Research proof index

For $0\le t\le1$, the reciprocal of $1-2^{-t}$ is controlled in the extended nonnegative reals by

$(1-2^{-t})^{-1}\le 2t^{-1}.$

For an $L^2$ function on a circle of period $T$, the squared $L^2$ norm of its $N$th partial Fourier sum equals the sum of the squared magnitudes of its Fourier coefficients from $-N$ through $N$:

$\lVert S_Nf\rVert_2^2=\sum_{n=-N}^{N}|\widehat f(n)|^2.$

predictableConvexStep is left-continuous in time.

The predictable part is additive for integrable processes.

On the mesh cell Ioc (pred u) u containing t, the step process predictableSeqStep is constant, equal to the discrete predictable part at the cell's right endpoint u.

The mesh step-extension of the discrete predictable part is left-continuous in time.