ABSTRACT
In many practical and theoretical problems, we often meet functoins with periodic property. Let L 2 ∘ [ 0 , T ] https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003067498/6ba0c26b-f086-40cc-86ed-1693dc596221/content/eq7126.tif"/> denote the class of T-periodic and square integrable functions. It is easy to show that L 2 ∘ [ 0 , T ] https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003067498/6ba0c26b-f086-40cc-86ed-1693dc596221/content/eq7127.tif"/> is a Hilbert space when the appropriate inner product is introduced. One of the basic problems in the study of periodic functions is to decompose the Hilbert space L 2 ∘ [ 0 , T ] https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003067498/6ba0c26b-f086-40cc-86ed-1693dc596221/content/eq7128.tif"/> into a sequence of subspaces, which are orthogonal each other, and to construct the bases of L 2 ∘ [ 0 , T ] https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003067498/6ba0c26b-f086-40cc-86ed-1693dc596221/content/eq7129.tif"/> by constructing the bases of each subspace.
