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(I have asked this question on SE but could not get any answer and hope this is o.k for MO) Let $X=\mathcal{F}L^{p}=\{f\in L^{\infty}(\mathbb R):\hat{f}\in L^{p}(\mathbb R)\},$ and $\|f\|_{X}= ... 0answers 90 views ### Swapping sums and integration for a kernel in Fourier space (the non absolutely convergent case) Under what conditions on$c_{r}^{m}$does $$\int_0^{2\pi} k(p,q)\exp(-inq)dq=\sum_{r=0}^{\infty}c_{r}^{m}\exp\left(-imp\right) \text{ in } L^2_{per}$$ hold for ... 0answers 70 views ### Is there a space in which the$\vec a$in$\sin(a_1\cdot x)+\sin(a_2\cdot x)$is linear? Suppose one has equations of the form$\sin(a_1\cdot x_i)+\sin(a_2\cdot x_i)=y_i$for$i = 1, \dots, n$(there are also amplitudes and phase shifts, but let's ignore these for now). I want to solve ... 1answer 363 views ### Proof of a Fourier pair with Bessel functions? How can we prove that the Fourier transform of the function $$f(x) = \begin{cases} (a^2-x^2)^{c/2} BesselJ[c,b\sqrt{a^2-x^2}] & \text{for }x^2 < a^2\\ 0 & \text{otherwise} \end{cases}$$ ... 1answer 77 views ### Elaboration of a certain section of a paper by Thanigasalam In section 11 of this paper by Thanigasalam, it says "... we get$G(10)\le 105$, and this implies that$H(10) \le 107$". However, it is very unclear how this follows. Why is it the case that$G(10)\le ...
I'm trying to answer this problem: Consider a real function f, bandlimited by frequency $\omega$, which satisfy $$\int_{-\infty}^\infty f(x)^2dx=c.$$ (For pure mathematicians: "bandlimited" means ...