LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034      M.Sc. DEGREE EXAMINATION – MATHEMATICS

AA 19

FIRST SEMESTER – NOV 2006

         MT 1805 – REAL ANALYSIS

 

 

Date & Time : 28-10-2006/1.00-4.00         Dept. No.                                                       Max. : 100 Marks

 

 

1. a)(1) When does the Riemann-Stieltjes integral reduce to Riemann integral. Explain with usual notations.

OR

(2) If a < s < b, f ÎÂ (a) on [a,b] and a (x) = I (x – s), the unit step function, then prove that = f (s).                                                                                                             (5)

b)(1) Let f be a bounded function on [a,b] having finitely many points of discontinuity on [a,b]. Let a be continuous at every point at which f is discontinuous. Prove that f ÎÂ(a).                                                                                                                                                (8)

(2) Suppose f is strictly increasing continuous function that maps an interval [A.B] onto [a,b]. Suppose a is monotonically increasing on [a,b] and f ÎÂ (a) on [a,b]. Define b and g on [A,B] by b (y) = a (f (y)), g (y) = f (f (y)). Then prove that g ÎÂ (b) and .                                                                                                     (7)

OR

(3) Let a be monotonically increasing function on [a,b] and let a¢ Î R on [a,b]. If f is a bounded real function on [a,b] then prove that f ÎÂ (a) on [a,b] Û f a¢ ÎÂ (a) on [a,b].(8)

(4) Let f ÎÂ (a) on [a,b]. For a £ x £ b, define F(x) = , then prove that F is continuous on [a,b]. Also, if f is continuous at some x _o Î (a,b) then prove that F is differentiable at x _o and F¢ ( x _o ) = f (x _o ).                                                                           (7)

1. a) Let : [a,b] ® R ^m and let x Î (a,b). If the derivatives of exist at x then prove that it is unique.

OR

(2) Suppose that  maps a convex open set E Í Rⁿ into R^m,  is differentiable on E and there exists a constant M such that  M, ” x Î E, then prove that

ú  (b) –  (a)ú £ M ú b – aú , ” a, b Î E.                                                                      (5)

 

1. b) (1) Suppose E is an open set in R ⁿ ; maps R into R ^m ; is differentiable at x _o Î E,  maps an open set containing    (E) into R ^k and  is differentiable at f (x_o). Then the mapping of E into R ^k, defined by is differentiable at x_o and .                                                                                                  (8)

(2) Suppose  maps an open set EÍ Â ⁿ into  ^m. Let   be differentiable at x Î E, then prove that the partial derivatives (D_j f _i) (x) exist and , 1£ j £ m, where {e ₁, e  _2, e  _3, …, e _n} and {u ₁, u _2, u _3, …, u _m} are standard bases of R ⁿ and R ^m.  (7)

(3) If X is a complete metric space and if f is a contraction of X into X, then prove that there exists one and only one x ÎX such that f (x) = x.                                          (15)

 

III.  a) (1) Prove:  where {f _n} converges uniformly to a function f on E and x is a limit point of a metric space E.

OR

(2) Suppose that {f _n} is a sequence of functions defined on E and suppose that                  ½f _n (x)½£ M _n, x ÎE, n = 1,2,… Then prove that converges uniformly on E if converges.                                                                                                                (5)

1. b) (1) Suppose that K is a compact set and

* {f _n} is a sequence of continuous functions on K

** {f _n} converges point wise to a continuous function f on K

*** f _n (x) ³ f _n+1 (x), ” n ÎK, n= 1,2,… then prove that f _n ® f  uniformly on K. (7)

(2) State and prove Cauchy criterion for uniform convergence of complex functions defined on some set E.                                                                                                                         (8)

OR

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

IV a) (1)Show that  converges if and only if n >0.

OR

(2) Prove that G  = .                                                                                         (5)

b)(1) Derive the relation between Beta and Gamma functions.                                       (7)

(2) State and prove Stirling’s formula.                                                                          (8)

OR

3) If f is a positive function on (0,¥) such that f (x+1) = x f (x);  f (1) =1 and log f is convex then prove that f (x) = G (x).                                                                                               (8)

(4) If x >0 and y >0  then                                        (7)

1. a) (1)If f (x) has m continuous derivatives and no point occurs in the sequence x ₀, x ₁, ..,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 ₀, x ₁, …, x _n.

OR

2) Show that the error estimation for sine or cosine function f in linear interpolation is given by the formula ½f(x)-P(x)½£ .                                                                    (5)

b)(1) Let x₀, x₁, …, x_n be n+1 distinct points in the domain of a function f and let P be the interpolation polynomial of degree £ n, that agrees with f at these points. Choose a point x in the domain of f and let [a,b] be any closed interval containing the points x ₀, x ₁, …, x _n  and x. If f has a derivative of order n+1 in the interval [a,b], then prove that there is at least one point c in the open interval (a,b) such that  where A (x) = (x – x₀) (x – x₁)…(x – x _n).            (7)

(2) 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 £ 1.  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.                                          (8)

OR

3) 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

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

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

 

 

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – MATHEMATICS

CV 28

FIRST SEMESTER – APRIL 2007

MT 1805 – REAL ANALYSIS

 

 

 

Date & Time: 27/04/2007 / 1:00 – 4:00Dept. No.                                              Max. : 100 Marks

 

 

Answer all the questions. Each question carries 20 marks.

 

1. (a). (i). Prove that refinement of partitions decreases the upper Riemann Stieltjes sum.

 

(OR)

 

(ii). If f is monotonic on [a, b], and if  is continuous on [a, b], then prove                                that  on [a, b].                                                                      (5)

 

(b). (i). Suppose c_n ≥ 0, for n = 1, 2, 3 …, converges, and { s_n} is a sequence of

distinct points in [a, b]. If  and f is continuous on [a, b], then prove that .

                  (ii). Suppose that  on [a, b], m ≤  f  ≤ M,  is continuous on   [m,M],and  on [a, b]. Then prove that  on [a, b].                                 (7+8)

 

(OR)

 

(iii). Assume that  increases monotonically and  on [a, b]. Let f be a

bounded real function on [a, b]. Then prove that  if and only if  and in

that case.

(iv). State and prove the fundamental theorem of Calculus.                                            (8+7)

 

2. (a). (i). Prove that a linear operator A on a finite dimensional vector space X is

one-to-one if and only if the range of A is all of X.

 

(OR)

 

(ii). If  then prove that  and .               (5)

 

(b). (i). Let  be the set of all invertible linear operators on R^k. If

, and  then prove that .

(ii). Obtain the chain rule of differentiation for the composition of two

functions.                                                                                                                  (7+8)

 

(OR)

 

                  (iii). Suppose  maps an open set E Rⁿ into R^m. Then prove that

if and only if the partial derivatives  exist and are continuous on E for , .

(iv). If X is a complete metric space and if  is a contraction of X into X,

then prove that there exists one and only one x in X such that .                 (8+7)

 

 

1. (a). (i).Show by means of an example that a convergent series of continuous functions

may have a discontinuous sum.

(OR)

(ii). State and prove the Cauchy criterion for uniform convergence.                               (5)

 

(b). (i). Suppose on a set E in a metric space. Let x be a limit point of E

and suppose that . Then prove that converges and that .

(ii). Let  be monotonically increasing on [a, b]. Suppose on [a, b],

for n = 1, 2, …, and suppose that  uniformly on [a, b]. Then prove that      on [a, b] and that.                                                            (8+7)

 

(OR)

(iii). If f is a continuous complex function on  [a, b], then prove that there

exists a sequence of polynomials P_n such that uniformly on [a, b]. (15)

 

1. (a). (i). Define the exponential function and obtain the addition formula.

(OR)

(ii). If , prove with usual notation that E(it) 1.                     (5)

 

(b). (i). Given a double sequence, i = 1,2,…,  j = 1,2,…, suppose that

and  converges. Then prove that .

(ii). Suppose that the series and converges in the segment

S = (–R, R). Let E be the set of all x in S at which  = . If E has a limit point in S, then prove that for all n.                                                                 (7+8)

(OR)

 

            (iii). State and prove the Parseval’s theorem.                                                                   (15)

 

 

 

 

 

 

 

1. (a). (i). If f has a derivative of order n at a point x₀, then prove that the Taylor

polynomial  is the unique polynomial such that

for any polynomial Q of degree ≤ n.

 

(OR)

 

(ii). Define the Chebychev polynomial T_n and prove that it is of degree n and that

the coefficient of xⁿ is 2^n–1.                                                                 (5)

 

(b). (i). State and prove the construction theorem.

(ii). Let where  is a polynomial of degree ≤ n, and let

. Then prove that , with equality if and

only if  where  is the Chebychev polynomial of degree n+1.                                                                                                                                     (8+7)

(OR)

 

(iii). Let x₀, x₁, …, x_n be n+1 distinct points in the domain of a function f and let P

be the interpolating polynomial of degree ≤ n, that agrees with f at these points. Choose a point x in the domain of f and let [a, b] be any closed interval containing the points x₀, x₁, …, x_n and x. If f has derivative of order n+1 in [a, b] then prove that there is a point c in (a, b) such that , where .

 

(iv). If f(x) has m continuous derivatives and no point occurs in the sequence x₀,

x₁, …, x_n more than m+1 times, then prove that there exists one polynomial  P_n(x) of degree ≤ n which agrees with f(x) at x₀, x₁, …, x_n.                                                                                                                                                                                                      (8+7)

 

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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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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – MATHEMATICS

FIRST SEMESTER – NOVEMBER 2012

MT 1816 – REAL ANALYSIS

(12 BATCH STUDENTS ONLY)

 

Date : 05/11/2012            Dept. No.                                        Max. : 100 Marks

Time : 1:00 – 4:00

Answer all the questions. Each question carries 20 marks.

I.a) 1) Prove that refinement over an interval increases the lower sum and decreases the   upper sum.

OR

a)2. Using the notion of Upper sums and Lower sums of a bounded function, when do you say that  on [a,b]. When does the Riemann-Stieltjes integral reduces to Riemann integral?                                                                                      (5)

 

b)1) Suppose f is bounded on [a,b]. f has finitely many points of discontinuity on [a,b] and is continuous at every point at which f is discontinuous the prove that  .

b)2) Prove: 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                                                                                   (5 +10)

OR

c)1) Prove: on [a,b]

c)2) Suppose c Î (a,b) and two of the three integrals ,  and  exist. Then prove that the third also exists and

c)3) If f is monotonic on [a,b] and if   is continuous on [a,b], then  prove that .                                   (4+4+7)

1. a)1) State and prove the theorem on Cauchy criterion for uniform convergence.

OR

    a)2) Using a suitable example show that the limit of the integral need not be equal to the integral of the limit.                                                                                           (5)

b)1) If X is a metric space and denote the set of all complex valued, continuous, bounded functions with domain X. Choosing a suitable distance function between the elements of   prove that is a metric space  and also a complete metric space.

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

OR

c)1) Let . Verify whether .

c)2) Suppose  converges uniformly to a function f on E, where E is a set as a metric space. Let x be a limit point of  E and suppose that . Then prove that {A_n}conveges and                                                               (5+10)

III. a)1) State and prove Parseval’s formula.

OR

      a)2) State and prove Dini’s theorem.                                                                          (5)

      b)1) State and prove Jordan’s theorem.

      b)2)State and prove Riemann- Lebesgue Lemma.                                                   (6+9)

OR

c)1) State and prove Riesz – Fischer’s theorem.

c)2) State and prove Riemann Localization theorem.                                                   (6+9)

 

1. a)1) Suppose E is an open set in Rⁿ and f maps E into R^m and x is an element in E such that when , . Then prove that A is unique.

OR

a)2) Let be the set of all invertible linear operators on Rⁿ. If then prove that .                                                                                                                                                                           (5)

1. b) State and prove Inverse function theorem.                           (15)

OR

c)1) If  and c is a scalarthen prove that  . And with the distance between A and B defined as   , prove that is a metric space.

c)2) Define Contraction principle and prove the following theorem: Let X is a complete metric space and if  is a contraction of  X into X then  there exists only one xX  such that (x)=x.                                                                                                                        (5+10)

 

Va)1). Derive the rectilinear co ordinates.

OR

 a)2) Derive the sum of powers of .                                                                         (5)

 

 b)1) Explain how the product and quotient rule are derived for functions f(x) and g(x)?

OR

b)2) Derive the expression for D’ Alembert’s wave equation for a vibrating string.

 

 

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