QuestionQuestion

Transcribed TextTranscribed Text

1. Let f be a function on the circle. For each N ≥ 1 the discrete Fourier coefficients of f are defined by aN (n) = 1 N X N k=1 f(e 2πik/N )e −2πikn/N , for n ∈ Z. We also let a(n) = Z 1 0 f(e 2πix)e −2πinx dx denote the ordinary Fourier coefficients of f. (a) Show that aN (n) = aN (n + N). (b) Prove that if f is continuous, then aN (n) → a(n) as N → ∞. G 4. Let e be a character on G = Z(N), the additive group of integers modulo N. Show that there exists a unique 0 ≤ ` ≤ N − 1 so that e(k) = e`(k) = e 2πi`k/N for all k ∈ Z(N). Conversely, every function of this type is a character on Z(N). Deduce that e` 7→ ` defines an isomorphism from ˆ to G. [Hint: Show that e(1) is an Nth root of unity.] 5. Show that all characters on S 1 are given by en(x) = e 2πinx with n ∈ Z, and check that en 7→ n defines an isomorphism from Sc1 to Z. [Hint: If F is continuous and F(x + y) = F(x)F(y), then F R is differentiable. To see this, note that if F(0) 6= 0, then for appropriate δ, c = δ 0 F(y) dy 6= 0, and cF(x) = R δ+x x F(y) dy. Differentiate to conclude that F(x) = e Ax for some A.

Solution PreviewSolution Preview

These solutions may offer step-by-step problem-solving explanations or good writing examples that include modern styles of formatting and construction of bibliographies out of text citations and references. Students may use these solutions for personal skill-building and practice. Unethical use is strictly forbidden.

    By purchasing this solution you'll be able to access the following files:
    Solution.pdf.

    $25.00
    for this solution

    or FREE if you
    register a new account!

    PayPal, G Pay, ApplePay, Amazon Pay, and all major credit cards accepted.

    Find A Tutor

    View available Advanced Math Tutors

    Get College Homework Help.

    Are you sure you don't want to upload any files?

    Fast tutor response requires as much info as possible.

    Decision:
    Upload a file
    Continue without uploading

    SUBMIT YOUR HOMEWORK
    We couldn't find that subject.
    Please select the best match from the list below.

    We'll send you an email right away. If it's not in your inbox, check your spam folder.

    • 1
    • 2
    • 3
    Live Chats