It is well known that the Fourier series is not summable by harmonic means almost everywhere. This paper aimed to introduce the harmonic Banach product summability factors of Fourier series and conjugate Fourier series in order to establish two new approximation theorems. Moreover, some illustrative examples are considered to obtain the sum of certain oscillatory and non-oscillatory sequences and demonstrated that their convergence are well behaved under the proposed mean. Also, the findings are validated and simulated through MATLAB programming.

No abstract