Abstract
In this chapter, we consider a scalar-valued binary operation on a linear space called an inner product. It leads to the concepts of the length of a vector and the orthogonality between vectors, which give to a linear space the structure of Euclidean geometry. Also, we learn the meaning of orthogonality between functions in a function space.
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Notes
- 1.
A method corresponding to the vdot function is not defined as a method of array class.
- 2.
W must be finite dimensional but V can be infinite dimensional in general. If W is infinite dimensional, it may not have the orthogonal projection.
- 3.
A procedure that achieves a certain purpose for a finite number of operation steps is called an algorithm.
- 4.
The order of vectors may affect the result of an orthonormal system.
- 5.
\(f\geqq 0\) (resp. \(f=0\)) means that \(f\left( x\right) \geqq 0\) (resp. \(f\left( x\right) =0\)) for all \(x\in \left[ a,b\right] \).
- 6.
See a textbook on calculus. In order to analyze a function space with a more elaborate theory, the concept of Lebesgue integration, which is a more general-purpose integration method, is needed rather than Riemann integration of a definite integral in calculus.
- 7.
Other numerical integration methods include the trapezoidal rule and Simpson’s rule. See the textbook on numerical analysis [15] in the Bibliography.
- 8.
See the textbooks on functional analysis in the Bibliography.
- 9.
We use here a sample function of one-dimensional Brownian motion created by simulation using the module random of NumPy.
- 10.
This is a problem deeply related to the definition of an integral.
- 11.
These are also defined in SciPy.
- 12.
Basic theory is mainly the work of mathematicians. Applications are the work of engineers, and typical applications include the design of filters that eliminate noise in electrical engineering and acoustics. Physicists are just halfway between mathematicians and engineers. Ideas from physics often contribute to both mathematical theory and engineering applications. Fourier analysis is a typical example.
- 13.
The Fourier transform of a function defined on \(\left( -\infty , \infty \right) \) is similarly calculated by considering its domain as the set of a sufficiently large number of sampling points of sufficiently wide finite interval.
- 14.
The discrete Fourier transform is obtained by multiplying a vector of dimension n by the basis transformation matrix, so if calculated normally, it takes time proportional to the number \(n^2\) of elements of the matrix of order n. In contrast, the fast Fourier transform is known to take time proportional to \(n\log n\). This method utilizes the fact that the basis transformation matrix has a special form. See the textbook on numerical analysis [15] in the Bibliography.
- 15.
These are peculiar variables of the object which are referred to with the object name as a prefix, such as S.something.
- 16.
A method is a function defined in a class of objects. Label it with the name of the object as a prefix, such as S.doing.
- 17.
In the context of object-oriented programming, this is called a constructor, and an object created in the class is called its instance.
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Tsukada, M., Kobayashi, Y., Kaneko, H., Takahasi, SE., Shirayanagi, K., Noguchi, M. (2023). Inner Product and Fourier Expansion. In: Linear Algebra with Python. Springer Undergraduate Texts in Mathematics and Technology. Springer, Singapore. https://doi.org/10.1007/978-981-99-2951-1_6
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DOI: https://doi.org/10.1007/978-981-99-2951-1_6
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