Silly proofs 2

I swear this was supposed to be Silly proofs three, but obviously my memories of having done two silly proofs are misleading.

This proof isn’t actually that silly. It’s a proof of the L^2 version of the Fourier inversion theorem.

We start by noting the following important result:

Int_{-inf}^{inf} e^{itx} e^{-x^2/2} = sqrt{2 pi} e^{-t^2/2}

Thus if we let h_0 = e^{-x^2}/2 then we have h_0^ = h_0 (where ^ denotes the fourier transform)

Let h_n(x) = (-1)^n e^{x^2/2} d^n/dx^n (e^{-x^2})

This satisfies:

h_n’ – xh_n = -h_{n+1}

So taking the Fourier transform we get

ix h_n^ – i (h_n^)’ = -h_{n+1}^

So, h_n and (-i)^n h_n^ satisfy the same recurrence relation. Further h_0^ = h_0.

Hence we have that h_n^ = (-i)^n h_n

Now, the functions h_n are orthogonal members of L^2, and so form an orthonormal basis for their span.

On this span we have the map h -> h^ is a linear map with each h_n an eigenvector. Further h_n^^ = (-1)^n h_n. Thus the fourier transform is a linear isometry from this space to itself.

Now, h_n is odd iff n is odd and even iff n is even. i.e. h_n(-x) = (-1)^n h_n

Thus h_n^^(x) = (-1)^n h_n(-x)

And hence h^^(x) = (-1)^n h(-x) for any h in the span. As both sides are continuous, it will thus suffice to show that the span of the h_n is dense.

Exercise: The span of the h_n is precisely the set of functions of the form p(x) e^{-x^2 / 2}

It will thus suffice to prove the following: Suppose f is in L^2 and Int x^n e^{-x^2 / 2} f(x) dx = 0 for every x. Then f = 0.

But this is just an easy application of the density of the polynomial functions in L^2[a, b] (pick a big enough interval so that the integral of |f(x)|^2 over that interval is within epsilon^2 of ||f||^2, and this shows that the integral of |f(x)|^2 over that interval is 0. Thus ||f||_2 < epsilon, which was arbitrary, hence ||f||_2 = 0). I’ve dodged numerous details here, like how the L^2 Fourier transform is actually defined, but this really can be turned into a fully rigorous proof – nothing in this is wrong, just a little fudged. The problem as I see it is that – while the L^2 Fourier theory is very pretty and cool – this doesn’t really convert well to a proof of the L^1 case, which is the more important one.

The Glory of Salads

Today I’m going to talk to you about salads. This is a subject I feel quite strongly about, so the post is going to be full of hyperbole and over the top language. But, let’s be honest, when are my posts not?

You probably think salads are boring. Some lettuce, cucumber, maybe tomatoes and carrots if you’re lucky. In order to make them interesting you need to pile them with dressing. Right?

Wrong! Wrong wrong wrong.

Indeed, mere words cannot express how wrong this is. So instead I am going to have to refer you to some higher authorities.

Here is what the bible has to say on the subject:

“Thou shalt not put the cucumber and a measly supermarket tomato on the lettuce and call it a salad, for that is an abomination.”

After a consultation with the Eschaton, it was convinced that the matter was of sufficiently great importance that the following appeared across the galaxy.

“4. Thou shalt not make boring salads within my historic light cone. Or else.”

Finally, if these have not convinced you of the severity of the situation, if you make boring salads then these cute kittens will cry.

So, on reflection, if you make boring salads then you will go to hell, your civilisation will be wiped out by a passing asteroid, and kittens will cry. Moreover, you will have a boring salad.

Now, I must explain how one goes about making an interesting salad.

The first myth to be disposed of is that a lot of dressing will make an interesting salad. If you put dressing on a boring salad then what you have is a boring salad covered in dressing. This might be edible, but it’s not an interesting salad.

The second thing to bear in mind about dressing is that, given a decent salad, it isn’t neccesary. I’m not saying it’s a bad thing – I really like salad dressing. But the more ingredients your salad has, the more the dressing becomes just an accompaniment to the salad – it’s not an integral part of it, and can quite happily be left out.

Implicit in this is that real salads should have a number of different ingredients. If you’re only going to have a few ingredients then they should be interesting ones.

For example, the salad I had for lunch today contained the following ingredients: Romaine lettuce (never ever use iceberg lettuce. It is the devil’s leaf.), sundried tomatoes, half an orange sweet pepper, two hard boiled eggs and a banana (do not doubt the use of banana in salad until you have tried it. It is awesome.).

I considered this to be a fairly basic salad. Ideally I’d have added some avocado, maybe a few capers, some seared red onion, etc. to it, but I didn’t have the first two and couldn’t be bothered to cook the third.

So, here is a list of some worthwhile salad ingredients. It is in no way exhaustive, and I’m definitely not suggesting you use all of them in a single salad.

Lettuce of course. My favourites are Romaine and little gems, but there is a wide variety of opinion on this. However the people who think iceberg lettuce is appropriate are simply wrong.

Cucumber fulfills a similar role to lettuce – it’s nice, bulks up the salad a bit, and has a simple flavour to it. It isn’t however in itself very interesting.

Carrots. Good quality carrots have a wonderful texture to them, and chopped or shredded (this is distinct from grated) carrot in a salad is very nice.

Good quality fresh tomatoes. None of those boring tasteless default supermarket tomatoes.

Avocado.

Sundried tomatoes.

Capers

Egg. Either scrambled (to the point where it’s dry rather than runny) or hard boiled.

Roast squash.

Sweet peppers. Either raw or cooked.

Good cheeses. Especially feta or mozzarrela.

Banana.

Raisins or sultanas.

Green beans.

Chickpeas.

Kidney beans.

Seared onion. Red is best here. You can also include them raw, but I don’t like it.

Tuna fish.

Anchovies.

Artichoke hearts.

Just about anything else that’s edible cold.

One particular combination (which I can’t eat any more) that works really well is that of banana, sundried tomatoes and feta. I know you’re probably looking skeptical at this, but try it anyway and then come back and yell at me if you’re still not convinced.

Having put together these salads, you can then drizzle dressing over them – vinaigrette, honey-mustard, sesame and soy sauce, whatever you feel like as long as it’s interesting.

So, spread the word. Salads can – and should – be interesting, and people who make boring salads will be the first against the wall when the revolution comes.

Silly proofs 2

I swear this was supposed to be Silly proofs three, but obviously my memories of having done two silly proofs are misleading.

This proof isn’t actually that silly. It’s a proof of the [tex]L^2[/tex] version of the Fourier inversion theorem.

We start by noting the following important result:

[tex]\int_{-\infty}^{\infty} e^{itx} e^{-\frac{1}{2}x^2} = \sqrt{2 \pi} e^{-\frac{1}{2}t^2}[/tex]

Thus if we let [tex]h_0 = e^{-\frac{1}{2}x^2}[/tex] then we have [tex]\hat{h_0} = h_0 [/tex] (where [tex]\hat{f}[/tex] denotes the fourier transform of [tex]f[/tex])

Let [tex]h_n(x) = (-1)^n e^{x^2/2} \frac{d^n}{dx^n} e^{-x^2}[/tex]

This satisfies:

[tex]h_n – xh_n = -h_{n+1}[/tex]

So taking the Fourier transform we get

[tex]ix \hat{h_n} – i \frac{d}{dx} \hat{h_n} = -\hat{h_{n+1}}[/tex]

So, [tex]h_n[/tex] and [tex](-i)^n h_n^[/tex] satisfy the same recurrence relation. Further [tex]\hat{h_0} = h_0[/tex]

Hence we have that [tex]\hat{h_n} = (-i)^n h_n[/tex].

Now, the functions [tex]h_n[/tex] are orthogonal members of [tex]L^2[/tex], and so form an orthonormal basis for their closed span.

On this span we have the map [tex]h \to \hat{h}[/tex] is an isometric linear map with each [tex]h_n[/tex] an eigenvector. Further [tex]\hat{\hat{h_n}} = (-1)^n h_n[/tex]. Thus the fourier transform is a linear isometry from this space to itself.

Now, [tex]h_n[/tex] is odd iff n is odd and even iff n is even. i.e. [tex]h_n(-x) = (-1)^n h_n[/tex]

Thus [tex]\hat{\hat{ h_n(x)} } = (-1)^n h_n(-x)[/tex].

And hence [tex]\hat{\hat{h}}(x) = (-1)^n h(-x)[/tex] for any [tex]h[/tex] in the span. As both sides are continuous, it will thus suffice to show that the span of the [tex]h_n[/tex] is dense.

Exercise: The span of the [tex]h_n[/tex] is precisely the set of functions of the form [tex]p(x) e^{- \frac{1}{2} x^2 }[/tex], where [tex]p[/tex] is some polynomial.
It will thus suffice to prove the following: Suppose [tex]f[/tex] is in [tex]L^2[/tex] and [tex]\int x^n e^{-\frac{1}{2}x^2 } f(x) dx = 0[/tex] for every x. Then [tex]f = 0[/tex].

But this is just an application of the density of the polynomial functions in [tex]L^2[a, b] [/tex]: pick a big enough interval so that the integral of [tex]|f(x)|^2[/tex] over that interval is within [tex]\epsilon^2[/tex] of [tex]||f||^2[/tex], and this shows that the integral of [tex]|f(x)|^2[/tex] over that interval is [tex]0[/tex]. Thus [tex]||f||_2 < \epsilon[/tex], which was arbitrary, hence [tex]||f||_2 = 0[/tex]. (Note: When editing this for the new blog site I noticed that this proof is wrong. I haven't been able to fix it yet, but will update this when I do).
I’ve dodged numerous details here, like how the [tex]L^2[/tex] Fourier transform is actually defined, but this really can be turned into a fully rigorous proof – nothing in this is wrong, just a little fudged. The problem as I see it is that – while the [tex]L^2[/tex] Fourier theory is very pretty and cool – this doesn’t really convert well to a proof of the [tex]L^1[/tex] case, which is in many ways the more important one.

Cooking lessons 3

This is going to get hopelessly out of order very quickly, as there are now two lessons I’ve missed out on actually posting. Never mind.

Yesterday I took Michael spice shopping. I was actually a bit unimpressed with where we went, so if anyone can tell me some good places to go spice shopping in London I’ll be really grateful. We bought ground cumin, ground red chilli, garam masala and (gasp) curry powder. I went for all powders because they’re a bit easier to work with to start with. Also because the selection of whole spices was crap. We can get on to the subtleties of spice usage once we’ve covered the basics, and once I’ve found a decent place to go shopping for them.

Here’s one of the first Indian recipes I ever learned to cook, about four years back when I was first trying out this ‘vegetarian’ thing and Tariq came into our kitchen and found my attempts at making a curry. They were umm… not very impressive. Ask nicely and I’ll tell you about it some time. Here is a very simplified and tinkered with version of the recipe he showed me.

Ingredients

One small bowl of lentils (about a serving size in order to make two to three servings). Preferably green, brown, puy, etc. Black or red won’t really work here.

One medium-large onion.

Sunflower oil.

One large spoon of ground cumin.

One spoon of garam masala.

A quarter of a spoon of curry powder.

Water.

Salt to taste.

Instructions

Mix the dried spices and salt. Dry fry them on a medium heat for about a minute, stirring constantly, and then transfer to a bowl.

Add enough oil to the bottom of the pan to just cover it. Dice the onion and fry it on medium heat until soft, and then add the spices. Fry for a few minutes more.

Now add the lentils and fry for about a minute. Cover with boiling water (say two to three times as much water as you had lentils) and bring to a boil. Reduce the heat to simmer, cover the pot, and leave it.

It will probably take about half an hour to cook. Be more worried about undercooking it than overcooking, but check on it every now and then to see if it’s getting dry and needs more water. The end result can either be dryish or soupy as you prefer. If it’s going to be soupy you might want to consider adding more salt and/or (horrors) half a stock cube. When the lentils are soft to eat it is ready.

Cooking lessons 2

Right, here’s the first real cooking lesson. There would have been one before this in which I discussed how to make the garlic lentils in a slightly more sane manner, but stuff got in the way. So I didn’t.

At the moment Michael is still spice free, causing me to need to be vaguely creative to work with the limited flavourings and ingredients available. Here’s what I came up with as a suggested dinner for tonight. It’s basically a mixture of things which are easy to make.

Pan fried potatoes

These have, for all intents and purposes, no nutritional value. However they taste great and are cheap and easy to make, so why complain?

First of all, you’ll need boiled potatoes for this. Cold is better, because hot potatoes are a nuisance to cut and don’t retain their shape as well, but either is fine. So, first we need to boil some potatoes. I recommend making excess potato and storing the rest in those sandwich bags I told you to get, as these are always useful to have to hand and if you have precooked potato in the fridge then this becomes even easier to make.

This is a complicated procedure akin to rocket science. You take a large quantity of unpeeled potatoes, given them a rinse and put them in a pot. You cover it with water. Put the heat on high until the water boils and then reduce the heat to medium and leave it to cook. Check them after about 20 minutes, and remove when cooked. They need to be soft when you stick a knife into them (check several as this will vary with potato size and they’re better slightly overcooked than undercooked).

Once the potatoes are cooked, let them cool for a bit (it may help to leave them to sit in a pot of cold water for a few minutes), then cut them up into roughly cm cubes (you don’t need to be too careful when doing this).

Now get a frying pan and add enough oil to cover the bottom in a thin layer. You may want to add a small amount of butter as well. Let it get hot and then add the potato cubes and sprinkle a fair bit of salt over the potatoes.

This will need to fry for quite a while until the cubes get crispy. Stir it every minute or so, but not constantly – prolonged contact with the frying pan is needed to crip the potatoes.

Spicy tomato sauce

Ingredients: One medium sized onion, garlic puree, lazy chillies, tomato passata, oil, salt

What to do:

Dice the onion fairly finely. Fry on medium heat with a little bit of oil and a pinch of salt.

After about five minutes, add a small spoon of lazy chillies and garlic puree (slightly more garlic than chillies if anything) to the onions and continue stirring until the onions are soft. Then add about half a bottle of passata, mix it up thoroughly and reduce the heat. Leave it on a low heat, stirring occasionally, until the sauce thickens.

Serve with the pan fried potatoes (you probably don’t want to put the sauce directly on them, as it will soak in and they’ll lose their crispiness. Serve on the side and mix as you eat).

Next post: Salads.