Reconstruction of a noncausal function from its SFCs by Bohr convolution

ABSTRACT

In this note, we study the reconstruction problem in wide sense of a noncausal function $f(t,\omega )$ from its stochastic Fourier coefficients (SFCs in abbr.). We employ Ogawa integral and orthonormal basis of exponential functions $\left\{e_n\right(t)=\exp (2\pi \sqrt{-1}nt\left)\right\}$ to define SFCs. We show that Bohr convolution of SFCs and $\left\{{\int }_0^1\overline{e_n\left(t\right)}\mathrm{d}{W}_t\right\}$ generates every Fourier coefficient of $f(t,\omega )$, $\overline{z}(z\in \mathbb{C})$ denoting the complex conjugate of z. We also note that $f(t,\omega )$ can be reconstructed from infinite SFCs, even if the other (finite or infinite) SFCs are absent.

KEYWORDS:

• Stochastic Fourier coefficients
• reconstruction problem in wide sense
• Bohr convolution
• Ogawa integral
• noncausal function

AMS SUBJECT CLASSIFICATIONS:

• 60H05
• 42A61
• 42A85

Acknowledgements

The authors would like to express our sincere thanks to the reviewer for his/her valuable comments.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was partially supported by Japan Society for the Promotion of Science KAKENHI [grant number 26400152].