Generalized eigenfunction
Well kiddo, let's start with some basics. Do you remember learning about functions in math class? A function is something that takes an input and gives an output - like a machine that turns apples into applesauce.
Now, let's talk about eigenvalues and eigenvectors. These are special values and vectors that come up when we're studying linear transformations. A linear transformation is like a rule that takes in a vector and spits out another vector - kind of like a recipe for making a salad.
So, a generalized eigenfunction is a special function that pops up when we're studying linear transformations in a certain way. It's a function that satisfies a certain equation that involves both an operator (which is a fancy way of saying "a thing that does something to another thing") and a constant.
This equation involving the operator and the constant is called an eigenfunction equation. What makes it "generalized" is that it applies to more than just one specific operator - it applies to a whole bunch of them!
So, just like how an apple can be turned into applesauce by different machines or recipes, the same eigenfunction equation can apply to many different operators or linear transformations. And that's pretty nifty, isn't it?
Now, let's talk about eigenvalues and eigenvectors. These are special values and vectors that come up when we're studying linear transformations. A linear transformation is like a rule that takes in a vector and spits out another vector - kind of like a recipe for making a salad.
So, a generalized eigenfunction is a special function that pops up when we're studying linear transformations in a certain way. It's a function that satisfies a certain equation that involves both an operator (which is a fancy way of saying "a thing that does something to another thing") and a constant.
This equation involving the operator and the constant is called an eigenfunction equation. What makes it "generalized" is that it applies to more than just one specific operator - it applies to a whole bunch of them!
So, just like how an apple can be turned into applesauce by different machines or recipes, the same eigenfunction equation can apply to many different operators or linear transformations. And that's pretty nifty, isn't it?