I am trying to prove this kernel is valid, $$ K(x) = \frac{1}{2}I(-1 < x < 1) $$ So far I can integrate to 1, but how do I prove $$k(x) = k(-x)$$ Also, how do we satisfy that k(x) is $\ge$ 0 for all x?
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2$\begingroup$ Since the function is constant for all valid values it should be pretty straightforward, isn’t it? $\endgroup$– TimCommented May 27, 2022 at 5:39
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2$\begingroup$ The term "kernel" is used for many different entities. Can you please state what you are referring to by "kernel"? $\endgroup$– frankCommented May 27, 2022 at 5:51
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1$\begingroup$ Proof exercises are frustrating because one's temptation is to declare the theorem "obvious." In fact, they are exercises in a disciplined reading of definitions, axioms and other previously proven theorems. Take the definition of a kernel verbatim. Then show through simple mathematical relations that those definitions are true for this kernel. If you feel silly while you are doing it, you are doing it right! $\endgroup$– Peter LeopoldCommented May 27, 2022 at 5:51
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$\begingroup$ statistical definition of kernel en.wikipedia.org/wiki/Kernel_(statistics). So from further study I gathered k(x) = k(-x) means that when you graph the function on a (x,y) plane it is symmetrical. Like how when you graph a normal distribution it is symmetrical. Now I have to see how k(x) is greater than or equal to 0 for all x. $\endgroup$– user359211Commented May 27, 2022 at 8:55
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$\begingroup$ For the reason we ask for your meaning of "kernel," see the "Mathematics" section of the Wikipedia disambiguation page. However, the two questions you ask are so trivial that we have to suspect there might be an issue just understanding the notation. Could you tell us how you define the function "$I$"? $\endgroup$– whuber ♦Commented May 27, 2022 at 13:58
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