Talk by Prof De Carli, "Exponential Riesz bases and frames on two dimensional trapezoids"

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Venue:GL 137

Speaker:  Prof Laura De Carli,

Date: Tuesday, Aug. 28,  4pm, DM 409

Title: "Exponential Riesz bases and frames on two dimensional trapezoids"

Abstract: Let D be a domain of R^n; we say that L^2(D) has an exponential basis if there exists a  sequence  of functions { e^{i sm x}} m in Z , ( where sm is in  C^n and x =  (x1, ... xn)) which is   such that every function in L^2(D) can be represented in a unique way as  Sum[cm  e^{i sm x}],  with cm  in  C. For example,  it is well known that  L^2(-Pi, Pi)  has an exponential basis, { e^{i nx})} m in Z, but is the same true if (- Pi, Pi) is replaced e.g, by the union of two or more intervals, or by a general domain of R^n? In a joint project with Andy  Kumar, a former FIU graduate student, we have investigated exponential bases of 2-dimensional domains called trapezoid. In particular, we have  considered  domains that are union of finitely many rectangles, and we have found exponential bases  for these domains. In this talk I will  describe   my joint project with Andy,  and I will also discuss  the problem of finding an exponential  basis on L^2 of the union of segments in R, and how it is related to our problem. I will also give the definition of exponential frame, which is a generalization of exponential basis,  and a brief introduction to the theory of frames.