Editing Relative strength index (section)

==General definitions==
{{Unreferenced section|date=February 2023}}
In analysis,
* If <math>g\colon T\to{\mathbb R}</math> is a [[real function]] with positive [[Lp_space#Lp_spaces_and_Lebesgue_integrals|<math>L^1</math>-norm]] on a set <math>T</math>, then <math>|g|</math> divided by its norm integral, the normalized function <math>p_g(t):=\tfrac{|g(t)|}{\|g\|_1}</math>, is an associated [[probability distribution|distribution]].
* If <math>p\colon T\to[0, 1]</math> is a distribution, one can evaluate measurable subsets <math>S\subset T</math> of the domain. To this end, the associated [[indicator function]] <math>\mathbf{1}_S\colon T\to\{0,1\}</math> may be mapped to an "index" <math>\langle p,\mathbf{1}_S\rangle\in [0, 1]</math> in the real [[unit interval]], via the [[integral|pairing]] <math>\langle p,\mathbf{1}_S\rangle :=\int_T p(t)\cdot\mathbf{1}_S(t)\,{\mathrm d}t=\int_S p(t)\,{\mathrm d}t</math>.

Now for <math>\mathbf{1}_{g>0}</math> defined as being equal to <math>1</math> [[if and only if]] the value of <math>g</math> is positive, the index <math>\langle\tfrac{g(t)}{\|g\|_1},\mathbf{1}_{g>0}\rangle</math> is a quotient of two integrals and is a value that assigns a weight to the part of <math>g</math> that is positive. 
A ratio of two averages over the same domain <math>T</math> is also always computed as the ratio of two integrals or sums.

Yet more specifically, for a real function on an [[ordered set]] <math>T</math> (e.g. a price curve), one may consider that function's [[gradient]] <math>g</math>, or some weighted variant thereof. In the case where <math>T = \{1,\dots, n\}</math> is an ordered [[finite set]] (e.g. a sequence of timestamps), the gradient is given as the [[finite difference]]. 

In the ''relative strength index'', <math>\tfrac{\text{SMMA}(U)}{\text{SMMA}(U+D)}</math> evaluates a sequence <math>g</math> derived from closing prices observed in an interval of <math>n</math> so called market periods <math>t \in \{1, \dots, n\}</math>. The positive value <math>|g(t)|</math> equals the absolute price change <math>| \text{close}(t) - \text{close}(t-1) |</math>, i.e. <math>U+D</math> at <math>t</math>, multiplied by an [[Exponential smoothing|exponential factor]] according to the SMMA weighting for that time. The denominator <math>\|g\|_1</math> is the sum of all those numbers. In the numerator one computes the SMMA of only <math>U</math>, or in other words <math>g</math> multiplied by the indicator function for the positive changes. The resulting index in this way weighs <math>U</math>, which includes only the positive changes, over the changes in the whole interval.