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?
Answer: 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.
8:30 as he spake these words, many believed on him.

How would the previous sentence be correctly capitalized?
Answer: 8:30 As he spake these words, many believed on him.
fasten the yarn end on the back side.

How would the previous sentence be correctly capitalized?
Answer: Fasten the yarn end on the back side.
mel is melissa's little sister and lives up to its high standards in all respects!

How would the previous sentence be correctly capitalized?
Answer: Mel is Melissa's little sister and lives up to its high standards in all respects!