Abstract
Fourier series is a well-established subject and widely applied in various fields. However, there is much less work on double Fourier coefficients in relation to spaces of general double sequences. We understand the space of double Fourier coefficients as an abstract space of sequences and examine relationships to spaces of general double sequences: p-power summable sequences for p = 1, 2, and the Hilbert space of double sequences. Using uniform convergence in the sense of a Cesàro mean, we verify the inclusion relationships between the four spaces of double sequences; they are nested as proper subsets. The completions of two spaces of them are found to be identical and equal to the largest one. We prove that the two-parameter Wiener space is isomorphic to the space of Cesàro means associated with double Fourier coefficients. Furthermore, we establish that the Hilbert space of double sequence is an abstract Wiener space. We think that the relationships of sequence spaces verified at an intermediate stage in this paper will provide a basis for the structures of those spaces and expect to be developed further as in the spaces of single-indexed sequences.
Highlights
Fourier series is one of the well-established subjects in mathematics and is widely applied in many fields of science and engineering
We are interested in the space of sequences composed of double Fourier coefficients, whose member is from a Fourier series of a two-variable function, often called a double Fourier series
There are very few closely related works; the space of Fourier coefficients relating to an analog of Wiener space is studied in [6], whereas the authors of [7] discuss them in relation to an abstract Wiener space
Summary
Fourier series is one of the well-established subjects in mathematics and is widely applied in many fields of science and engineering. We derive an isomorphic relationship mapped from the space of square-integrable functions to the space of 2-power summable sequences; the isomorphism maps the twoparameter Wiener space to the space of double Fourier coefficients. This enables us to achieve an abstract Wiener space of the Hilbert space of double sequences. There are very few closely related works; the space of Fourier coefficients relating to an analog of Wiener space is studied in [6], whereas the authors of [7] discuss them in relation to an abstract Wiener space Both are for the spaces of single-indexed sequences. Conclusions and discussions are given in the last section
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