonometric sums and doubly infinite series of Bessel functions. The two identities are intimately connected with the classical circle and divisor problems, respectively. For each of Ramanujan’s identities, there are three possible interpretations for the double series. In two earlier papers, t...

In this paper we introduce some new modified cosine sums and then using these sums we study L1-convergence of trigonometric cosine series.