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    Real 2 Trialtwo 2024 Chapter 3 Only

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    mirumbeismail
    The document contains 6 questions regarding real analysis and Fourier series. Question 1 involves finding orthogonal polynomials and using them to approximate a function. Question 2 involves determining values to make a set of functions orthogonal and using it to approximate another function. Questions 3-5 involve properties of orthogonal sequences including Parseval's equality. Question 6 asks to find the Fourier series for several functions.

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    Real 2 Trialtwo 2024 Chapter 3 Only

    Uploaded by

    mirumbeismail
    19 views1 page
    The document contains 6 questions regarding real analysis and Fourier series. Question 1 involves finding orthogonal polynomials and using them to approximate a function. Question 2 involves determining values to make a set of functions orthogonal and using it to approximate another function. Questions 3-5 involve properties of orthogonal sequences including Parseval's equality. Question 6 asks to find the Fourier series for several functions.

    Original Description:

    These are questions on Fourier series

    Original Title

    Real2trialtwo2024chapter3only

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    The document contains 6 questions regarding real analysis and Fourier series. Question 1 involves finding orthogonal polynomials and using them to approximate a function. Question 2 involves determining values to make a set of functions orthogonal and using it to approximate another function. Questions 3-5 involve properties of orthogonal sequences including Parseval's equality. Question 6 asks to find the Fourier series for several functions.

    Copyright:

    © All Rights Reserved
    Download as PDF, TXT or read online from Scribd
    Download as pdf or txt
    19 views1 page

    Real 2 Trialtwo 2024 Chapter 3 Only

    Uploaded by

    mirumbeismail
    The document contains 6 questions regarding real analysis and Fourier series. Question 1 involves finding orthogonal polynomials and using them to approximate a function. Question 2 involves determining values to make a set of functions orthogonal and using it to approximate another function. Questions 3-5 involve properties of orthogonal sequences including Parseval's equality. Question 6 asks to find the Fourier series for several functions.

    Copyright:

    © All Rights Reserved
    Download as PDF, TXT or read online from Scribd
    Download as pdf or txt
    of 1

    MTH3201: REAL ANALYSIS II − Sem II TRIAL QUESTIONS set III 2023/2024 1

    Makerere University
    Department of Mathematics
    Date:14th March, 2024

    1. (a) Show that the polynomials f0 (x) = 1, f1 (x) = x and f2 (x) = 23 x2 − 1


    2 are orthogonal on the closed
    interval [−1, 1].
    (b) Let (
    0, if −1 ≤ x < 0
    f (x) =
    1, if 0 ≤ x ≤ 1
    Find the constants c0 , c1 and c2 such that s2 (x) = c0 f0 + c1 f1 + c2 f2 gives the best approximation
    in the mean to f on [−1, 1]
    2. (a) Let ϕ0 (x) = 1, ϕ1 (x) = x − a1 , ϕ2 (x) = x2 − a2 x − a3 . Determine the values of a1 , a2 and a3 such
    that the set of functions {ϕo , ϕ1 , ϕ2 } is orthogonal on [0, 1].
    (b) Find the polynomial of degrees less or equal to two(2) that best approximates f (x) = sin πt in the
    mean on [0, 1].
    3. (a) Show that the sequence {sin nx}, n ∈ N is orthogonal on [0, π].
    (b) For f ∈ R[0, π], show that the Fourier series of f with respect to sequence {sin nx}, n ∈ N is given
    ∞ Rπ
    bn sin nx where bn = π2 0 f (x) sin nxdx
    P
    by
    n=1
    (c) Find the Fourier series of f (x) = 3x on [0, π] in terms of the orthogonal sequence {sin nx}, n ∈ N.
    4. (a) Show that the sequence of {1, cos nx}∞
    n=1 is orthogonal on [0, π]
    ∞
    (b) For f ∈ R[0, π] show that the Fourier series of the above sequence is given by 12 a0 +
    P
    an cos nx
    n=1
    2 π 2 π
    R R
    where a0 = π 0 f (x)dx and an = π 0 f (x) cos nxdx
    (c) Find the Fourier series expansion of f (x) = 3x on [0, π] in terms of the orthogonal sequence
    {1, cos nx}.
    5. (a) Assuming that the orthogonal system {sin nx} is complete on [0, π]. Show that parseval’s equality
    ∞ Rπ Rπ
    b2n = π2 0 (f (x))2 dx where bn = π2 0 f (x) sin xdx.
    P
    becomes
    n=1
    (b) Use Parseval’s equality and the indicated function to find the sum of the given series
    ∞
    X 1
    (i) , f (x) = 1
    (2k − 1)2
    k=1
    ∞
    X 1
    (ii) , f (x) = x.
    k2
    k=1

    6. Find the Fourier series of each of the following functions f on [−π, π],
    (a) (
    −1, if −π < x < 0
    f (x) =
    1, if 0 ≤ x < π

    (b) f (x) = |x|


    (c) f (x) = 1 + x.

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