APROKSIMASI FUNGSI KONTINU TERBATAS DENGAN KONVOLUSI
Keywords: approximation, aproksimasi, bounded continuous functions, convolution, fungsi kontinu terbatas, konvolusi
Abstract
Convolution is a mathematical operation on two functions that produces a new function that can be seen as a modified version of one of its original functions. The convolution operator has no identity element. However, it has an approximate identity. It can be found as a sequence of gk such that convolution of f and gk converges to f for k→∞. It implies that convolution can be used to approximate a function. In this article, we have proven basic theorems about approximation function by convolution for a bounded function in C(Rd).
Konvolusi adalah suatu operasi pada dua fungsi dan menghasilkan suatu fungsi baru yang dapat dipandang sebagai versi modifikasi dari salah satu fungsi aslinya. Operasi konvolusi tidak memiliki unsur identitas. Namun, operasi konvolusi memiliki identitas hampiran, yakni dapat ditemukannya suatu barisan fungsi gk sehingga konvolusi dari f dan gk konvergen ke f untuk k→∞. Hal ini mengakibatkan konvolusi dapat digunakan untuk aproksimasi fungsi. Pada artikel ini dibuktikan teorema-teorema yang mendasari aproksimasi fungsi dengan konvolusi bagi fungsi terbatas di C(Rd) .
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References
Anastassiou. (2004). Fuzzy approximation by fuzzy convolution type operators. Elsevier: Computers and Mathematics with Applications, 48, 1369-1386.
Avramidou, P. (2004). Convolution operators inducted by approximate identities and pointwise convergence in Lp (R) spaces. Proceedings of American Mathematical Society, 175-184.
Bhaya, S. E. (2019). Approximation of function in Lp spaces for p<1 using radial basis function neural network. Journal of University of Babylon for Pure and Applied Sciences, 27(3), 400-405.
Cheney, W. L. (2009). A Course in Approximation Theory. United State: American Mathematical Society.
Folland, G. (1992). Fourier Analysis and Its Application. California: Wadswoth & Brooks/Cole Advanced Book & Sofware.
Gao, Z., Liang, J. &. Xu, Z. (2020). Kernel-independent sum-of-exponentials with application to convolution quadrature. arXiv:2012.13477v1 [math.NA].
Gunawan, H. (2017). Analisis Fourier dan Wavelet. Bandung: FMIPA ITB.
Kahar, E. R. (2016). Aproksimasi Fungsi dengan Konvolusi. Bandung: (Tesis ITB) Tidak diterbitkan.
Lahr, C. D. (1973). Approximate identities for convolution measure algebra. Pasific Journal of Mathematics, 47(1), 147-159.
Leila, A. G.-C. (2007). Function approximation of tasks by neural networks. Conference Paper: 6th Conference on Nuclear and Particle Physics.
Madden, M. G. (2004). The genetic evolution of kernel for support vector machine classifiers. Proceeding of AICS 2004, 15th Irish Conference on Artificial Intelligence and Cognitive Science.
Paul-Escande, P. W. (2017). Approximation of integral operators using product-convolution expansions. Journal of Mathematical Imaging and Vision, Springer Verlag, 58 (3), pp.333-348.
Shawe-Taylor, J. (2004). Kernel Methods for Pattern Analysis. Cambridge: Cambridge University Press. doi:10.1017/CBO9780511809682.
Stein, E. M. (2003). Fourier Analysis: An Introduction. New Jersey: Princeton University Press.
Wu, K.-P. & Wang S-D. (2009). Choosing the kernel parameters for support vector machines by the inter-cluster distance in the feature space. Pattern Recognition.42, 710-717.
Xu, K. (2018). Spectral approximation of convolution operators. SIAM Journal on Scientific Computing, 40.10. 1137/17M1149249.