‎The sequences of the form ${E_{mb}g_{n}}_{m‎, ‎ninmathbb{Z}}$,‎ ‎where $E_{mb}$ is the modulation operator‎, ‎$b>0$ and $g_{n}$ is the‎ ‎window function in $L^{2}(mathbb{R})$‎, ‎construct Fourier-like‎ ‎systems‎. ‎We try to consider some sufficient conditions on the window‎ ‎functions of Fourier-like systems‎, ‎to make a frame and find a dual‎ ‎frame with the same structure‎. ‎We also extend the given two Bessel‎ ‎Fourier-like systems to make a pair of dual frames and prove that‎ ‎the window functions of Fourier-like Bessel sequences share the‎ ‎compactly supported property with their extensions‎. ‎But for‎ ‎polynomials windows‎, ‎a result of this type does not happen.

Keywords: Fourier-like systems, Shift-invariant systems, A pair of dual frames, Polynomials.

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