Q 5.5-13E Question: For a function f聽in C... [FREE SOLUTION]

91影视

Linear Algebra With Applications

Otto Bretscher

$Math Studyset 91影视 Explanations$ Math

5th Edition 路 442 pages

ISBN: 9780321796974

Chapter 5: Q 5.5-13E (page 261)

Question: For a function fin $C [- 蟺, 蟺]$ (with the inner product defined on page 255), consider the sequence of all its Fourier coefficients. Is this infinite sequence in I₂ ?f so, what is the relationship between
|| f || (the norm in $C [- 蟺, 蟺]$ )
And $| | (a_{0}, b_{1}, c_{1}, b_{2}, c_{2}, . ....) | |$ (the norm in I₂ ) The inner product space I₂ was introduced in Example 2.

Short Answer

Expert verified

The sequence is $(a_{0}, b_{1}, c_{1}, b_{2}, c_{2}, ....., b_{n}, c_{n} ....,) 鈭� l_{2}$ and $| | (a_{0}, b_{1}, c_{1}, b_{2}, c_{2}, ....., b_{n}, c_{n} ....,) | | = | | f | |$ .

Step by step solution

Consider the theorem below.

$({a_{0}}^{2}, {b_{1}}^{2}, {c_{1}}^{2}, . ...., {b_{n}}^{2}, {c_{n}}^{2} + . ...,) = | | f | |^{2}$

The infinite series of the squares of the Fourier coefficients of a piecewise continuous function fconverges to $| | f | |^{2}$ .

Now, observe that $({a_{0}}^{2}, {b_{1}}^{2}, {c_{1}}^{2}, ....., {b_{n}}^{2}, {c_{n}}^{2} + ....,) = | | f | |^{2}$ .

Which implies the sequence $(a_{0}, b_{1}, c_{1}, b_{2}, c_{2}, ....., b_{n}, c_{n} ....,)$ is square sum able and hence is in $l_{2}$ .

Also, observe that

$| | (a_{0}, b_{1}, c_{1}, b_{2}, c_{2}, ....., b_{n}, c_{n} ....,) | | = \sqrt{{a_{0}}^{2}, {b_{1}}^{2}, {c_{1}}^{2}, ....., {b_{n}}^{2}, {c_{n}}^{2} + ....} \begin{array}{l} = \sqrt{| | f | |^{2}} \\ = | | f | | \end{array}$

Thus, we proved that the sequence $(a_{0}, b_{1}, c_{1}, b_{2}, c_{2}, ....., b_{n}, c_{n} ....,) 鈭� l_{2}$ and .

$| | (a_{0}, b_{1}, c_{1}, b_{2}, c_{2}, ....., b_{n}, c_{n} ....,) | | = | | f | |$

Hence, the sequence are

$(a_{0}, b_{1}, c_{1}, b_{2}, c_{2}, ....., b_{n}, c_{n} ....,) 鈭� l_{2}, | | (a_{0}, b_{1}, c_{1}, b_{2}, c_{2}, ....., b_{n}, c_{n} ....,) | | = | | f | |$ , .

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91影视

Short Answer

Step by step solution

Consider the theorem below.

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