Question

Masaaaa

問題としてはこのURLのやつでexercise2.2.9の問題です。

2.2.9. Define T : ℓ^2(Zn ) → ℓ^2(Zn ) by
(T(z))(n) =z(n + 1) − z(n).
Find all eigenvalues of T.

https://www.researchgate.net/profile/Seyed-Yahya-Moradi/post/Can-anyone-suggest-a-wavelet-analysis-textbook/attachment/59d63a0cc49f478072ea6628/AS%3A273723943260161%401442272281897/download/An+Introduction+to+Wavelets+Through+Linear+Algebra.pdf

16:22マ
l 全
の
Exerc:
164/520
matrices, convolution operators, and Fourier r
operators.
2.2.9. Define T:l'(Zn) -
→ e°(ZN) by
ニ
Find all eigenvalues of T.
2.2.10. Let T(m):e'(Z4)
→ '(Z) be the Fourier multipliei
(mz)' where m = (1,0, i, -2)
defined by T (m)(2) =
i. Find be l(Z4) such that T(m) is the convolutior
Tb (defined by Th(Z) = b*z).
ii. Find the matrix that represents T(m) with resp
standard basis.
2.2.11. i. Suppose Ti, T2:l(ZN)
→ e(ZN) are tra
invariant linear transformations. Prove that th
sition T, o T, is translation invariant.
ii. Suppose A and B are circulant NxN matric
directly (i.e., just using the definition of a
matrix, not using Theorem 2.19) that AB is
Show that this result and Theorem 2.19 imp
Hint: Write out the (m + 1,n+1) entry of
the definition of matrix multiplication; compare
hint to Exercise 2.2.12 (i).
iii. Suppose b,, bz e l'(Zn). Prove that the cor
Tb, o Tb, of the convolution operators Tb, and
convolution operator T, with b =
2
bz * b.. E
Exercise 2.2.6.
iv. Suppose m,, mz € l"(Z). Prove that the cor
T(m2) ° T(m)
and T(m) is the Fourier multiplier operator T)
m(n) = m2(n)m」(n) for all n.
v. Suppose Ti, T2:l"(Zw) → e'(Zn) are linear tra
tions. Prove that if Ti is represented bya matri
respect to the Fourier basis F (i.e., [T; (z)]F =A
Tz is represented by a matrix Az with respect t
the composition T20T, is represented by the ma
with respect to F. Deduce part i again.
Remark:ByTheerem 2.19, we have just proved
of the Fourier multiplier operat
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Answer 1

Crystal Clear

Tを行列の形にすると巡回行列になります。
https://ja.wikipedia.org/wiki/%E5%B7%A1%E5%9B%9E%E8%A1%8C%E5%88%97
このwikiでの記法だと、c[0]=-1, c[n-1]=1, 他は0　のバージョンです。
https://manabitimes.jp/math/2289
たぶん本文にも書いてあると思います。

巡回行列 離散フーリエ変換