in a series of articles he generalized results in the classical fourier theory to more general spaces (hilbert spaces) by investigating various orthonormal systems of functions, their summability and the representation of functions in such systems by fourier series and fourier integrals and more general integral transformations.

How would the previous sentence be correctly capitalized?
In a series of articles he generalized results in the classical Fourier theory to more general spaces (Hilbert spaces) by investigating various orthonormal systems of functions, their summability and the representation of functions in such systems by Fourier series and Fourier integrals and more general integral transformations.