Advanced Calculus II as of

Homework

1. Homework I on Sequences and Series: Due We, Jan 17 in class

   
   For this set of questions, you should try all the questions 1-4. But turn in only one well written solution of your choice. (Be sure to include "all" the details!). Due We, Jan 17

   1. If $(x_n)$ is a sequence in $\mathbf{R}^p$ such that $x_n\to 0$, show that the sequence of arithmetic means converges to the same limit, $\lim_{n\to\infty} \frac{1}{n}\sum_{k=1}^n x_k =0$.
   2. The root test is stronger than the ratio test. To confirm this, prove one of the following claims:
      • If $(a_n)$ is a bounded sequence of strictly positive real numbers, show that $\limsup_{n\to\infty}\sqrt[n]{a_n} \leq \limsup_{n\to\infty} \frac{a_{n+1}}{a_n}$. (Bartle Exercise 18.F:)
      • Alternative version: If $(a_n)$ is a bounded sequence of strictly positive real numbers such that $r=\lim_{n\to\infty} \frac{a_{n+1}}{a_n}$ exists, then $\lim_{n\to\infty}\sqrt[n]{a_n}=r$.
   3. The Cauchy Condensation Test (Barlet Exercise 34.K, or Rudin Theorem 3.27) says that if a numerical sequence $(a_n)$ is monotone decreasing and positive then $\sum_{n=1}^\infty a_n$ converges iff $\sum_{k=0}^\infty 2^k a_{2^k}$ converges.

      Use The Cauchy Condensation Test to prove both of the following statements:

      • For $p\in\mathbf{R}$ the series $\sum_n\frac{1}{n^p}$ converges iff $p>1$.
      • For $p\in\mathbf{R}$ the series $\sum_{n=2}^\infty\frac{1}{n (\log n)^p}$ converges iff $p>1$.
   4. Barlet Exercise 34.K has a hint in the back of the book. Can you use it to write the proof of The Cauchy Condensation Test without looking into Rudin's book?
2. Homework II on uniform convergence Due We, Jan 24 in class

   1. Recall that $\lim_{x\to\infty} f(x)=L$ means that $\forall_{\epsilon>0} \exists_{M>0} \forall_{x>M} |f(x)-L|<\epsilon$.
      Let $(f_n)$ be a family of functions on $[0,\infty)$ such that $f_n\to f$ uniformly on $[0,\infty)$ and for every $n$ we have $\lim_{x\to\infty} f_n(x)=0$. Prove that $\lim_{x\to\infty} f(x)=0$.
   2. Let $(f_n)$ be a family of functions on interval $[0,1]$ such that $|f_n(x)|\leq 2024$ for $x\in[0,1]$ and $f_n\to f$ pointwise on $[0,1]$. Prove that $f$ is bounded on $[0,1]$. Note: We do not assume that $f_n$ or $f$ are continuous.
   3. Let $(f_n)$ be a family of functions on interval $[-1,1]$ defined by $f_n(x)=\frac{1}{1+n x^2}$. Show that $(f_n)$ converges pointwise, but not uniformly on $[-1,1]$.
3. Homework III due We, Jan 31 in class

   1. [Inspired by Bartle Exercise 24.M] Consider a bounded continuous function $f(x)=x$ on the interval $[0,1]$. What polynomial approximation do we get from Bernstein's proof of Stone-Weierstrass theorem? What approximation do we get from "Rudin's proof", based on the kernel $Q_n(x)=\frac{(2n+1)!}{2^{2n+1}(n!)^2}(1-x^2)^n$? [Somewhat coincidentally, for our $f$ the answers turn out to be the same for both methods. However, they are not the same for $x^2$.]
   2. Bartle, Exercise 26.N: Show that if a sequence $(f_n)$ of continuous functions on a compact set $K$ converge uniformly, then the family $\mathcal{F}=\{f_n: n\in\mathbb{N}\}$ is uniformly equicontinuous on $K$.
   3. Bartle Exercise 24.R. Prove that $e^x$ is not the uniform limit on $[0,\infty)$ of a sequence of polynomials.
4. Homework IV due We, Feb 7 in class: Choose 3 of the following 4 questions.

   1. A question someone asked in class about uniform convergence versus absolute convergence: Show that the series $\sum_{n=1}^\infty (-1)^n \frac{x^2+n}{n^2}$ converges uniformly in every bounded interval but does not converge absolutely for any value of $x$.
   2. Bartle, Exercise 37.H: Determine the radius of convergence for as many of the following as you can: (a) $\sum_{n=1}^\infty \frac{x^n}{n^n}$, (b) $\sum_{n=0}^\infty \frac{n^\alpha x^n}{n!}$, (c) $\sum_{n=0}^\infty \frac{n^n x^n}{n!}$, (d) $\sum_{n=2}^\infty \frac{x^n}{\log n}$, (e) $\sum_{n=0}^\infty \frac{(n!)^2}{(2n)!}x^n$, (e) $\sum_{n=0}^\infty \frac{x^n}{n^{\sqrt{n}}}$.
   3. Prove by mathematical induction Leibnitz formula: $$\frac{d^n}{dx^n}(f(x)g(x)) =\sum_{k=0}^n \binom{n}{k} f^{(k)}(x) g^{(n-k)}(x)$$ Use this formula to deduce formula for the $n$-th coefficient in the Taylor polynomial for the product $f(x)g(x)$ in terms of the coefficients of the Taylor polynomials (or the power series, assuming convergence) for the infinitely differentiable functions $f$ and $g$.
   4. Bartle Exercise 37.T: Use Abel's theorem 37.20 to prove that if $A=\sum a_n$, $B=\sum b_n$ and $C=\sum c_n$ converge, where $c_n=\sum_{k=0}^n a_k b_{n-k}$, then $C=AB$. Try to do get the proof without looking at "a solution" in the back of the book. Even if you do look there, be sure to add missing details and quote the actual theorem that you need to use! Why do the power series converge absolutely on $(0,1)$?

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5. Homework V due We, Feb 21 in class

   • 3 Grad students: Choose one: Bartle Exercise 21.I. (Correct a typo and solve) Bartle 21.J, 21.K, 21.O
   • 1 Grad Student and all undergrad students are exempted from this Hwk. Of course, you can do one of the problems above. Or you can do one of the "coordinate-free" derivatives listed for Thursday.
6. Homework VI due We, Feb 28 in class.

   In the questions, $E$ is an open subset of $\mathbb{R}^p$

   1. Bartle Exercises 39 A-E (choose one)
   2. Bartle 39.T: Suppose $f$ is real valued function defined on an $E$ and that the partial derivatives $D_1 f,\dots,D_pf$ are bounded on $E$. Use the method of proof of Thm 39.9 to show that $f$ is continuous on $E$.
   3. If $f,g: E\to\mathbb{R}$ are differentiable, show that $\nabla (f g)=f\nabla g+g\nabla f$
   4. A version of Bartle 39.V: If $f:E\to \mathbb{R}^q$ and $g:E\to\mathbb{R}$ are differentiable, show that the function $\phi:E\to\mathbb{R}^{q+1}$ defined by $\phi(\vec x)=(f(\vec x),g(\vec x))$ is differentiable and $D\phi(\vec x)\vec h=(D f(\vec x)\vec h,Dg(x)\vec h)$. Note: I'd re-write these as column vectors.
7. Hwk VII, due We, March 6 in class

   1. Bartle Exercise 40.D
   2. Consider $\Omega=(0,\infty)\times \mathbb{R}$ and function $f:\Omega\to\mathbb{R}^2$ given by $f(x_1,x_2)=(x_1 \cos x_2, x_1\sin x_2)$
      • Use the inverse function theorem to show that $f$ has a local inverse at any point $(x_1,x_2)\in\Omega$ (That is there is a neighborhood of every point where $f$ is invertible
      • Show that $f$ is not one-to-one so it cannot have an inverse.
      • Find an explicit formula for the local inverse $f^{-1}$ near point $x_1=1$, $x_2=\pi$. (Use the formula to determine largest possible open sets $U,V$.)
      • Find an explicit formula for the local inverse $f^{-1}$ near point $x_1=1$, $x_2=3\pi$. (Use the formula to determine largest possible open sets $U,V$.)
   3. Bartle, Exercise 41.F.
8. Hwk VIII, due We, March 20 in class

   1. The Lambert $W$ function $\mathbb{R}\to\mathbb{R}$ is defined by implicit equation $F(x,W(x))=0$, where $$F(x,y)=y e^{y}-x $$ According to Wikpiedia, it has two brachnes: $W_0,W_1$, shown on the graph there . Show that the Lambert function is well defined near point $(x,y)=(e,1)$ (this is $W_0$, with the range $(-1,\infty)$) and near point (corrected Tue, March 19!!!) $(x,y)=(-2/e^2,-2)$ (this is $W_{-1}$ with the range $(-\infty,-1)$)

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      For a more challenging (optional) version, of this homework consider $W:\mathbb{R}^2\to\mathbb{R}^2$ defined by the implicit equation $W e^W=z$ with $W=w_1+i w_2$ and $z=x_1+ix_2$.

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   2. Bartle 42A alike: Find the critical points and determine their nature for $f(x,y)=x^2+4x y+y^2+2 y^3$.
   3. A version of Bartle 42.N: We wish to choose real $A,B,C$ so that $$\int_0^\pi \left(f(x)-A \sin x - B \sin 2 x - C \sin 3 x\right)^2 dx$$ is minimized. Show that $A,B,C$ can be expressed in terms of the integrals $\int_0^\pi f(x)\sin x$, $\int_0^\pi f(x)\sin 2x$, $\int_0^\pi f(x)\sin 3x$.

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9. Hwk IX, due We, April 3 in class

   1. Compute the Fourier series for $f(x)=x^2$ for $|x|\leq \pi$, extended to periodic function. (You may use WolframAlpha to compute the integrals)
   2. Use the Fourier series for the above $f$ to evaluate the sum of the series $\sum_{n=1}^\infty \frac{(-1)^{n-1}}{n^2}$ and to confirm that $\sum_{n=1}^\infty \frac{1}{n^2}=\frac{\pi^2}{6}$
   3. Use the Parserval's identity for the above $f$ to evaluate the sum of the series $\sum_{n=1}^\infty \frac{1}{n^4}$.
10. Hwk X, due Th, April 11 in class (assuming we cover the change of variable material)

    1. Barle Exercise 45.H alike: Determine the area of the region in the first quadrant bounded by the four curves $$ xy=1, \quad xy=2, \quad y=x, \quad y=2x,$$ by introducing an appropriate change of variable. (To compute $\int_{[a,b]\times [c,d]} f(x,y)$ for a continuous $f$ use the iterated integrals $\int_a^b \left(\int_c^d f(x,y)dy\right) dx$. This is Bartle Thm 44.12)
       (You may substitute Barle Exercise 45.H, which is similar and has the change of variable and the answer in the back of the book.)
    2. For $s>0$, let $B_s=\{(x,y): x^2+y^2\leq s\}$ be the circle of radius $\sqrt{s}$. Use an appropriate change of variable to compute the integral $$I(s):= \frac1{2\pi} \int_{B_s} e^{-(x^2+y^2)/2}$$ (In statistics, $I(s)$ is called the cumulative distribution function of the chi-squared law with two degrees of freedom. The same formula arises in queuing theory, where it is called an exponential law.)

• Instructor: Wlodek Bryc
  Office French West 4316
  E-mail:
  Office Hours
  Phone: 513-556-4098
• Main Textbook: R. Bartle, The Elements of Real Analysis, 2nd edition. Secondary: W. Rudin, Principles of Mathematical Analysis, 3rd edition.

Exams and important dates. See also UC Academic Calendars

• Exam 1: Tue, Feb 13, 1:30-3PM. Seminar Room 4206 French Hall
• Exam 2: Tue March 26, 1-3PM. Room 115 W. Charlton
• Final Exam: Mo, Apr 22, 10:15-12:15, Edwards 7120

Course Grade Calculation

Topics

According to course description, material covered includes: linear maps, differentiability, partial derivatives, differentiability of functions whose partial derivatives are continuous, chain rule, Jacobian, inverse and implicit function theorems. Uniform convergence of sequences of functions, Arzela-Ascoli theorem. Basics of Fourier series.

1. We shall start with some of the topics from the previous semester:

   • Sequences $(x_n)$ in metric spaces: Dr. Speight notes Ch 4, Bartle Ch III (pg 90), Rudin Ch 3 and Ch 7
   • Numerical Series $\sum_n a_n$: Covered root test, Dirichlet test
     Dr. Speight notes Ch 4, Bartle Ch VI, Rudin Ch 3 and Ch 7
   • Sequences and series of functions $f_n:\mathbf{R}^p\to\mathbf{R}^q$. Uniform convergence, completeness of $C[0,1]$. Bartle Sect 26, Sect. 37, Rudin Ch 7, Dr. Speight notes end of Ch 5,.

   Some more topics that were not covered last Semester:

   • The Stone-Weierstrass Theorem (Rudin Thm 7.26, Bartle Thm 24.7/24.8. Do we want Bartle, Thm 26.2? No! $C[0,1]$ is a complete and separable metric space!
   • Equicontinuity and The Arzela-Ascoli Theorem. Bartle Thm 26.7 Rudin Ch 7, Thm 7.25
   • Power series Bartle Sect. 37, Rudin Ch. 8, Dr. Speight notes Ch 6 pgs $\geq 22$
     • Radius of convergence, term-wise integration, term-wise differentiation
     • Multiplication of series
     • Taylor's Thm Bartle 28.6.
   • Fourier series: Bartle Section 38. (Moved after Exam 1 and after differentiation...)
   • Improper integrals Bartle Section 32.
   • We will conclude improper integrals with Dirichlet's Test (Bartle Thm 32.9), prove "integral test for series" (Thm. 35.12), and work on exercises 32.F, 32.G.
2. Exam 1 will be over the material that we manage to cover. You should know how to do problems similar to homework as well as proofs of selected theorems.

   Required proofs:

   1. Uniform convergence of continuous functions: Bartle, Theorem 24.1
   2. Absolute convergence of series of vectors: Bartle Thm 34.7
   3. Root test for absolute convergence of numerical series: Bartle Thm 35.3+Corollary 35.4
   4. Dirichlet's test for series: Bartle, Thm 36.2
   5. Radius of convergence for power series: Bartle, Thm 37.13 (Cauchy-Hadamard)
   6. Uniform convergence: Bartle, Thm 37.14
   7. Term by term Differentiation: Bartle, Thm 37.16
   8. Uniqueness: Bartle, Thm 37.17
   9. Improper integrals:
      • Thm 32.5 Cauchy Criterion
      • Thm 32.7 Comparison Test
      • Thm 32.8 Limit Comparison Test
3. Derivatives for functions $\mathbb{R}^p\supset E \to \mathbb{R}^q$.
   • Linear functions (and linear functionals): Bartle Section 21, Rudin Ch 9, Dr. Speith notes Ch 5 pg 20-24
   • The derivative in $\mathbb{R}^p$: Bartle Section 39, Rudin Ch9
   • The chain rule and MVT: Bartle Section 40, Rudin Ch 9
   • Mapping Theorems, Inversion theorem, Implicit functions: Bartle Section 41, Rudin Ch 9
   • Extrema: Bartle Section 42
4. Orthogonal expansions - a prequel to Fourier series.
5. Exam 2 March 26, 1-3:30PM, W. Charlton 115

   The exam will be over the material that we manage to cover. You should know how to do problems similar to homework as well as proofs of selected theorems.
   Required proofs:

   1. Banach fixed point theorem (Bartle Thm 23.5 )
   2. Continuity theorem (Bartle, Lemma 39.5)
   3. Partial derivatives theorem (Bartle Thm 39.6 and Corollary 39.7)
   4. $C^1$ Differentiability criterion (Bartle, Thm 39.9)
   5. Chain Rule (Bartle Thm 40.2)
   6. EZ Mean Value Thm (Bartle Thm 40.4)
   7. First derivative test : Bartle Thm 42.1
   8. Second derivative test: Bartle Thm 42.5
6. Fourier series: Bartle Section 38.
7. Informal info: Determinants, Jacobians, and changing the variables in multiple integrals
8. Application of Banach contractive mapping theorem to ODE. Picard iterations
9. A sequel to Fourier series - Intro to orthogonal polynomials.

   Position marker: We are here, more or less

10. Final Exam Mo, Apr 22, 10:15-12:15, Edwards 7120
    The final exam will be cumulative over all the material covered. You should know how to do problems similar to homework as well as proofs of selected theorems. (4 questions)
    Required proofs:
    1. List of proofs from Exam 1
       1. Uniform convergence of continuous functions: Bartle, Theorem 24.1
       2. Absolute convergence of series of vectors: Bartle Thm 34.7
       3. Root test for absolute convergence of numerical series: Bartle Thm 35.3+Corollary 35.4
       4. Dirichlet's test for series: Bartle, Thm 36.2
       5. Radius of convergence for power series: Bartle, Thm 37.13 (Cauchy-Hadamard)
       6. Uniform convergence: Bartle, Thm 37.14
       7. Term by term Differentiation: Bartle, Thm 37.16
       8. Uniqueness: Bartle, Thm 37.17
       9. Improper integrals:
          • Thm 32.5 Cauchy Criterion
          • Thm 32.7 Comparison Test
          • Thm 32.8 Limit Comparison Test
    2. List of proofs from Exam 2
       1. Banach fixed point theorem (Bartle Thm 23.5 )
       2. Continuity theorem (Bartle, Lemma 39.5)
       3. Partial derivatives theorem (Bartle Thm 39.6 and Corollary 39.7)
       4. $C^1$ Differentiability criterion (Bartle, Thm 39.9)
       5. Chain Rule (Bartle Thm 40.2)
       6. EZ Mean Value Thm (Bartle Thm 40.4)
       7. First derivative test : Bartle Thm 42.1
       8. Second derivative test: Bartle Thm 42.5
    3. Fourier Series
       • Bartle Lemma 38.3
       • Bartle Theorem 38.4 (Bessel Inequality)

• Jan 11:

  Norms etc are in exercises/projects for Section 8 in Bartle. Limits in $\mathbb{R}^p$ are in exercises/projects for Secton 15. Series are in exercises for Section 34. Chose an exercise that you find interesting/puzzling and be ready to present it in class. Here is a puzzler for you to consider: Suppose the $n$-th parial sum of the series $\sum_{n=1}^\infty a_n$ is $s_n=2-\frac{1}{n+1}$. Then $ \sum_{n=1}^\infty a_n=\lim_{n\to\infty}s_n=2$. But $a_n=s_n-s_{n-1}=\frac{1}{n(n+1)}$ and WolframAlpha says $\sum_{n=1}^\infty\frac{1}{n(n+1)}=1$ not $2$. Is this a paradox that shatters the foundations of analysis? Or of computer science? (It is known that there are problems with $\infty$ that may one day create a paradox that will destroy calculus.)
  Binomial law: something behind a proof of Theorem 24.7 in Bartle. (Done)

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• Jan 18: Lets choose one of the topics to look at: (I suggest first things first!)

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  Homework I We may want to discuss the questions

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  Exercise 24.H in Bartle Let $(f_n)$ be a monotone decreasing sequence of continuous functions on $[-1,1]$. (That is $f_1(x)\geq f_2(x)\geq \dots$ for all $x\in[-1,1]$.) If $f_n$ converges pointwise to a function $f$ that is continuous on $[-1,1]$ then convergence is uniform. Hint: WLOG $f\equiv 0$.

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  One can try to make examples that illustrate the assumptions in the above Dini's theorem: (I think one can cook up piecewise linear functions for each case)

  • Give an example of a monotone decreasing sequence $(f_n)$ of continuous functions on $[-1,1]$ such that $f_n$ converges pointwise to a function $f$ that is not continuous on $[-1,1]$.
  • Give an example of a sequence $(f_n)$ of continuous functions on $[-1,1]$ such that $f_n$ converges pointwise to a function $f$ that is continuous on $[-1,1]$ but the convergence is not uniform.
  • Give an example of a monotone decreasing sequence $(f_n)$ of continuous functions on $(-1,1)$ such that $f_n$ converges pointwise to a function $f$ that is continuous on $(-1,1)$ but the convergence is not uniform.

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  Define function $\exp(x)=\sum_{n=0}^\infty \frac{x^n}{n!}$. How many of these can we prove?

  • $ \tfrac{1}{dx}\exp(x)=\exp(x)$
  • $\exp(x+y)=\exp(x)\exp(y)$
  • $\exp(x)>0$ for all $x$
  • Denote $e=\exp(1)$. If $x$ is a rational number, then $\exp(x)=e^x$. How about irrational numbers, like $x=\sqrt{2}$ or $x=\pi$?
  • Taylor's identity: $\exp(x)-\sum_{k=0}^n \tfrac{x^k}{k!}=\frac{1}{n!}\int_0^x e^t (x-t)^n dt $
  • Euler's formula: $\exp\left (\begin{bmatrix}0 & t \\ -t&0\end{bmatrix}\right)=\begin{bmatrix}\cos t & \sin t \\ - \sin t & \cos t\end{bmatrix} $. Or a better known version $e^{i x}=\cos x+ i \sin x$
  • G. Boole (1859?) forgotten formula for $e^{\frac{d}{dx}}$: The series $\sum_{n=0}^\infty \frac{1}{n!} f^{(n)}$ on many functions converges to $f(x+1)$

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  Separable complete metric spaces are nicest spaces to work with. (What is "metric space"? What is "separable"? What is "complete"?)

  • Consider $C[0,1]$, the space of all continuous functions on $[0,1]$ with the norm $\|f\|_\infty=\max_{x\in[0,1]}|f(x)$. Can we prove that $C[0,1]$ with metric $d(f,g)=\|f-g\|_\infty$ is a separable complete metric space? [Yes, we can!]
  • Consider $C[0,1]$, the space of all continuous functions on $[0,1]$ with the norm $\|f\|_1=\int_0^1|f(x)|dx$. Can we prove that $C[0,1]$ with metric $d_1(f,g)=\|f-g\|_1$ is a separable complete metric space? [No, we cannot prove completeness] Hint: One can find a sequence $(f_n)$ that is Cauchy in metric $d_1$ but does not converge to a continuous function.
• Jan 25:

  Let's look at things we did not have time to do last week. ( $e^{\frac{d}{dx}}$ was added later!) (more to be posted...)

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• Feb 1:

  • Moved from previous week: If you never saw a proof of Holder's inequality perhaps we should go over some of Bartle's Project 8$\beta$.
    Given $p,q>1$ such that $\frac{1}{p}+\frac{1}{q}=1$, for continuous $f,g$ we have $$\left|\int_0^1 f(x)g(x) dx\right|\le \sqrt[p]{\int_0^1 |f(x)|^p} \sqrt[q]{\int_0^1 |f(x)|^q}$$ Given sequences $(a_n)$ and $(b_n)$ of real numbers such that $\sum_n |a_n|^p<\infty$ and $\sum_n |b_n|^q<\infty$, the series $\sum_na_nb_n$ converges and $$\left|\sum_{n=1}^\infty a_n b_n\right|\leq \sqrt[p]{\sum_n |a_n|^p}\sqrt[q]{\sum_n |b_n|^q}$$ The hard part is to prove inequality in Bartle's Project 8$\beta$(a). But we can just accept "an elementay inequality" $a b\leq \frac{a^p}{p}+\frac{b^q}{q}$ for $a,b>0$ without proof, and use it.
    Note: The inequalities hold also for $p=1,q=\infty$. The best known and most important case is of course $p=q=2$.

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  • Exercises for Euler's formula and geometric sum that we may want for Fourier series: Formulas for $\sum_{k=1}^n \sin kx$ and $\sum_{k=1}^n \cos k x$ so that $$\tfrac12+\sum_{k=1}^n \cos k x=\frac{\sin (n+\tfrac12)x}{2 \sin \tfrac12 x}$$

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  • Convolution in the (complete separable) normed vector space $\ell_1=\{(a_k): \sum_{k=0}^\infty |a_k|<\infty\}$. Given $\mathbf{a},\mathbf{b}\in\ell_1$, define convolution $\mathbf{c}=\mathbf{a}*\mathbf{b}$ by $c_n=\sum_{k=0}^n a_k b_{n-k}$. Show that
    • $\mathbf{c}\in\ell_1$ and $\|\mathbf{c}\|_1\leq \|\mathbf{a}\|_1\|\mathbf{b}\|_1$
    • Convolution is a commutative, $\mathbf{a}*\mathbf{b}=\mathbf{b}*\mathbf{a}$, bi-linear, and associative operation.
    • Food for thought: $(\ell_1,*)$ is an example of Banach algebra. We can define convolution powers $\mathbf{a}^{*n}$ recurrently by $\mathbf{a}*\mathbf{a}^{*(n-1)}$. (What is $\mathbf{a}^{*0}$ in this formula?) If $\|\mathbf{a}\|_1<1$ then the series $\sum_{n=0}^\infty \mathbf{a}^{*n}$ converges (to what?). Also the series $\sum_{n=0}^\infty \frac{1}{n!}\mathbf{a}^{*n}$ converges (to what?).
  • Show that pointwise multiplication of sequences in $\ell_1$ is a well defined bilinear, commutative, associative operation on $\ell_1$. Does this multiplication satisfy the "Banach algebra" condition $\|\mathbf{a}\mathbf{b}\|_1 \leq \|\mathbf{a}\|_1\|\mathbf{b}\|_1$? Is there a multiplicative identity?
• Feb 8+Feb 9+Feb 12: I will try not to cover any new material before Exam 1.

  • Questions. Discussion of homework.
  • Improper integral exercises from Bartle:
    • Exercise 32.F: Discuss convergence/divergence/absolute convergence:
      • (a) $\int_1^\infty \frac{dx}{x(1+\sqrt{x})}$
      • (b) $\int_1^\infty \frac{x+2}{x^2+1}dx$
      • (c) $\int_1^\infty \frac{\sin(1/x}{x} dx$
      • (d) $\int_1^\infty \frac{\cos x}{\sqrt{x}}dx$
      • (e) $\int_0^\infty \frac{x \sin x}{1+x^2} dx$
      • (f) $\int_0^\infty\frac{\sin x \sin 2 x}{x} dx$
    • Exercise 32.G: For what values of $p$ and $q$ are the following convergent? Absolutely convergent?
      • (a) $\int_a^\infty \frac{x^p}{1+x^q}dx$
      • (b) $\int_1^\infty \frac{\sin x}{x^q} dx$
      • (c) $\int_1^\infty \frac{\sin x^p}{x} dx$
      • (d) $\int_1^\infty \frac{1-\cos x}{x^q} dx$
  • Improper integrals: for what real $p$, the integral $\Gamma(p)=\int_0^\infty x^{p-1} e^{-x}dx$ converges?
  • Improper integrals: for what real $\alpha,\beta$, the integral $B(\alpha,\beta)=\int_0^1 x^{\alpha-1} (1-x)^{\beta-1}dx$ converges?
  • A nice question from Dilani: If $f(x)$ is an increasing function on $[0,\infty)$ such that $\int_0^\infty f$ exists, prove that $\lim_{x\to\infty} f(x)=0$.
  • A version of the above from Bartle's Exercise 32.J: If $f$ is monotone and the improper integral $\int_0^\infty f$ exists, prove that $\lim_{x\to\infty} x f(x)=0$
  • Improper integrals: Does $\int_0^\infty \frac{\sin x}{x} dx$ converge? Converge absolutely? Note: according to WolframAlpha $\int_0^\infty \frac{\sin x}{x} dx=\pi/2$.

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  • If you like special functions then you should check that $B(\alpha,\beta)=B(\beta,\alpha)$. If you are a conesseur, then you should also check (at least heuristically) that $B(\alpha,\beta)=\frac{\Gamma(\alpha)\Gamma(\beta)}{\Gamma(\alpha+\beta)}$. If special functions are not to your liking, skip this.

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  • Leftover Exercises for Euler's formula that we may want for Fourier series: Use the geometric sum formula to obtain formulas for $\sum_{k=1}^n \sin kx$ and $\sum_{k=1}^n \cos k x$ so that $$\tfrac12+\sum_{k=1}^n \cos k x=\frac{\sin (n+\tfrac12)x}{2 \sin \tfrac12 x}$$ (This can also be proved by induction. But how would you know what to prove if you don't remember the formula?)

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• Feb 15 Linear Algebra. Continuity.

  Note:(*) marks recommended questions.

  1. Find matrix representations for the linear transformations $\mathbb{R}^p\to\mathbb{R}^q$ described below:
     • (*) $R_\alpha:\mathbb{R}^2\to \mathbb{R}^2$: rotation by angle $\alpha$
     • Orthogonal projection onto the line spanned by $\begin{bmatrix}1\\1\\1\\1\end{bmatrix}$ in $\mathbb{R}^4$.
     • Orthogonal projection onto the hyperplane $x_1+x_2+x_3+x_4=0$ in $\mathbb{R}^4$. (The same question for the plane $x+y+z=0$ in $\mathbb{R}^3$ is harder!)
  2. Linear functions
     • (*) If there is a linear transformation $L:\mathbb{R}^p\to\mathbb{R}^q$ that is one-to-one (injective) then $p\leq q$.
     • If there is a linear transformation $L:\mathbb{R}^p\to\mathbb{R}^q$ that is onto (surjective) then $p\geq q$.
     • If a linear transformation $L:\mathbb{R}^p\to\mathbb{R}^p$ is invertible, then $\|L^{-1}\|\geq \frac{1}{\|L\|}$
     • (*) If a linear transformation $L:\mathbb{R}^p\to\mathbb{R}^p$ is invertible, then there is $\delta>0$ such that $\|L(\vec x)\| \geq \delta \|x\|$ for all $\vec x$.
     • (*) If a linear transformation $L:\mathbb{R}^p\to\mathbb{R}^p$ is close to identity map, $\|L-id\|<1$, then $L$ is invertible.
     • More generally, the set of all invertible transformations is an open set in the normed vector space $\mathcal{L}(\mathbb{R}^p)$ of the linear transformations $L:\mathbb{R}^p\to\mathbb{R}^p$
  3. For $p>1$, consider a normed vector space $\ell_p=\{(x_i): \sum |x_i|^p<\infty\}$. If a linear functional $\phi:\ell_p\to\mathbb{R}$ is given by a "dot product" $\phi(\vec{x})=\vec a \cdot \vec x=\sum a_j x_j$ and is continuous, show that its "operator norm" $\|\phi\|_*$ is equal to $\|\vec a\|_q$ where $\frac1p+\frac1q=1$.
  4. Determine continuity at $(0,0)$ of the functions $f:\mathbb{R}^2\to \mathbb{R}$ given below.
     • (*) $f(x,y)=\begin{cases} \frac{x^2}{x^2+y^2} & x^2+y^2>0 \\ 0 & x^2+y^2=0 \end{cases}$.
     • $f(x,y)=\begin{cases} \frac{xy}{x^2+y^2} & x^2+y^2>0 \\ 0 & x^2+y^2=0 \end{cases}$.
     • $f(x,y)=\begin{cases} \frac{x+y}{x^2+y^2} & x^2+y^2>0 \\ 0 & x^2+y^2=0 \end{cases}$.
     • (*) $f(x,y)=\begin{cases} \frac{x^2 y^2 }{x^2+y^2} & x^2+y^2>0 \\ 0 & x^2+y^2=0 \end{cases}$.
     • $f(x,y)=\begin{cases} \frac{x^4 + y^4 }{x^2+y^2} & x^2+y^2>0 \\ 0 & x^2+y^2=0 \end{cases}$.
  5. The image (range) of a function $f:(a,b)\to \mathbb{R}^q$ is called a curve. For a differentiable $f$, let $\vec v(x)=f'(x)$. The length of the curve is then $\int_a^b \|\vec v(x)\|dx$.
     • The helix of radius $r$ and pitch $h$ is the graph of function $f(x)=(r\cos( x), r\sin (x), \frac{h x}{2\pi})$. Find the length of one spiral, $0\leq x\leq 2\pi$.
• Feb 22 Derivatives

  • Coordinate-free differentiation in (finite dimensional) vector spaces. Recall that if $\dim V=n$ then $V$ "is" $\mathbb{R}^n$:
    • (*) If $V=\mathcal{M}_{m\times n}$ is the set of matrices with norm $\||A\||=\sqrt{tr(AA^T)}$ and $f:V\to\mathcal{M}_{k\times k}$ is $f(X)=AX+XB$ with fixed $A\in\mathcal{M}_{k\times m}$ and $B\in \mathcal{M}_{n\times k}$ then $f'(X)(H)=?$
    • If $V=\mathcal{M}_{n\times n}$ matrices, what are the derivatives of $f:V\to V$ given by
      • $f(X)=X^T$
      • $f(X)=XX^T$
    • Consider vector space $V=\mathcal{M}_{n\times n}$ of matrices with norm $\||A\||=\sqrt{tr(AA^T)}$. The derivative of the detereminant at $I$ is then "the trace".
      Verify this claim for $2\times 2$ matrices. Better yet, prove that for an invertible $X\in\mathcal{M}_{2\times 2}$ and $f(X)=\det X$ we have $f'(X)(H)=\det X tr(X^{-1}H)$. (Which is the trace when $X=I$.)
    • (*) If $\mathcal{P}_n\simeq \mathbb{R}^{n+1}$ is the vector space of polynomials of degree $\leq n$ (with any norm), what are the derivatives of
      • $f:\mathcal{P}_n\to\mathcal{P}_{n-1}$ given by $f(p)=\frac{d}{dx}p$
      • $g:\mathcal{P}_n\to\mathcal{P}_{2n}$ given by $f(p)=p^2$
      • $h:\mathcal{P}_n\to\mathcal{P}_{2n-1}$ given by $f(p)=p \frac{d}{dx}p$
  • Partial and directional derivatives:
    • Compute the partial derivative $D_{\vec u}f(\vec x)$ for $f(x,y)=(x^2-y^2 , x y)$ in the direction $\vec u=(a,b)$
    • If $f:E\to\mathbb{R}$ is differentiable, find the direction of its steepest growth: find unit vector $\vec u=\vec{u}(\vec x)\in\mathbb{R}^p$ such that $\frac{d}{dt} f(\vec x+t \vec u)$ is largest. (Hint: the answer is $\vec u= \nabla f (\vec x)/\|\nabla f (\vec x)\|$, so just prove that this works. For more hints, see Bartle Exercise 39.J)
    • (*) Determine $D_{\vec u}\phi(0,0)$ for $\phi(x,y)=xy^2/(x^2+y^2)$, $\phi(0,0)=0$. Does $\phi'(0,0)$ exist? (no) Is $\phi$ continuous? (yes, use $|x|<\eps$ in the numerator, $x=0$ in the denominator )
    • 
  • (*) Banach fixed point theorem (Bartle, Thm 23.5, Rudin Thm 9.23, Dr Speight notes Ch 5 pg 47++
• Feb 29 Mapping Theorems, Implicit functions, Inverse functions:

  1. More derivatives:
     • Let $f: \mathbb{R}^p\times \mathbb{R}^q\to\mathbb{R}^r$ be a bilinear function, i.e. $f(c \vec x+\vec y,\vec z)=c f(\vec x,\vec z)+f(\vec y,\vec z)$ and $f(\vec x, \vec y+c \vec z)=f(\vec x,\vec y)+c f(\vec x,\vec z)$.
       (If you do not like the abstract setup, consider $r=1$ and $f(\vec x,\vec y)=\vec x \cdot (A \vec y)$ for a fixed $p\times q$ matrix $A$. Or do $2\times 3$, it ain't much different.)
       Determine the expressions for the derivatives $f'$ and $f''$ and $f'''$.
       The formula for the first derivative is given in Bartle Exercise 40.M with the Hint: $\|f(\vec x,\vec y)\|\leq C \|\vec x\| \cdot \|\vec y\|$. (You can just use it. But why would this be true?)
     • A cleaner version of the above: $p=q$
     • An even cleaner version: given bilinear function $f: \mathbb{R}^p\times \mathbb{R}^p\to\mathbb{R}^r$, define a quadratic form $g(\vec x)=f(\vec x,\vec x)$. Then $g:\mathbb{R}^p\to\mathbb{R}^r$. For a given $\vec x\in\mathbb{R}^p$, determine the linear funtion $g'(\vec x)(\cdot):\mathbb{R}^p\to\mathbb{R}^q$, the bilinear function $g''(\vec x)(\cdot,\cdot):\mathbb{R}^p\times \mathbb{R}^p \to\mathbb{R}^q$ and the trilinear form $g'''(\vec x)(\cdot,\cdot,\cdot):\mathbb{R}^p\times \mathbb{R}^p\times \mathbb{R}^p \to\mathbb{R}^q$.
     • Bartle 40.U: Let $f:\mathbb{R}^2\to\mathbb{R}$ be defined by $f(x,y)=\frac{xy(x^2-y^2)}{x^2+y^2}$ with $f(0,0)=0$. Show that $\frac{\partial^2 f}{\partial x \partial y}(0,0)\ne \frac{\partial^2 f}{\partial y \partial x}(0,0)$
  2. Chain rule: Bartle 40.G gives a bunch of nice formulas
  3. Bartle 40.G alike: if $f:\mathbb{R}\to\mathbb{R}$ has derivatives of all orders and $g:\mathbb{R}^2\to\mathbb{R}$ is $g(x,y)=f(xy)$, what is $g'$? What is $g''$? What is $g'''$?
  4. Inverse functions (if we cover the theorem):
     • Rudin, Ch9 Exercise 17/18: For $f=(f_1,f_2):\mathbb{R}^2\to\mathbb{R}^2$ given below (a) Find the range of $f$, (b) Show that $f$ is locally one-to-one, but it is not one-to-one on $\mathbb{R}^2$ (c) What are the images under $f$ of lines parrallel to the coordinate axes?
       • $f_1(x,y)=e^x \cos y$, $f_2(x,y)=e^x \sin y$
       • $f_1(x,y)=x^2-y^2$, $f_2(x,y)=2 x y$
• March 7

  1. Implicit functions:
     • Rudin Ch 9 Exercise 19: Show that the system \begin{eqnarray} 3 x+ y -z +u^2=0 \\ x -y+2z+u=0\\ 2x+2y-3z+2u=0 \end{eqnarray} can be solved for $x,y,u$ in terms of $z$; for $x,z,u$ in terms of $y$; for $y,z,u$ in terms of $x$. But not for $x,y,z$ in terms of $u$
  2. Extrema
     1. Find the extrema of $f(x_1,\dots,x_n)=x_1^4+\dots+x_n^4-4(x_1+\dots+x_n)$ over $\mathbb{R}^n$
     2. Find the extrema of $$f(x_1,\dots,x_n)=\frac{1}{x_1}+\dots+\frac{1}{x_n}+\frac{1}{1-(x_1+\dots+x_n)}$$ on the set $x_1>0,\dots,x_n>0$
     3. Bartle 42K alike: Find the distance between the planes in $\mathbb{R}^4$ with parametric equation $\vec r=s_1\begin{bmatrix}1 \\ 0\\ 0 \\ -1\end{bmatrix}+s_2 \begin{bmatrix}0 \\ 1\\ -1 \\ 0\end{bmatrix}$ and $\vec r=\begin{bmatrix}1 \\ 1\\ 1 \\ 1\end{bmatrix}+s_1 \begin{bmatrix}0 \\ 1\\ -1 \\ 0\end{bmatrix} +s_2 \begin{bmatrix}-1 \\- 1\\ 1 \\ 1\end{bmatrix}$
        Hint: minimize $\|\vec r(s_1,s_2)-\vec r(t_1,t_2)\|^2$

        The expression to minimize is $\left(-s_1-t_2-1\right)^2+\left(-s_2-t_2-1\right)^2+\left(s_1+t_2-1\right){}^2+\left(s_2-t_1+t_2-1\right)^2$, so this is best done by symbolic software. Answer The planes are at distance $2$, but there is an entire line of critical points $s_2 = -1 + s_1$ , $t_1 = -2$, $t_2 = -s_1$.
        Note A version we worked on in class with (miscopied) $\vec r=s_1\begin{bmatrix}1 \\ 0\\ 0 \\ 1\end{bmatrix}+s_2 \begin{bmatrix}0 \\ 1\\ -1 \\ 0\end{bmatrix}$ led to the pair of plains that interseted at a single point $\begin{bmatrix}1 \\ -1\\ 1 \\ 1\end{bmatrix}$.

• March 21

  1. Discussion of Homework VIII
  2. Exam 2 info
  3. Bartle 42.Z-alike: If $p,q,r>1$ are such that $1/p+1/q+1/r=1$, show that if $a,b,c>0$ then $$abc\leq \frac{a^p}{p}+\frac{b^q}{q}+\frac{c^r}{r}$$ Prove that $\sum_{k=1}^n a_k b_k c_k\leq \sqrt[p]{\sum_{k=1}^n|a_k|^p}\sqrt[q]{\sum_{k=1}^n|b_k|^q}\sqrt[q]{\sum_{k=1}^n|c_k|^q}$
  4. Bartle 42.X Show that the maximum of $f(x_1,\dots,x_p)=x_1^2\dots x_p^2$ subject to the contraint $x_1^2+\dots+x_p^2=1$ is equal to $p^{-p}$. Use this to prove that $|y_1\dots y_p|\leq p^{-p/2}\|\vec y\|^p$.
  5. Bartle 42.Y Prove the geometric/arithmetric mean inequality $(a_1\dots a_n)^{1/n}\leq \frac{1}{n}(a_1+\dots+a_n)$ for positive real numbers.
  6. Orthogonal polynomials:
  • Jacobi Legendre polynomials are orthogonal with respect to the inner product $\langle f,g\rangle=\int_{-1}^1 f(x)g(x)dx$. Use the Gram-Schmid orthogonalization to find the first three Jacobi Polynomials. (WolframAlpha: JacobiP[n,0,0,x])
  • Three step recursions: if $p_n(x)$ are (monic, i.e. $p_n(x)=x^n+\dots$ has leading term $x^n$) orthogonal polynomials in the inner product $\langle f,g\rangle=\int f(x)g(x) w(x) dx$ with, lets say, continuous non-negative $w$ that integrates to 1, then there are sequences $b_n\in\mathbb{R}$ and $c_n \geq 0$ such that a three step recursion holds: $$x p_n(x)=p_{n+1}(x)+b_n p_n(x)+c_n p_{n-1}(x), \quad p_0=1, p_{-1}=0$$ Hint: $xp_n(x)$ is a linear combination of $p_{n+1},\dots,p_0$
  • If polynomials $p_n$ satisfy the three step recursion and $\int p_n(x)w(x)dx=0$ for $n=1,2,\dots$, then polynomials $\{p_n\}$ are orthogonal with respect to the inner product $\langle f,g\rangle=\int f(x)g(x) w(x) dx$.
  • Chebyshev polynomialsv of the first kind, $T_n(x)$, satisfy the three step recursion $$2 x T_n(x)=T_{n+1}(x)+T_{n-1}(x)$$ with the initial conditions $T_0(x)=1$, $T_1(x)=x$. Show that $T_n(\cos \theta)=\cos n \theta$ and therefore $T_n$ are orthogonal with respect to the inner product $\langle f,g\rangle=\int_{-1}^1f(x)g(x)\frac{dx}{\sqrt{1-x^2}}$.
  • Chebyshev polynomials of the second kind, $U_n(x)$, satisfy the three step recursion $$2 x U_n(x)=U_{n+1}(x)+U_{n-1}(x)$$ with the initial conditions $U_0(x)=1$, $U_1(x)=2x$. Show that $$U_n(\cos \theta)=\frac{\sin ((n+1) \theta)}{\sin \theta}$$ and therefore $U_n$ are orthogonal with respect to the inner product $\langle f,g\rangle=\int_{-1}^1f(x)g(x)\sqrt{1-x^2}dx$. Also show that $U_n(1)=n+1$. (It is known that $\max_{x\in[-1,1]}|U_n(x)|=n+1$.)
  • There has been some research on Sobolev-type orthogonality with respect to the inner products of the form $\langle f,g\rangle=\int f'(x)g'(x)\mu(dx)+\int f g \nu(dx)$. If you find this interesting, there is a 2015 survey.

  ---

• March 28

  We did not cover enough Fourier series to do anything serious about them. The questions about "identities" that were listed here are moved to the next week. For this week, we can do the following:

  • We had exercises for Euler's formula and geometric sum for Feb 1 that we did not have time to do. We now need to get two trig identities $$\tfrac12+\sum_{k=1}^n \cos k x=\frac{\sin (n+\tfrac12)x}{2 \sin \tfrac12 x}$$ $$\sum_{k=0}^{n-1} \sin (k+\tfrac12) x=\frac{\sin^2 (\tfrac{n}2)x}{2 \sin \tfrac12 x}$$
  • Added We, 3.27
    • Show that one can have $\|f_n\|_2\to 0$ while $\|f_n\|_\infty\to \infty$ for a sequence of periodic piecewise continuous functions.
    • A bit harder question: Give an example of $\|f_n\|_2\to 0$ such that $f_n(x)$ does not converge to 0 at any $x\in[-\po,\pi]$.
    • A bit harder question: Give an example of $\|f_n\|_2\to 0$ such that $\{f_n(x)\}$ is an unbounded sequence at every $x\in[-\po,\pi]$.
  • Periodic extensions of continuous functions may be continuous (like the case of $x^2$) or not (like the case of $x^3$).
  • Some Fourier series of interest (integration exercises):
    • $f(x)=x$, $|x|\leq \pi$
    • $f(x)=\frac12\pi-|x|$, $|x|\leq \pi$
    • $f(x)=x^2$, $|x|\leq \pi$
    • $f(x)=|\sin x|$
    • $f(x)=\begin{cases} \cos x & x>0 \\ - \cos x & x<0 \end{cases}$
    • $f(x)=\begin{cases} x (\pi -x) & x>0 \\ -x(\pi+x) & x<0\end{cases}$.

  Added Th 3.28 Some "paradoxes" that arise from neglecting the assumptions in the theorems. From the above exercises, $x\sim 2\sum_{n=1}^\infty\frac{(-1)^{n-1}}{n}\sin (n x)$

  • Paradox 1: If $f(x)=x$ then $f'(x)=1$ so $1\sim 2\sum_{n=1}^\infty (-1)^{n-1}\cos (n x)$. But the Fourier series for $1\sim 1$ consists of one term. Explanation: periodic extension of $f(x)$ is not continuous at $x=\pi$.
  • Paradox 2: if $f(x)=x^2$ then $f'(x)=2x$ so $f(x) \sim 4\sum_{n=1}^\infty \frac{(-1)^n}{n^2} \cos(n x)$. Putting $x=\pi$ we get $\pi^2 = 4 \sum_{n=1}^\infty \frac{1}{n^2}$. Furthermore, $f$ is continuous and $f'_-(\pi)=2\pi$ and $f'+)\pi)=-2\pi$ exist. But $\sum_{n=1}^\infty \frac{1}{n^2}=\pi^2/6$ not $\pi^2/4$. Explanation: The Fourier series for $f'(x)$ does not determine $a_0$ in the Fourier Series for $f$. The actual series is $f(x)\sim \frac{\pi^2}{3}+4\sum_{n=1}^\infty \frac{(-1)^n}{n^2} \cos(n x)$ . Putting $x=\pi$ we get $4\sum_{n=1}^\infty \frac{(-1)^n}{n^2}\cos(n \pi)=2\pi^2/3$ so $\sum_{n=1}^\infty \frac{1}{n^2}=\pi^2/6$ as expected.
• April 4

  • Show that $x=2\sum_{n=1}^\infty \frac{(-1)^{n-1}}{n}\sin n x$ for $|x|<\pi$. (Of course, the series is 0 at $|x|=\pi$.). [Bartle, Exercise 38.F(a)]
  • Non-trivial sums computed from Fourier series:
    • $\sum_{n=1}^\infty \frac{1}{n^2}=\pi^2/6$ [Bartle, Exercise 38.I(e)]
    • $\sum_{n=1}^\infty \frac{1}{(2n-1)^2}=\pi^2/8$ [Bartle, Exercise 38.E(c)]
    • $\sum_{n=1}^\infty \frac{(-1)^{n-1}}{n^2}=\pi^2/12$ [Bartle, Exercise 38.F(c)]
  • Bartle, Exercise 38N: Use Parserval's identity to show more identities:
    • $\sum_{n=1}^\infty \frac{1}{n^2}=\pi^2/6$
    • $\sum_{n=1}^\infty \frac{1}{n^4}=\pi^4/90$
    • $\sum_{n=1}^\infty \frac{1}{(2n-1)^2}=\pi^2/8$
    • $\sum_{n=1}^\infty \frac{1}{n^6}=\pi^6/945$
  • Math Qualifier review
    • Dillani asked about the following question in Bartle:
      If $$f(x)=\begin{cases}0 & x \not\in \mathbb{Q} \\ 1/n & x\in \mathbb{Q},\quad x=\frac{m}{n}, \mbox{ $\tfrac{m}{n}$ is irredicible} \end{cases}$$ show that $f$ is Rieman-integrable on $[0,1]$ and $\int_0^1 f(x)dx=0$.
    • Aug 2014: Let $\mathbb{R} \xrightarrow{f} \mathbb{R}$ be a continuous function.
      • Show that for each $x \in \mathbb{R}$, $$ \displaystyle f(x) - f(0) = \sum_{k=0}^\infty \left[f \left(\tfrac{x}{2^k}\right) - f\left(\tfrac{x}{2^{k+1}}\right) \right]\,. $$
      • Explain what it means to say that $f$ is differentiable at $x=0$.
      • Suppose that $\displaystyle \lim_{h\to 0} \frac{f(h) - f(h/2)}{h/2} = 0$. Use part (a), or some other method, to prove that $f$ is differentiable at $x=0$ with $f'(0) = 0$.
    • Aug 2014: Let $[0,1]\xrightarrow{f} \mathbb{R}$ be a continuous function.
      • Show that for each $\varepsilon\in(0,1)$, $$ \lim_{n\to +\infty} \int_0^{1-\varepsilon} f(x^n) \, dx = (1-\varepsilon)\,f(0). $$
      • Find $$ \lim_{n\to +\infty} \int_0^1 f(x^n) \, dx . $$ (Hint: Start by explaining why $f$ is bounded.)
    • Aug 2014 candidate (not used): For $x \in [-1,1]$ define $$ f(x) = \begin{cases} \tfrac12 x + x^2\sin(1/x), & \text{if } x\neq 0; \\ 0, & \text{if } x=0. \end{cases} $$
      1. Prove that $f$ is differentiable at $x=0$ and that $f'(0)>0$.
      2. Verify that $f$ is not increasing in any open interval containing $0$. Hint: compute $f'(x)$ near $x=\tfrac{1}{2\pi n}$ and near $x=\tfrac{1}{2n\pi+\pi/2}$

  ---

• April 11

  • Clarifications for the implicit function problem on Exam 2. The following version should clarify the issue.
    Consider a system of equations $$ x^4+y^3+z+w=4, \quad x+y+z+w=4 $$ Show that near $x=y=z=w=1$, the system can be solved for $(x,y)$, or for $(x,z)$, or for $(y,w)$, but not for $(z,w)$.
  • Aug 2014 (multivariate): Let $C:=\{(x,y)\in\mathbb{R}^2 \mid (x+y)^3=3x+5y \}$. Consider the point $p:=(1,1)$ in $C$. Prove that there is an open neighborhood $W$ of $p$ such that $C\cap W$ is the graph $y=f(x)$ of some smooth function $f$ defined near $x=1$, and calculate $f'(1)$.
  • Aug 2014 (multivariate): Define $\mathbb{R}^2\xrightarrow{f}\mathbb{R}$ by $$ f(x,y):=\begin{cases} \frac{xy^2}{x^2+y^2} & \text{when $(x,y)\ne0$}\,, \\ 0 & \text{when $(x,y)=0$}\,. \end{cases} $$ Prove that the partial derivatives $\frac{\partial f}{\partial x}(0,0)$ and $\frac{\partial f}{\partial y}(0,0)$ both exist, but $f$ is not differentiable at $(0,0)$.
  • Math Qualifier review
    • Aug 2014: Let $[0,1]\xrightarrow{f} \mathbb{R}$ be a continuous function.
      • Show that for each $\varepsilon\in(0,1)$, $$ \lim_{n\to +\infty} \int_0^{1-\varepsilon} f(x^n) \, dx = (1-\varepsilon)\,f(0). $$
      • Find $$ \lim_{n\to +\infty} \int_0^1 f(x^n) \, dx . $$ (Hint: Start by explaining why $f$ is bounded.)
    • Aug 2013: For each $n\in\mathbb{N}$, define $\mathbb{R}\xrightarrow{f_n}\mathbb{R}$ by $f_n(x):=x/n$; so, ($f_n)_1^\infty$ is a sequence of functions.
      1. Prove that $(f_n)_1^\infty$ converges pointwise to zero (the zero function) on $\mathbb{R}$.
      2. Prove that $(f_n)_1^\infty$ does not converge uniformly to zero on $\mathbb{R}$.
      3. Prove that $(f_n|_{[0,1]})_1^\infty$ does converge uniformly to zero on $[0,1]$.
      4. Prove that $(f_n)_1^\infty$ is not equicontinuous on $\mathbb{R}$.

    ---

  • More qualifier problems
  • May 1015: Let $\mathbb{R} \xrightarrow{f} \mathbb{R}$ be differentiable. Suppose that $f'$ is bounded on $\mathbb{R}$. Prove that there exist nonnegative constants $C$ and $D$ such that for all $x \in \mathbb{R}$, $\vert f(x) \vert \leq C \vert x \vert +D$.
  • May 1015: Define a sequence of functions $(f_n)_1^\infty$ by $\displaystyle{f_n(x):= e^{-n(n x-1)^2}}$.
    • Show that $(f_n)_1^\infty$ converges pointwise to 0 on $[0,1]$
    • Verify that the convergence is not uniform on $[0,1]$.
    • Prove that $ \lim_{n \to +\infty} \int_0^1 f_n(x) \, dx =0$.
  • May 1015 : Let $(X, \|\cdot\|)$ be a normed vector space and $S$ be a non-empty subset of $X$. Which of the following conditions always implies the other two conditions? Prove one of the two implications. Also, give one example illustrating that one condition does not imply some other.
    • S is closed and bounded.
    • S is sequentially compact
    • S is complete.

  ---

  • Dini's Theorem (Exercise 24.H in Bartle) Let $(f_n)$ be a monotone decreasing sequence of continuous functions on a compact set $K$. (That is $f_1(x)\geq f_2(x)\geq \dots$ for all $x\in K$.) If $f_n$ converges pointwise to a function $f$ that is continuous on $K$ then convergence is uniform. Hint: WLOG $f\equiv 0$.

    ---

    One can try to make examples that illustrate the assumptions in the above Dini's theorem: (I think one can cook up piecewise linear functions for each case)
    • Give an example of a monotone decreasing sequence $(f_n)$ of continuous functions on $[-1,1]$ such that $f_n$ converges pointwise to a function $f$ that is not continuous on $[-1,1]$.
    • Give an example of a sequence $(f_n)$ of continuous functions on $[-1,1]$ such that $f_n$ converges pointwise to a function $f$ that is continuous on $[-1,1]$ but the convergence is not uniform.
• Week of April 15 ...
  • Questions? Items to clarify?
  • Math Qualifier review (continued from previous week)
    1. 2015 (not used?) Let $(f_n)_1^\infty$ be a sequence of maps $\mathbb{R}\xrightarrow{f_n}\mathbb{R}$.
       1. Define what it means to say that $(f_n)_1^\infty$ converges pointwise to a function $f$.
       2. Define what it means to say that $(f_n)_1^\infty$ converges uniformly to a function $f$.
       3. Suppose that $(f_n)_1^\infty$ converges uniformly to $f$. Let $(x_n)_1^\infty$ be a sequence of real numbers that converges to $a\in\mathbb{R}$. Prove that if $f$ is continuous at $a$ then the sequence $(f_n(x_n))_1^\infty$ converges to $f(a)$.
       4. Find an example of a sequence $(f_n)_1^\infty$ that converges pointwise to a function $f$ and a sequence $(x_n)_1^\infty$ of real numbers that converges to $a\in\mathbb{R}$ such that $(f_n(x_n))_1^\infty$ does not converge to $f(a)$.
    2. 2015 (not used?) Let $(X,d)$ be a metric space. The \emph{distance} between two sets $A,B\subset X$ is $$ dist(A,B):=\inf\{ d(a,b) \mid a\in A, b\in B\}\,. $$
       • Suppose that $A,B\subset X$ are disjoint, $A$ is compact, and $B$ is closed. Prove that $dist(A,B)>0$.
       • Give an example of disjoint closed sets $A$ and $B$ in some metric space with $dist(A,B)=0$.
       (If you do not like abstract metric spaces, assume $X=\mathbb{R}^p$.)
    3. 2015 (not used?) Suppose that $f\colon \mathbb{R} \to \mathbb{R}$ is $\mathcal{C}^2$, $f'(0) = 0$ and $f''(0) <0$. Prove that there exists $\delta > 0$ such that $f(x) < f(0)$ for $0 < |x| < \delta$.
    4. 2015 (not used?) Suppose that $0< t < 1$. Prove that $f(x) = x^t$ is uniformly continuous on $[0,+\infty)$.
    5. 2015 (not used?) Suppose that $f_n \colon \mathbb{R} \to \mathbb{R}$ is a sequence of differentiable functions and $[a,b]$ is a compact interval. Assume that $f_n \to f$ pointwise on $[a,b]$ and $f_n' \to g$ uniformly on $[a,b]$.
       • Prove that $f_n \to f$ uniformly on $[a,b]$.
       • Prove that $f$ is differentiable and $f'=g$

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    6. Apr 2013 Let $a_0, a_1, \cdots, a_n$ be real numbers with property that \[ a_0 +\frac{a_1}{2}+ \frac{a_2}{3} + \cdots \frac{a_n}{n+1}=0.\] Prove that the equation \[ a_0 + a_1 x+ a_2 x^2 +\cdots a_n x^n=0 \] has at least one solution in $(0,1)$. (Hint: use Rolle's theorem.)
    7. Apr 2013 For each $n\in\mathbb{N}$, define $f_n (x):= e^{-n x^2}$. Show that for any given $a\in(0,2)$, $(f_n)_{n=1}^\infty$ converges uniformly on $[a,2]$ and \[ \lim _{n\to \infty} \int ^2_a f_n (x)\, dx =0 \,.\]
    8. Apr 2013 Let $(a_n)_1^\infty$ and $(b_n)_1^\infty$ be sequences of positive real numbers. Assume that the limit $$ L:=\lim_{n\to\infty}\frac{a_n}{b_n} $$ exists and is positive and finite. Prove that $\sum_1^\infty a_n$ and $\sum_1^\infty b_n$ either both converge or both diverge.

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    9. Final 2024 (not used): (two versions of the question were considered.)
       1. Let $g: \mathbb{R}\to \mathbb{R}$ be a $C^1$ function. Suppose that $(f_n)_1^\infty$ is a sequence of continuous functions $f_n:[0,1]\to[0,1]$ that converges uniformly on $[0,1]$ to some function $f:[0,1]\to[0,1]$. Prove that $(g\circ f_n)_1^\infty$ converges uniformly to $g\circ f$ on $[0,1]$.
       2. Let $g:[0,1]\to \mathbb{R}$ be a continuous function. Suppose that $(f_n)_1^\infty$ is a sequence of continuous functions $f_n:[0,1]\to[0,1]$ that converges uniformly on $[0,1]$ to some function $f:[0,1]\to[0,1]$. Prove that $(g\circ f_n)_1^\infty$ converges uniformly to $g\circ f$ on $[0,1]$.
    10. Final 2024 (not used): Let $F:\mathbb R^2\to \mathbb R^2$ be a differentiable map such that $F(1,1)=(1,0)$ and $F'(x,y)(a,b)=(ay+bx, by-ax)$.
        • Is $F$ a $C^1$ function? (Sketch of proof is required)
        • Show that $F$ is locally invertible at $(1,1)$
        • If $G=F^{-1}$ in a neighborhood of $(1,0)$, determine $G'(1,0)$
    11. Prove the B-L inequality $|\sqrt{x}-\sqrt{y}|\leq \sqrt{|x-y|}$ and use it to prove that if $f:\mathbb{R}\to \mathbb{R}$ is uniformly continuous on $\mathbb{R}$ then $g(x):=\sqrt{|f(x)|}$ is uniformly continuous on $\mathbb{R}$. (Also give an examaple that the converse does not hold.)
        Which other functions besides $\sqrt{x}$ satisfy $|\phi(x)-\phi(y)|\leq \phi(|x-y|)$?
    12. More to be posted ... if needed ...
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