Glossary—Terminology

Even Symmetry A real function \(f:{\mathbb{R}}\rightarrow{\mathbb{R}}\) is said to be pointwise even-symmetric when \(\forall x\in{\mathbb{R}}:f(x)=f(-x).\)

Odd Symmetry A real function \(f:{\mathbb{R}}\rightarrow{\mathbb{R}}\) is said to be pointwise odd-symmetric when \(\forall x\in{\mathbb{R}}:f(x)=-f(-x).\)

Support The support of a real function is the set of points of its domain where the function does not vanish.

Partition of Unity A real function \(f:{\mathbb{R}}\rightarrow{\mathbb{R}}\) is said to satisfy the pointwise partition-of-unity condition when it is such that \(\forall x\in{\mathbb{R}}:\sum_{k\in{\mathbb{Z}}}\,f(x-k)=1.\)

Interpolating Function A real function \(f:{\mathbb{R}}\rightarrow{\mathbb{R}}\) is said to be interpolating when its restriction to the integers is such that \(\forall k\in{\mathbb{Z}}:f(k)={\mathbf{[\![}}k=0\,{\mathbf{]\!]}},\) where the notation \({\mathbf{[\![}}P\,{\mathbf{]\!]}}\) is that of the Iverson bracket.

Integer-Shift Orthogonality The two functions \(f:{\mathbb{R}}\rightarrow{\mathbb{C}}\) and \(g:{\mathbb{R}}\rightarrow{\mathbb{C}}\) are said to be mutually integer-shift-orthogonal in terms of the integer shift \(k\) if \(\forall k\in{\mathbb{Z}}\setminus\{0\}:0=\int_{-\infty}^{\infty}\,f^{*}(x)\,g(x+k)\,{\mathrm{d}}x.\) As a special case, the zero function \(x\mapsto f(x)=0\) is integer-shift-orthogonal to every function \(g.\)

Integer-Shift Orthonormality The two functions \(f:{\mathbb{R}}\rightarrow{\mathbb{C}}\) and \(g:{\mathbb{R}}\rightarrow{\mathbb{C}}\) are said to be mutually integer-shift-orthonormal if they are integer-shift-orthogonal and if \(1=\int_{-\infty}^{\infty}\,f^{*}(x)\,g(x)\,{\mathrm{d}}x\).

Iverson Bracket The Iverson bracket \({\mathbf{[\![}}P\,{\mathbf{]\!]}}\) of the statement \(P\) is the indicator function of the set of values for which the statement is true.