I have the following two projections that are zero: $$\int_{-\infty}^\infty a(x, y) dy = \int_{-\infty}^\infty y a(x,y) dy = 0$$ Is there anything that can be said about $a(x,y)$? What family of functions satisfy the projections above?
What are the implications if a projection is zero
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calculus
real-analysis
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1It's just a rewording, but those are the functions orthogonal to all affine-linear functions ... – 2017-02-12
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0first thing to be said: $x$ does not play any role. second thing: $0=\int a(y)\mathrm{d}y$ implies $\int a^+(y)\mathrm{d_y}=\int a^-(y)\mathrm{d}y<\infty$, third thing: for $\int ya(y)\mathrm{d}y$ to exist $a$ has to drop at least superlinearly for $y\rightarrow +-\infty$ I would guess, might well be you even need quadratically. – 2017-02-12