LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – MATHEMATICS

XZ 26

FIRST SEMESTER – APRIL 2008

MT 1805 – REAL ANALYSIS

 

 

 

Date : 30-04-08                  Dept. No.                                        Max. : 100 Marks

Time : 1:00 – 4:00

 

1. a) 1) Let and  on [a,b] then prove that

        (i) on [a,b] and (ii)

OR

 2)   Define step function and prove: If a < s < b, on [a,b] and , the unit step

      function, then prove that                                                     (5)

1. b) 1) Let , n = 1,2,3,… . Suppose that is convergent and {s_n} is a sequence of distinct

numbers in (a,b). Let . Let  f  be continuous on [a,b] then prove that

   2) Let be monotonically increasing function on [a,b] and let on [a,b]. If f is a bounded real

       function on [a,b] then prove that

              OR

     3) Let on [a,b] for . Define , then prove that F is continuous on [a,b].

If F is continuous at some point , then prove that F is differentiable at x_o and .

    4) State and prove the fundamental theorem of Calculus and deduce the following result:

 

Suppose F and G are differentiable functions on [a,b], then prove that                                             (6 + 9)

1. a) 1) Let exists then prove that it is unique.

                     OR

    2)   Define a convex set and prove: Suppose that  maps a convex set ;  is

          differentiable on E and there exists a constant M such  that then prove that

(5)

1. b) 1) When do you say a function is continously differentiable? Letmaps an open set

         show that  if and only if the partial derivatives D_jf_i exists and are

continuous on E for                                               (15)

                OR

2) a)  State and prove the Contraction principle.

 

1. b) Let C(X) denote the set of all continuous, complex valued, bounded functions onX. Prove that C(X)

          is a complete metric space.                                                             (5+10)

III. a)1) Prove that every converging sequence is a Cauchy’s sequence. Is the converse true?

        OR

1. b) 1) State and prove the Cauchy criterion for uniform convergence.

2) Suppose {f_n} is a sequence of differentiable functions on [a,b]. Suppose that {f_n(x₀)} converges uniformly on [a,b] then prove that {f_n} converges uniformly on [a,b] to some function f and                                             (5 + 10)

        OR

3) State and prove Stone-Weierstrass theorem.                                                    (15)

1. a)1) Is the trignometric series a Fourier series? Justify your answer.

       OR

2) Define a Gamma function and state the three properties that characterize Gamma function completely.                                                                                                            (5)

b)1) State and prove the Parseval’s theorem.

2) If f is continuous (with period ) and if  then prove that there is a trignometric polynomial P such that  for all real x.                                          (10 + 5)

        OR

3) State and prove the Dirichlet’s necessary and sufficient condition for a Fourier series to converge to a sum s.                                                                                                      (15)

1. a)1) Write a note on Lagrange’s polynomial.

OR 

2) Write a note on Chebyshev polynomial.                                                                       (5)

b)1)  Let f be a continuous function on [a,b] and assume that T is a polynomial of degree  n that best approximates f on [a,b] relative to the maximum norm. Let R(x) = f(x) – T(x) denote the error in this approximation and let . Then prove

1. i) If D = 0 the function R is identically zero on [a,b].
2. ii) If D>0, the function R has at least (n+1) changes of sign on [a,b]. (15)

 OR

2)  If f(x) has m continuous derivatives and no point occurs in the sequence x_o, x₁, x₂, …, x_n more than (m + 1) times then prove that there exists exactly one polynomial P_n(x) of degree n which agrees with f(x) at x_o, x₁, x₂, …, x_n.

3) Let P _n+1 (x) = x ⁿ⁺¹+ Q(x), where Q(x) is a polynomial of degree n, and let . Then prove .                                                                              (10+5)

 

 

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

AB 27

M.Sc. DEGREE EXAMINATION – MATHEMATICS

FIRST SEMESTER – November 2008

    MT 1805 – REAL ANALYSIS

 

 

 

Date : 06-11-08                 Dept. No.                                        Max. : 100 Marks

Time : 1:00 – 4:00

 

Answer ALL the questions

I a)1) If  (with the usual notations) holds for some P and some prove that the same holds for every refinement of P.

OR

    2) If f is continuous on [a,b] then prove that   on [a,b].                                     (5)

1. b) Suppose on [a,b], , is continuous on [m,M], and h(x) = (f(x)) on [a,b]. Then prove that on [a,b]
2. c) State and prove the fundamental theorem of Calculus for a function on [a,b]. (9 + 6)

OR

1. d) Let be a monotonically increasing function on [a,b] and let on [a,b]. If f is a bounded real function on [a,b] then prove that on [a,b] on [a,b].              In this case
2. e) If f maps [a,b] into R^k and if for some monotonically increasing function on [a,b]  then prove that  and                                               (8+7)
3. II. a) 1) Let be the set of all invertible operators on .Then prove that is open and the mapping A A^-1 is continuous on .

OR

         2) Let f be a differentiable function from E into R^m where E is an open set contained in Rⁿ. Then prove that the linear transformation from Rⁿ to R^m is unique.                                  (5)

1. b) Define Convex set and prove: Suppose that  maps a convex set E  into ; is differentiable on E and there exists a constant M such that then . Also prove that if f’(x) = 0 for all x in E then f is constant.
2. c) State and prove the chain rule on the differentiability of a function.         (7+8)

OR

1. d) Suppose that maps a convex set E into . Let  is differentiable at x Then prove that the partial derivatives D_j f_i (x) exists and , where {e₁, e₂, …, e_n} and  {u₁, u₂, …, u_m} are the standard basis of  and respectively.                                                                     (15)

 

III.a) 1) Let denote the set of all continuous, complex valued, bounded functions on X. prove that is a complete metric space.

OR

2) If  is a sequence of continuous functions on E, and if uniformly on E, then prove that f is continuous on E. Is the converse true? Justify your answers.

 

 

1. b) State and prove the Weierstrass approximation theorem. (15)

OR

1. c) Let be monotonically increasing on [a,b]. Suppose on [a,b], for n = 1,2,3,… , and suppose uniformly on [a,b], Then prove that on [a,b] and
2. d) Define equicontinuity of a function and prove: If K is compact, if for n = 1,2,3,… and if {f_n} is pointwise bounded and equicontinuous on K, then

(i)   {f_n}  is uniformly bounded on K,

(ii)  {f_n}  contains a uniformly convergent subsequence.                                                       (6+9)

1. a)1) Prove that G = .

OR

2) If then prove that where E is a periodic function with period 2.       (5)

 

1. b) Define Gamma function and derive a simple approximate expression for when x takes on very large values.
2. c) Derive the relationship between Beta and Gamma function. (10+5)

OR

1. d) Explain with usual notations: Fourier series, orthogonal and orthonormal system. And prove the following theorem: Let {f_n } be orthonormal on [a,b]. Let S _n (x) = be the n^th partial sum of the Fourier series of f and suppose that t_n (x) = . Then prove that  and equality holds if and only if g_m =  c _m , m = 1,2, …,n.           (15)
2. V) a) 1) If f has a derivative of order n at a point x₀, then prove that the Taylor Polynomial is the unique polynomial such that whatever Q may be in P ^{( n )}.

OR

2) Define Chebyshev polynomial and list down its properties.                                              (5)

1. b) Given n+1 distinct points x ₀,x ₁, …, x _n and n+1 real numbers f (x₀), f (x₁),  …,       f (x _n) not necessarily distinct, then prove that there exists one and only one polynomial P of degree £ n such that P (x _j) = f (x _j) for each j = 0,1,2,…,n.  and the polynomial is given by the formula  where .
2. c) Let P _n+1 (x)= x ⁿ⁺¹ + Q(x) where Q is a polynomial of degree £ n and let maximum of ½P _n+1 (x)½, -1 £ x £ Then prove that we get the inequality . Moreover , prove that  if and only if , where T _n+1 is the Chebyshev polynomial of degree n+1.                                                                                                                               (7 + 8)

OR

1. c) Let f be a continuous function on [a,b] and assume that T is a polynomial of degree £ n that best approximates f on [a,b] relative to the maximum norm. Let R(x) = f (x) –T(x) denote the error in the approximation and let D = . Then prove that

(a) If D = 0 the function R is identically zero on [a,b].

(b) If D > 0, the function R has at least (n+1) changes of sign on [a,b].                               (15)

 

 

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