-
![]()
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
Aresearchgate.net - 非公開
