The only problem with what you call higher order functions that I've ever had is that such expressions are necessarily more abstract and hard to grasp because they are more complex. But I don't see why you would say that it seems unnatural to mathematicians, on the contrary it's the most natural thing in the world. That's why no special notation is needed.
Maybe I'm missing something, but I don't see an issue at all. Perhaps you can present some examples?
I understand where you're going with the Fourier example, you often need surrounding context to know what's going on, you need to know what it is used for.
This is perhaps the biggest difference between math and computer notation, but it's a feature, not a limitation. If you want to you could easily repeat all the variables on the left side of each expression, to make it clear what the frequency domain variable is. But you don't do that, because there is no need for each expression to stand on its own. It's not even needed in all computer languages, e.g. Swift is typesafe but type is inferred and often not even explicit anywhere.
Maybe I'm missing something, but I don't see an issue at all. Perhaps you can present some examples?
I understand where you're going with the Fourier example, you often need surrounding context to know what's going on, you need to know what it is used for.
This is perhaps the biggest difference between math and computer notation, but it's a feature, not a limitation. If you want to you could easily repeat all the variables on the left side of each expression, to make it clear what the frequency domain variable is. But you don't do that, because there is no need for each expression to stand on its own. It's not even needed in all computer languages, e.g. Swift is typesafe but type is inferred and often not even explicit anywhere.