Abstract
Let e 1, e 2, …, e l be the standard basis of an l-dimensional Euclidean space consisting of l unit vectors. Then any vector x can be expressed as
and such an expression is determined uniquely. The coefficients c 1, c 2, ⋯, c l are computed as c i = 〈x, e i〉 (inner product).
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- 1.
- 2.
In the case of a real vector space \(\mathfrak {H}\), an inner product is a real-valued function \(\langle \cdot, \cdot \rangle : \mathfrak {H}\times \mathfrak {H}\rightarrow \mathbb {R}\) such that (i)–(iv) are satisfied. Of course, (ii) should be rewritten as 〈x, y〉 = 〈y, x〉.
- 3.
- 4.
The term of the highest degree = (1∕2n n!){(2n)(2n − 1)⋯(n + 1)}x n.
- 5.
The first process of integration by parts is as follows:
$$\displaystyle \begin{aligned}f(x)\frac{d^{n-1}}{dx^{n-1}}(x^2 -1)^n \Big|{}_{-1}^1-\int_{-1}^1 f'(x)\frac{d^{n-1}}{dx^{n-1}}(x^2-1)^n dx \end{aligned}$$$$\displaystyle \begin{aligned}=-\int_{-1}^1f'(x)\frac{d^{n-1}}{dx^{n-1}}(x^2-1)^{n}dx.\end{aligned}$$ - 6.
We denote by \(\mathbb {C}_+\) the complex half-plane \(\mathbb {R}ez>0\). The function \((\varGamma : \mathbb {C}_+\rightarrow \mathbb {C})\) defined by
$$\displaystyle \begin{aligned} \varGamma(s)=\int_0^\infty x^{s-1}e^{-x}dx \end{aligned} $$(t)is called the gamma function. Γ is analytic on \(\mathbb {C}_+\). Sometimes the domain of Γ is taken to be (0, ∞) rather than \(\mathbb {C}_+\). In the text above, we also follow this policy. The function \(B: \mathbb {C}_+\times \mathbb {C}_+\rightarrow \mathbb {C}\) defined by
$$\displaystyle \begin{aligned} B(p,q)=\int_0^1 x^{p-1}(1-x)^{q-1}dx \end{aligned}$$is called the beta function. (Note that the integral is convergent.) The following formulas hold good (cf. Cartan [1] Chap. V, §3 and Takagi [9] Chap. 5, §68):
-
1∘
Γ(s + 1) = sΓ(s).
We also obtain Γ(n + 1) = n! for all \(n\in \mathbb {N}\) by induction.
-
2∘
B(p, q) = Γ(p)Γ(q)∕Γ(p + q).
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3∘
\(\varGamma (s)\varGamma (1-s)=\pi /\sin \pi s.\)
If s = 1∕2, in particular, we have \(\varGamma (1/2)=\sqrt {\pi }\).
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4∘
\(\varGamma \big (n+\frac {1}{\, 2\,}\big )=\frac {1}{\,2\,}\big (\frac {3}{\, 2\, }\big )\cdots \big (n-\frac {1}{\, 2\,}\big )\sqrt {\pi } \quad (n=0,1,2,\cdots ).\)
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1∘
- 7.
More rigorously, the Fourier coefficient corresponding to \((1/\sqrt {2\pi })e^{inx}\) is
$$\displaystyle \begin{aligned}\frac{1}{\sqrt{2\pi}}\int_{-\pi}^\pi f(x)e^{-inx}dx,\end{aligned}$$and the Fourier series is
$$\displaystyle \begin{aligned}\sum_{n=-\infty}^\infty\frac{1}{\sqrt{2\pi}}\int_{-\pi}^\pi f(x)e^{-inx}dx\cdot\frac{1}{\sqrt{2\pi}} e^{inx}. \end{aligned}$$ - 8.
- 9.
We acknowledge Yosida [12] p. 88 for the proof here.
- 10.
The injectivity of T is also verified by Theorem 1.6 (ii).
References
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Maruyama, T. (2018). Fourier Series on Hilbert Spaces. In: Fourier Analysis of Economic Phenomena. Monographs in Mathematical Economics, vol 2. Springer, Singapore. https://doi.org/10.1007/978-981-13-2730-8_1
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