{"id":320,"date":"2025-10-07T02:38:37","date_gmt":"2025-10-07T02:38:37","guid":{"rendered":"https:\/\/mrivas.su.domains\/gbe\/?p=320"},"modified":"2025-10-07T02:38:37","modified_gmt":"2025-10-07T02:38:37","slug":"concrete-functional-analysis","status":"publish","type":"post","link":"https:\/\/mrivas.su.domains\/gbe\/uncategorized\/concrete-functional-analysis\/","title":{"rendered":"Concrete functional analysis"},"content":{"rendered":"\n

The book Concrete Functional Analysis <\/em>by Richard M. Dudley<\/a> and Rimas Norvai\u0161a<\/a> presents some aspects of nonlinear analysis and their applications to probability.

I wanted to get an understanding of the book as it could have been taught to a high school student. Here, I found the following observations by examining via the lens of AI tools.

The idea behind the book is that there is a new measure or yardstick for variation referred to as wiggliness<\/em> and defined as p-variation<\/em><\/strong>. <\/em>It adds up the size of each little wiggle and raises it to the power of p<\/em>. Bigger p <\/em>pays little attention to tiny jitter and more to big moves. This gives a new family of wiggliness meters <\/em>not just one. <\/p>\n\n\n\n

The book demonstrates that with the right p<\/em>, we can make sense of integrals and calculus style operations even for quite jumpy signals. This lead to a result called the Love-Young inequality<\/strong>, which tells you when an integral \"\int f dg\" exists and how big it can be based on the p-variation of \"f\" and the q-variation of \"g\" with \"\frac{1}{p} + \frac{1}{q} > 1\". <\/p>\n\n\n\n

For Brownian motion<\/em>, a Brownian path has finite p-variation only when \"p > 2\", too wiggly for \"p \leq 2\".
<\/p>\n\n\n\n

\n

If one function is \u201cnot too wiggly\u201d in the p-variation sense and the other is \u201cnot too wiggly\u201d in q-variation with \"\frac{1}{p}+\frac{1}{q} > 1\" the integral \"\intf dg\" exists and is controlled (Love\u2013Young). This is like a cousin of Cauchy\u2013Schwarz\/H\u00f6lder, but tuned to rough signals<\/em>. In stats terms: it tells you when you can safely integrate (or \u201caccumulate\u201d) a noisy curve against another without things blowing up.<\/p>\n<\/blockquote>\n\n\n\n

The book also studies when composition of functions behave smoothly – like a data pipeline where you first transform your variable with \"G\", then apply \"F\". <\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

The Love\u2013Young<\/strong> \u201csafe integration\u201d region where \"\frac{1}{p}+\frac{1}{q} > 1\".Example: Brownian needs \"p > 2\"; pair it with something with, say, \"q < 2\" so the inequality holds.
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n\n\n\n

how the estimate \"\sum |\Delta x|^p\" changes as we cut the interval into more and more pieces. <\/p>\n\n\n\n

Brownian-like path<\/p>\n\n\n\n

p=3 shrinks<\/strong> \u2192 for \"p>2\", the roughness is \u201ctamed,\u201d and p-variation is finite.<\/p>\n\n\n\n

p=1.5 grows<\/strong> with refinement \u2192 too rough: p-variation \u201cblows up\u201d for \"p\le 2\".<\/p>\n\n\n\n

p=2 hovers around a constant (the borderline).<\/p>\n","protected":false},"excerpt":{"rendered":"

The book Concrete Functional Analysis by Richard M. Dudley and Rimas Norvai\u0161a presents some aspects of nonlinear analysis and their applications to probability. I wanted to get an understanding of the book as it could have been taught to a high school student. Here, I found the following observations by examining via the lens of […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[],"class_list":["post-320","post","type-post","status-publish","format-standard","hentry","category-uncategorized"],"_links":{"self":[{"href":"https:\/\/mrivas.su.domains\/gbe\/wp-json\/wp\/v2\/posts\/320","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/mrivas.su.domains\/gbe\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/mrivas.su.domains\/gbe\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/mrivas.su.domains\/gbe\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/mrivas.su.domains\/gbe\/wp-json\/wp\/v2\/comments?post=320"}],"version-history":[{"count":4,"href":"https:\/\/mrivas.su.domains\/gbe\/wp-json\/wp\/v2\/posts\/320\/revisions"}],"predecessor-version":[{"id":327,"href":"https:\/\/mrivas.su.domains\/gbe\/wp-json\/wp\/v2\/posts\/320\/revisions\/327"}],"wp:attachment":[{"href":"https:\/\/mrivas.su.domains\/gbe\/wp-json\/wp\/v2\/media?parent=320"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mrivas.su.domains\/gbe\/wp-json\/wp\/v2\/categories?post=320"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mrivas.su.domains\/gbe\/wp-json\/wp\/v2\/tags?post=320"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}