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Frames provide unconditional basis-like, but generally nonunique, representations of vectors in a Hilbert space H . The redundancy of frame expansions allows the flexibility of choosing different dual sequences to employ in frame representations. In particular, oblique duals, Type I duals, and Type II duals have been introduced in the literature because of the special properties that they possess. A Type I dual is a dual such that the range of its synthesis operator is contained in the range of the synthesis operator of the original frame sequence, and a Type II dual is a dual such that the range of its analysis operator is contained in the range of the analysis operator of the original frame sequence. This paper proves that all Type I and Type II duals are oblique duals, but not conversely, and characterizes the existence of oblique and Type II duals in terms of direct sum decompositions of H , as well as characterizing when the Type I, Type II, and oblique duals will be unique. These results are also applied to the case of shift-generated sequences that are frames for shift-invariant subspaces of L 2 (R d ).

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Stock prices and bond interest rates are inversely related. The current account plays the most important role in the rise of composite stock index. When the trade surplus increases, stock prices rise sharply. Conversely, when the exchange rate and government bond interest rates increase, the composite stock index decreases. The stock index is influenced by several economic variables, politics, the economy, and investment sentiment. In real situations, such financial variables are mostly fuzzy variables. For example, the Korea Composite Stock Price Index (KOSPI) is recorded based on the closing price. However, the daily KOSPI has many values, such as the open price, low price, and high price; therefore, if only the closing price is used for analysis, only partial information is. To analyze the relationship between KOSPIrelated variables, the KOSPI, stock deposits, government bond interest rates, and foreign security investment data are applied. We investigated the mediated relation of variables, known as mediation analysis. In particular, we implemented fuzzy mediation analysis owing to the ambiguity of given data. In the data analysis, we proposed several fuzzy mediation models using fuzzy transformed data.