Uploaded October 2021 | Updated September 2026, 15 hours ago
Ever wonder how research is done in mathematics? A mathematical analyst is a mathematician that spends his flipping through the literature looking for inequalities. At least, that's what I was told when I first started studying analysis. A decade later, and I'm still learning new tricks of the trade.
I assembled this fun video to talk about several inequalities that have helped me in my research over the years. It starts by talking about the triangle inequality, and then the next obvious inequality to include is the Cauchy Schwarz inequality. It starts at a nice and easy undergrad level, and then builds up to a couple of simple theorems I proved in my research.
One gap I found I had in my own knowledge came up a couple of years ago, when I was working on Weighted Composition Operators over a Hilbert space I developed with my colleagues called the Mittag-Leffler RKHS. It is a one parameter generalization of Bargmann-Fock spaces, and my co-author and I were trying to transport some characterizations of these operators to our new space. Turned out for some parameters (alpha between 0 and 1), the usual arguments worked as expected, but for others, we had to employ a slightly different strategy.
This came in the form of the Hausdorff-Young inequality. I was literally flipping through Hardy, Littlewood, and Polya's book when I found it. There are a lot of different expressions of it, but the one we used intertwined the coefficients of entire functions and Fourier coefficients.
If you want to see how we used it, then check out this video! I tried to make it fun, and talk broadly about inequalities first.
What sorts of inequalities have you found useful in your work, whether you are an undergrad or a seasoned researcher?
The triangle inequality and Cauchy Schwarz are part of my blood now.
Weighted Composition Operators over the Mittag-Leffler space link.springer.com/article/10.1007/s11785-020-01064-5
0:00 Start
0:48 It's a BIG gap
1:40 It's what we do
2:16 It's a walk off! Hilbert Spaces and Their Inequalities
3:44 Holder me close
4:56 Youth Today and a Young Inequality
6:40 A satisfying proof by picture
7:22 It's a convex relationship
8:30 Frequencies and their Bounds
9:07 Inequalities in Action
10:28 Taylor coefficients ARE Fourier coefficients???
12:32 Proving that bound
14:58 Beauty in mathematics on the runway
Ever wonder how research is done in mathematics? A mathematical analyst is a mathematician that spends his flipping through the literature looking for inequalities. At least, that's what I was told when I first started studying analysis. A decade later, and I'm still learning new tricks of the trade.
I assembled this fun video to talk about several inequalities that have helped me in my research over the years. It starts by talking about the triangle inequality, and then the next obvious inequality to include is the Cauchy Schwarz inequality. It starts at a nice and easy undergrad level, and then builds up to a couple of simple theorems I proved in my research.
One gap I found I had in my own knowledge came up a couple of years ago, when I was working on Weighted Composition Operators over a Hilbert space I developed with my colleagues called the Mittag-Leffler RKHS. It is a one parameter generalization of Bargmann-Fock spaces, and my co-author and I were trying to transport some characterizations of these operators to our new space. Turned out for some parameters (alpha between 0 and 1), the usual arguments worked as expected, but for others, we had to employ a slightly different strategy.
This came in the form of the Hausdorff-Young inequality. I was literally flipping through Hardy, Littlewood, and Polya's book when I found it. There are a lot of different expressions of it, but the one we used intertwined the coefficients of entire functions and Fourier coefficients.
If you want to see how we used it, then check out this video! I tried to make it fun, and talk broadly about inequalities first.
What sorts of inequalities have you found useful in your work, whether you are an undergrad or a seasoned researcher?
The triangle inequality and Cauchy Schwarz are part of my blood now.
Weighted Composition Operators over the Mittag-Leffler space link.springer.com/article/10.1007/s11785-020-01064-5
0:00 Start
0:48 It's a BIG gap
1:40 It's what we do
2:16 It's a walk off! Hilbert Spaces and Their Inequalities
3:44 Holder me close
4:56 Youth Today and a Young Inequality
6:40 A satisfying proof by picture
7:22 It's a convex relationship
8:30 Frequencies and their Bounds
9:07 Inequalities in Action
10:28 Taylor coefficients ARE Fourier coefficients???
12:32 Proving that bound
14:58 Beauty in mathematics on the runway
