Some curious results related to a conjecture of Strohmer and Beaver

We study results related to a conjecture formulated by Strohmer and Beaver about optimal Gaussian Gabor frame set-ups. Our attention will be restricted to the case of Gabor systems with standard Gaussian window and rectangular lattices of density 2. Although this case has been fully treated by Faulhuber and Steinerberger, the results in this work are new and quite curious. Indeed, the optimality of the square lattice for the Tolimieri and Orr bound already implies the optimality of the square lattice for the sharp lower frame bound. Our main tools include determinants of Laplace–Beltrami operators on tori as well as special functions from analytic number theory, in particular Eisenstein series, zeta functions, theta functions and Kronecker’s limit formula. We note that our results also carry over to energy minimization problems over lattices and a heat distribution problem over flat tori.

Angklung Tradisional Sunda: Intangible, Cultural Heritage of Humanity, Penerapannya dan Pengkontribusiannya Terhadap Kelahiran Angklung Indonesia

Angklung consists of two to four bamboo tubes suspended in a bamboo frame, bound with rattan cords. The tubes will produce certain notes when the frame is shaken or tapped. Each angklung produces a single note or chord, so several players must collaborate in order to play melodies. Traditional Angklungs use the pentatonic scale, but in 1938 musician Daeng Soetigna introduced Angklungs using the diatonic scale, known as angklung padaeng. Angklung is closely related to traditional customs, arts and cultural identity in Indonesia, played during ceremonies such as rice planting and harvest. Angklung education is passed down orally from generation to generation, and increasingly in educational institutions (Prodi Angklung and Musik Bambu ISBI Bandung. Angklung has been included in the UNESCO’s (United Nations Educational, Scientific, Cultural Organization) list of intangible cultural heritage of humanity. This paper discusses the interesting things about the angklung. Especially the process of traditional angklung that developed into the modern angklung and then both has been worldwide as Indonesian culture heritage.

WEIGHTED AND CONTROLLED FRAMES: MUTUAL RELATIONSHIP AND FIRST NUMERICAL PROPERTIES

Weighted and controlled frames have been introduced recently to improve the numerical efficiency of iterative algorithms for inverting the frame operator. In this paper, we develop systematically these notions, including their mutual relationship. We will show that controlled frames are equivalent to standard frames and so this concept gives a generalized way to check the frame condition, while offering a numerical advantage in the sense of preconditioning. Next, we investigate weighted frames, in particular their relation to controlled frames. We consider the special case of semi-normalized weights, where the concepts of weighted frames and standard frames are interchangeable. We also make the connection with frame multipliers. Finally, we analyze weighted frames numerically. First, we investigate three possibilities for finding weights in order to tighten a given frame, i.e. decrease the frame bound ratio. Then, we examine Gabor frames and how well the canonical dual of a weighted frame is approximated by the inversely weighted dual frame.

ON COARSE QUANTIZATION OF TIGHT GABOR FRAME EXPANSIONS

This paper presents a coarse quantization algorithm (TFΣΔ-II) for tight Gabor frame expansions of certain functions in L2(ℝ), an alternative to the TFΣΔ of Ref. 11. By using some a priori information about the function to be quantized and compromising the translation invariance of the TFΣΔ, TFΣΔ-II produces an approximation in L2(ℝ), as opposed to the weak type approximations of TFΣΔ. In particular, for a tight Gabor frame with frame bound A, we prove that the L2-approximation error corresponding to a kth-order TFΣΔ-II quantizer is of order O(A-k). Furthermore, motivated by TFΣΔ-II, we construct an algorithm to coarsely quantize the Fourier coefficients of certain compactly supported functions.