Problem-solving recipes

A step-by-step procedure for every problem type this course teaches — what to do first, what to do next, and how to tell which case you're in.

What are you being asked to do?

Start with what you're actually being asked to produce.

Readiness review — classify a function's symmetry, find the fundamental period of a sum, test an improper integral for convergence, project one vector onto another, or find a matrix's eigenvalues and diagonalizability: whichever prerequisite skill the problem is actually testing.

Given two functions on an interval, need their inner product, or a single norm — the inner-product and norm recipes below; need all three of $L^1$ , $L^2$ , and $L^\infty$ side by side instead, or a parallelogram-law check on which norm actually comes from an inner product — the function-space-geometry recipes further down.

A periodic function, need its Fourier series — the real-form coefficient recipe, or the complex-form recipe if $c_n$ is what's asked for (a genuinely different computation, not just a rewrite of the real form).

Given a function and an orthogonal family, need the best approximation — the orthogonal-projection-onto-a-family recipe.

A Fourier series near a jump, need to know whether it can converge uniformly — the uniform-convergence test; need an actual ringing-free approximation through that jump instead of just a yes/no verdict — the Cesàro/Fejér mean.

A frequency-domain filter, need to classify its type or build its mask — the filter recipe.

A sampled frequency, need every alias it could be mistaken for — the aliasing recipe.

A function, need to know if it's an eigenfunction of $d/dt$ , or a forced linear ODE, need its frequency-domain steady state — the two operator-viewpoint recipes.

Readiness

Classify a function as even, odd, or neither

Use when: you're given a formula for $f$ and need to know whether $f(-x)$ equals $f(x)$ , equals $-f(x)$ , or matches neither

Goal: a settled even, odd, or neither label, backed by an identity that holds for every $x$ , not a check at one value

Periodic functions and even/odd symmetry, revisited

Find the fundamental period of a sum of periodic terms

Use when: you have a sum of two or more periodic terms, each with its own known period, and need the period of the whole combination

Goal: the smallest positive number that is a common period of every term, found without graphing

Periodic functions and even/odd symmetry, revisited

Test an improper integral for convergence

Use when: the integral has an infinite limit of integration, or the integrand is undefined (blows up) somewhere on the interval

Goal: either the exact value the integral converges to, or a converges-or-diverges verdict backed by a valid comparison

Improper integrals, revisited

Find the orthogonal projection of one vector onto another

Use when: you have a nonzero vector $u$ and any vector $v$ , and need the piece of $v$ that lies exactly along $u$ 's own direction

Goal: the projection vector $\operatorname{proj}_u v$ , and if asked, confirmation that the leftover residual is perpendicular to $u$

Orthogonal projection, revisited

Find a matrix's eigenvalues and determine whether it's diagonalizable

Use when: you have a square matrix $A$ and need its eigenvalues, plus a settled yes or no call on whether $A=PDP^{-1}$ for some invertible $P$

Goal: every eigenvalue of $A$ , plus a verdict on whether $A$ has enough independent eigenvectors to diagonalize

Eigenvalues, eigenvectors, and diagonalization, revisited

Periodic functions and orthogonality

Compute the inner product of two functions on an interval

Use when: you're given two piecewise-continuous functions and an interval and asked for $\langle f,g\rangle$

Goal: a single real number equal to $\int_a^b f(x)g(x)\,dx$ , with no $x$ left in it

The inner product of two functions on an interval

Compute a norm of a function on an interval

Use when: you're given a function and an interval and asked for $\|f\|_1$ , $\|f\|_2$ , or $\|f\|_\infty$

Goal: a single nonnegative number measuring the function's size, by the specific rule the requested norm uses

Norms and "energy" of a function

The $L^2$ energy norm, built from the inner product above.

Find the best approximation of a function using an orthogonal family

Use when: you have a target function $f$ and a finite orthogonal family $u_1,\dots,u_p$ , and need the closest combination of them to $f$

Goal: the coefficients $c_i=\langle f,u_i\rangle/\langle u_i,u_i\rangle$ that make $\sum c_iu_i$ the best $L^2$ approximation to $f$ in that family's span

Why an orthogonal family is the right thing to expand a function in

The Fourier series

Compute the Fourier coefficients of a periodic function

Use when: you have a periodic function with period $T$ and need $a_0$ , $a_n$ , and $b_n$ for its Fourier series

Goal: explicit formulas or numbers for $a_0$ , $a_n$ , and $b_n$ that determine the function's Fourier series

Deriving the coefficient formulas from orthogonality

The real form, $a_0$ / $a_n$ / $b_n$ .

Compute the complex Fourier coefficients $c_n$

Use when: you have a periodic function and need $c_n$ for its complex-exponential series $f(t)=\sum_{n=-\infty}^{\infty} c_ne^{in\omega_0t}$

Goal: explicit values or a formula for every $c_n$ , with $n$ ranging over all integers

The complex form of the Fourier series

A genuinely different computation from the real form above, not a rewrite of it.

Convergence of Fourier series

Determine whether a Fourier series can converge uniformly on an interval

Use when: you have a Fourier series and a specific interval, and need to know whether convergence there can be uniform

Goal: a yes-or-no conclusion for that interval, backed by a proof by contradiction whenever the answer is no

Why the distinction matters for a Fourier series specifically

Compute a Cesàro (Fejér) mean of a Fourier partial sum

Use when: you need a Fourier approximation that converges uniformly through a jump, with no Gibbs ringing, and can accept averaging across several partial sums instead of relying on a single one

Goal: the tapered trigonometric polynomial $\sigma_N(t)$ , ringing-free near any jump, built entirely from the same coefficients an ordinary partial sum already uses

Cesàro (Fejér) averaging as a fix for non-uniform convergence

A ringing-free approximation traded against slower convergence away from a jump — not a strict upgrade over the uniform-convergence test above.

Fourier series as function-space geometry

Compute the L¹, L², and L∞ norms of a function

Use when: you have an explicit, possibly piecewise, formula for $f$ on an interval and need one or more of its three norms

Goal: exact values for $\lVert f\rVert_1$ , $\lVert f\rVert_2$ , and $\lVert f\rVert_\infty$ , computed directly from $f$ 's formula

The L1, L2, and L∞ norms compared

All three of $L^1$ , $L^2$ , $L^\infty$ side by side, distinct from the single $L^2$ energy norm above.

Test whether a norm comes from an inner product

Use when: you have two specific functions $f,g$ and a candidate norm, and need to know whether that norm could ever support orthogonal projection

Goal: a verified answer, backed by one concrete check, on whether an inner product could possibly sit behind this norm

Why L2 is the right space for Fourier series

Convolution and filtering

Classify or build an ideal frequency-domain filter mask

Use when: you're told which frequencies should pass and which should be blocked, and need $H(\omega)$ 's exact low-pass, high-pass, or band-pass form

Goal: the piecewise formula for $H(\omega)$ , or, run the other way, the correct classification of a given $H(\omega)$

Low-pass, high-pass, and band-pass filters, defined by their frequency response

Sampling and aliasing

Find every alias of a frequency at a given sampling rate

Use when: you know a true frequency $f$ and a sampling rate $f_s$ , and need every frequency that produces the identical samples

Goal: the complete alias family $\{kf_s\pm f\}$ and the single apparent frequency any reconstruction actually returns

Aliasing: two frequencies, one set of samples

The operator viewpoint

Test whether a function is an eigenfunction of the differentiation operator

Use when: you're handed a specific function and asked whether $D$ returns a scalar multiple of it

Goal: either a confirmed eigenvalue $ik$ , or a concrete reason the candidate function fails to be an eigenfunction of $D$

Differentiation as an operator, and the Fourier basis as its eigenfunctions

Solve a forced constant-coefficient ODE for its steady state in frequency space

Use when: you have a constant-coefficient linear differential equation forced by a sinusoid, and only need the steady-state response

Goal: the steady-state $y(t)$ , found by dividing one complex number by another instead of solving a differential equation directly

Constant-coefficient differential equations become algebra in frequency space