Some analysis problems. Let f be a continuous function on R and let for n =,2,..., F n (x) = x (x t) n f(t)dt. Prove that F n is n times differentiable, and prove a simple formula for its n-th derivative. 2. Let f(x,y) = n= (i) Show that for each y >, the it x x 2 +yn2, y >. g(y) := x + f(x,y) exists. Evaluate the it function g. (ii) Determine iff(x,y)converges to g(y)uniformlyfor y (, )as x +. 3. (i) Find an explicit value of ǫ > such that for every x [, ] x x+ǫ 2. (ii) Find an explicit integer N such that there exists a polynomial P of degree at most N such that for every x [, ] x P(x). Hint: Although this may not be clever, you can use the expansion of x+ǫ in power series in (x ). 4. Determine all complex-valued functions f, which are continuous in [, 2] and satisfy the condition 2 f(x)xn dx = for n =,,2,3,... 5. Does the series converge uniformly on [, )? Prove or disprove. n= e x/n( )n n Mostly from previous qualifying exams, some slightly modified
2 6. Let {a n } n= be a numerical sequence and let b n = n k 5 a n 6 k. k= (i) Does the it exist when a k = a is constant? (ii) Prove or disprove: If a n converges then b n converges. (iii) Prove or disprove: If b n converges then a n converges. 7. Given a sequence {c n } n= of complex numbers, we let s n = n k= c k denote the partial sums and σ N = s +...+s N their arithmetic means. We say that the N+ series n c n is Cesáro summable to σ if N σ N = σ. Show that if n c n is Cesáro summable to σ and n nc n = then the series c n converges and n= c n = σ. 8. Let s N (x) = N n= ( ) n x3n n 2/3. Prove that s N (x) converges to a it s(x) on [,] and that there is a constant C so that for all N the inequality holds. sup s N (x) s(x) CN 2/3 x [,] 9. Let α R, and let u be the function defined on (, ), by u(x) = x α. For which values of α does u(x+h) u(x) u (x), h uniformly on (, ), as h (h ).. Let f be a real-valued differentiable function defined on the entire real line. Assume that f(x+h) f(x) f (x), uniformly as h. Show that f is h uniformly continuous. Must f itself be uniformly continuous?
. Recall the following definition: a function g : (,) R has bounded variation if sup N 2 sup g(x ) g(x 2 ) +...+ g(x N ) g(x N ) <, <x N <...<x < where the second sup is taken over all strictly decreasing sequences x N < < x with x i (,). Find the exponents p for which the function f : (,) R, has bounded variation. f(x) = x p sin(/x) 2. Let λ, λ 2,..., λ n,... be real numbers. Argue that e iλnx f(x) = n 2 defines a continuous bounded function on R and then show that the it exists. T T n= T T f(x)dx 3. For every b R prove or disprove that the improper integral converges. x b cos(e x )dx 4. (i) Let {f n } be a sequence of C functions on a compact interval I such that f n (x) + f n (x) M for all x I. Show that there is a subsequence {f n k } which converges uniformly on I. (ii) Is the preceding statement still true if we drop the assumption that I is compact? (Proof or counterexample) (iii) Can one also show that under the assumptions in (i) the sequence f n has a subsequence whose derivatives converge uniformly? (Proof or counterexample) 5. Let n= a n be the rearrangement of ( ) n n= n are followed by N negative terms, i.e. n= 3 where M positive terms a n = 2 + 4 + + 2M 3 2N + 2(M +) +... Compute n= a n.
4 6. (i) Determine for which η R the indefinite integral exists. (ii) Determine whether the it η exists. (iii) Determine whether the it exists. η t iη sintdt t iη sintdt t iη sintdt 7. Let p < q <. Prove or disprove the following statements. (i) L p (R n ) L q (R n ). (ii) L q (R n ) L p (R n ). (iii) L q ([,]) L p ([,]). (iv) L p ([,]) L q ([,]). (v) l p (Z) l q (Z). (vii) l q (Z) l p (Z). (viii) l p (Z) l q (Z). (ix) L p (R n ) L q (R n ) L s (R n ) for all s [p,q]. 8. Suppose α, and let f be a bounded function on the real line with the property that f(x+h) f(x) A h α for all h R and almost all x R. Show that there is a constant C and for each t > a C -function g t such that sup f(x) g t (x) Ct α x and sup g t (x) Ctα. x Hint: Use an approximation of the identity.
9. Let X, Y be normed spaces, and let Ω be an open subset of X. A function f : Ω Y is differentiable at a Ω if there is a bounded linear transformation L : X Y so that f(a+h) f(a) L(h) Y =. h h X (i) Show that this linear transformation is unique. We call it the derivative of f at a, also denoted by f (a). (ii) Let M n be the set of n n matrices. Specify the norm on M n that you work with. Define g : M n M n by g(a) = A 3. Show that g is differentiable everywhere and determine the derivative g (A). (iii) Let Ω M n be the subset of invertible matrices. Show that Ω is open Define f : Ω Ω by f(a) = A. Show that f is differentiable on Ω and determine f (A). (iv) Let M n be the set of n n matrices and define f : M n R by f(a) = det(a). Show that f is differentiable everywhere and determine f (A). 2. Let b. The sequence {a n } n= satisfies a n+ = a n 2 + b for n. 2a n (i) Suppose that a n converges to L. Then determine L. (ii) Does a n converge? 2. Let f be a continuous function on [,]. Prove that ( ) /p f(x) p dx exists and identify this it. p 22. For which exponent p > is the function f(x,y) = x p +y 2 integrable in a neighborhood of in R 2? Hint: you can decompose the domain of integration in regions defined by 2 m x p +y 2 < 2 m. 23. Let a > and b >. Prove that there is a unique differentiable function f defined on (, ) satisfying f() = and f (x) = a b f(x) 3/2 for all x. Also show that x f(x) exists and determine this it. 5
6 24. Let g be the solution on some interval [, T) to the problem: Show that T <. g() = g (t) = [g(t)] 2( 2+sin(e t +g(t)) ) ]. 25. Let f L (R) (meaning that f(x) dx < ). Let G(λ) = e iλt2 f(t)dt. Prove that G is a continuous function and that λ G(λ) =. 26. (i) Does p N = N n=2 (ii) Does q N = N n=2 ( )n (+ n ( )n (+ R ) tend to a nonzero it as N? n ) tend to a nonzero it as N? 27. Let a be a decreasing C -function in [, ) such that t a(t) =. N (i) Show that N a(t)sin(tx)dt exists for all x >. N (ii) For ǫ > show that N a(t)sin(tx)dt converges uniformly for x [ǫ, ). (iii) Show that uniform convergence fails in (, ), for a suitable choice of a. 28. For n let a n = [log(2+n)]. (i) For which complex numbers z does the series n= a nz n converge? (ii) For which complex numbers z does the series n= a nz n converge absolutely? (iii) On which compact sets of the complex plane does the series n= a nz n converge uniformly? 29. Give an example of a Riemann integrable function f: [,] [,] which has a dense set of discontinuities. Verify all conclusions. 3. Let s n (x) = n sin(kx). Show that there exists a constant C, independent of N,x, such that N s n (x) < C, < x < π, N =,2,3,... n 2 n= k= (Hint: Estimate s n (x) for n x and for n > x separately.)
3. For λ >, define H(λ) = + Prove that, for some constant C >, e λ(x3 +x 5) dx. H(λ) = Cλ /3 +O(λ ). Hint : Evaluate + e λx3 dx in terms of λ and + e x3 dx. Use the same change ofvariables, andestimate thedifference + e λ(x3 +x 5) dx + e λx3 dx, dealing separately with large and small values of x. 32. Let M(n,R) bethe vector space of n n matrices with real entries. Denote by a norm on M(n,R). For A M(n,R) let tr(a) be the trace of A (that is the sum over all entries on the diagonal). Show that there are neighborhoods U, V of the identity matrix I such that for every A V there is a unique B U with n tr(b)b 3 = A. 33. Let u : R 3 R denote a smooth function and let u = 2 x u+ 2 y u+ 2 z u be the Laplacian of u. Suppose that u = on R 3 and u(x,y,z) = x 3 y 3 on the sphere of radius R centered at the origin. Find u(,,). 34. Calculate y 3 dx+xy 2 dy C (x 2 +y 2 ) 2 where C is the plane curve given by the equation x 2 + 22y 8 = 24, with the positive orientation. 35. Let D R d, d 2 be a compact convex set with smooth boundary D so that the origin belongs to the interior of D. For every y D let α(x) [,π) be the angle between the position vector x and the outer normal vector n(x). Let ω d be the surface area of the unit sphere in R d. Compute ω d D cos(α(x)) x d dσ(x) where dσ denotes surface measure on D. Does (a reasonable interpretation of) your result hold true if d =? 36. Let f be a function in C (R) with compact support and let b >. Show that the it A b (x) = ε + R\[ ǫ,bǫ] exists for all x R. How do A b (x) and A c (x) differ for b c? f(x y) dy y 7
8 37. Prove or disprove the following statement: ε x 2 +y 2 ε 2 f(x,y) (x+iy) 3 dxdy exists for every function f C 2 (R 2 ) with compact support. Hint: For < a < b, what are the values of x dxdy and (x+iy) 3 a 2 <x 2 +y 2 <b 2 38. Let X be a metric space with metric d. (i) Define ρ : X X R by a 2 <x 2 +y 2 <b 2 y (x+iy) 3 dxdy? ρ(x,y) = d(x,y) +d(x,y). Prove that ρ is a metric on X. (ii) Show that a subset U of X is open with respect to the metric d if and only if it is open with respect to the metric ρ. 39. Let f : R R be a continuous function with f(x) for all x. Consider the two statements f(x) dx converges, () f(n) converges. (2) n= (i) Discuss the truth of the implications () = (2) and (2) = (). (ii) Assume f is continuously differentiable and satisfies f (x) A for some constant A < ; again discuss the truth of the implications () = (2) and (2) = (). (iii) Finally, assume f (x) A f(x) for some constant A < and once more discuss the truth of the implications () = (2) and (2) = (). 4. Assume that a k > and + k= a k = +. Assume that (b k ) is a bounded sequence. Show that one can choose an increasing sequence of integers k(n) such that + a k(n) = + n= and the sequence ( b k(n) ) has a it.
4. (i) What is the volume of the region Ω in R n, defined by Ω = {(x,...,x n ) R n x j >, < x +x 2 + +x n < }? (ii) What is the area of the parallelogram spanned by the vectors (,,,) and (2,,2,) in R 4? (iii) What is the (3 dimensional) volume of the box (parallelepiped) spanned by the vectors (,,,,), (,,,,), and (,,,,) in R 5? 42. (i) Let ψ C (R) be a compactly supported continuous function. Show that Let N N J N = Does J N exist? N + (ii) Let χ C(R). 2 Prove that N + N + ψ(x/n) +x dx =. e ix +xdx. χ( x N )eix +xdx exists. (iii) To what extent does the it in part (3) depend on the choice of the function χ? 43. Let f be a positive decreasing function defined on (, ). This means that if < a < b <, then f(a) f(b) >. Let ǫ > be a fixed positive number. (i) Suppose that for all < x <, f(2x) 2 ǫ f(x). Prove that there is a constant C depending only on ǫ so that for a >, a f(x)dx Caf(a). (ii) Suppose that for all < x <, f(x) 2 + ǫ f(2x). Prove that there is a constant C depending only on ǫ so that for a >, a f(x)dx Caf(a). (iii) Suppose that for all < x <, f(2x) 2 f(x). Prove that the improper integral f(x)dx diverges. 9
44. For a,b >, let F(a,b) = For which p > is it true that + dx x 4 +(x a) 4 +(x b) 4. F(a,b) p dadb < +? Hint: Do not try to evaluate the integral defining F(a,b) directly. Instead, first suppose a b and show that there are positive constants C and C 2 so that C b 3 F(a,b) C 2. 45. (i) State and prove the Baire category theorem for complete metric spaces. (ii)let{f n } n beasequence ofrealvaluedcontinuousfunctionsontheinterval [,], and let E be the set of x [,] for which sup n f n (x) =. Show that E cannot be [,] Q. 46. Let {f n } n= be a sequence of continuous functions on [,] and assume that sup n f n (x) < for every x [,]. Show that there exists an interval (a,b) [,] and an M R so that f n (x) M for all x (a,b) and all n =,2,... 47. Consider a function f : R R. (i) Prove: If the second derivative f (x ) exists then f(x +h) 2f(x )+f(x h) = f (x h h 2 ). (ii) Suppose that h f(x +h) 2f(x )+f(x h) h 2 exists. Is it true that the second derivative of f exists at f (x )? Give a proof or a counterexample!
48. Let f be a function defined in the interval [ 2,2] satisfying f(b) f(a) f(c) f(b) A b a c b whenever 2 a < b < c 2 (i.e., f is a convex function). Show that there is C such that for h f(x+h)+f(x h) 2f(x) dx Ch 2. Hint: Show first that if x < x < < x N and x i x i = h then N f(x i+ ) 2f(x i )+f(x i ) C h 49. Let K be a continuous function on the unit square Q = [,] [,] with the property that K(x,y) < for all (x,y) Q. Show that there is a continuous function g defined on [,] so that g(x)+ K(x,y)g(y)dy = ex +x2, x. 5. Prove that there is a unique C function f defined on [,] which satisfies the integral equation x t cos(tx)f(t) f(x)+ dt = +f(t) 2 for all x [,].