4 votes

Is the Ball-Multiplier for the Fourier Transform auto-adjoint?

This is a simple corollary of Plancharel's theorem: $$\langle Sf,g\rangle=\langle \mathcal{F}^{-1}(\chi_{[-1,1]}\hat f),g\rangle= \langle \chi_{[-1,1]}\hat f,\hat g\rangle=\langle \hat f,\chi_{[-1,1]}\...
Pelota's user avatar
  • 548
1 vote

Orthogonal orthornomal bases imply pair-orthogonal vectors

To simplify the analysis assume $\def\R{\mathbb{R}}A,B \in \R^{n \times n}$ so that the Frobenius orthogonality condition can be written as, $$ \def\C{\mathbb{C}} {{A}_i}^j {B^i}_j =0 $$ (the proof is ...
Ted Black's user avatar
  • 636
1 vote

Space of square-integrable functions and its scalar product

Properties of the conjugate needed to prove it: $\hspace{6cm}\textbf{1.}\;\left(\displaystyle\sum_i z_i\right)^\ast=\displaystyle\sum_i z_i^\ast\sim \left(\displaystyle\int z\right)^\ast=\displaystyle\...
Joan S. Guillamet F.'s user avatar

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