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In mathematics, the qualifier pointwise is used to indicate that a certain property is defined by considering each value () of some function. An important class of pointwise concepts are the pointwise operations, that is, operations defined on functions by applying the operations to function values separately for each point in the domain of definition.
This concept is often contrasted with uniform convergence.To say that = means that {| () |:} =, where is the common domain of and , and stands for the supremum.That is a stronger statement than the assertion of pointwise convergence: every uniformly convergent sequence is pointwise convergent, to the same limiting function, but some pointwise convergent sequences are not uniformly convergent.
MI is the expected value of the pointwise mutual information (PMI). The quantity was defined and analyzed by Claude Shannon in his landmark paper "A Mathematical Theory of Communication", although he did not call it "mutual information". This term was coined later by Robert Fano. [2] Mutual Information is also known as information gain.
In statistics, probability theory and information theory, pointwise mutual information (PMI), [1] or point mutual information, is a measure of association. It compares the probability of two events occurring together to what this probability would be if the events were independent .
To say that the sequence of random variables (X n) defined over the same probability space (i.e., a random process) converges surely or everywhere or pointwise towards X means ∀ ω ∈ Ω : lim n → ∞ X n ( ω ) = X ( ω ) , {\displaystyle \forall \omega \in \Omega \colon \ \lim _{n\to \infty }X_{n}(\omega )=X(\omega ),}
In other words, the output transform is the pointwise product of the input transform with a third transform (known as a transfer function). See Convolution theorem for a derivation of that property of convolution. Conversely, convolution can be derived as the inverse Fourier transform of the pointwise product of two Fourier transforms.
There exist continuous functions whose Fourier series converges pointwise but not uniformly. [8] However, the Fourier series of a continuous function need not converge pointwise. Perhaps the easiest proof uses the non-boundedness of Dirichlet's kernel in L 1 (T) and the Banach–Steinhaus uniform boundedness principle.
We say that ~ is (pointwise) conformal to . Evidently, conformality of metrics is an equivalence relation. Evidently, conformality of metrics is an equivalence relation. Here are some formulas for conformal changes in tensors associated with the metric.